A table of $n$ -component handlebody links of genus $n+1$ up to six crossings

Giovanni Bellettini Dipartimento di Ingegneria dell’Informazione e Scienze Matematiche, Università di Siena, 53100 Siena, Italy, and International Centre for Theoretical Physics ICTP, Mathematics Section, 34151 Trieste, Italy bellettini@diism.unisi.it , Giovanni Paolini California Institute of Technology and Amazon Web Services, Pasadena CA, United States (work done while at University of Fribourg, Department of Mathematics, 1700 Fribourg, Switzerland) paolini@caltech.edu , Maurizio Paolini Dipartimento di Matematica e Fisica, Università Cattolica del Sacro Cuore, 25121 Brescia, Italy maurizio.paolini@unicatt.it and Yi-Sheng Wang National Center for Theoretical Sciences, Mathematics Division, Taipei 106, Taiwan yisheng@ncts.ntu.edu.tw

Abstract.

A handlebody link is a union of handlebodies of positive genus embedded in $3$ -space, which generalizes the notion of links in classical knot theory. In this paper, we consider handlebody links with one genus $2$ handlebody and $n-1$ solid tori, $n>1$ . Our main result is the complete classification of such handlebody links with six crossings or less, up to ambient isotopy.

1. Introduction

Knot tabulation has a long and rich history. Some early work, motivated by Kelvin’s vortex theory, dates back to as early as the late 19th. Over the past decades, more effort has been put into it by many physicists and mathematicians; all prime knots up to $16$ crossings are now classified [8]. In recent years knot tabulation has been further generalized to other contexts. [17] and [19] tabulate all prime theta curves and handcuff graphs up to seven crossings, [10] enumerates all irreducible handlebody knots of genus $2$ up to six crossings, and [5] classifies all alternating Lengendrian knots up to seven crossings.

The aim of this paper is to extend the Ishii-Kishimoto-Moriuchi-Suzuki handlebody knot table [10] to handlebody links with $n>1$ components having total genus $n+1$ . We call such a handlebody link an $(n,1)$ -handlebody link; it consists of exactly one genus $2$ handlebody and $n-1$ solid tori. The following theorems summarize the main results of the paper.

Theorem 1.1.

Table 1 enumerates all non-split¹¹1A handlebody link ${\rm HL}$ is split if there is a $2$ -sphere $\mathfrak{S}\subset\mathbb{S}^{3}$ with $\mathfrak{S}\cap{\rm HL}=\emptyset$ separating ${\rm HL}$ into two parts., irreducible²²2A handlebody link ${\rm HL}$ is reducible if there is a $2$ -sphere $\mathfrak{S}$ in $\mathbb{S}^{3}$ with $\mathfrak{S}\cap{\rm HL}$ an incompressible disk in ${\rm HL}$ . $(n,1)$ -handlebody links, up to ambient isotopy and mirror image, by their minimal diagrams, up to six crossings.

$4_{1}$ and $5_{1}$ in Table 1 are the only non-split, irreducible $(n,1)$ -handlebody links with four and five crossings, respectively. There are $15$ handlebody links with six crossings, among which $8$ have two components ( $n=2$ ), $6$ have three components ( $n=3$ ), and $1$ has four components ( $n=4$ ). As a side note, $6_{5}$ in Table 1 also represents the famous figure eight puzzle devised by Stewart Coffin [3]. Thus, its unsplittability implies the impossibility of solving the puzzle (Remark 3.2). Also, $6_{9}$ in Table 1 is an irreducible handlebody link with a $\partial$ -irreducible complement; such phenomenon cannot happen when $n=1$ (Remark 3.3).

Our task with respect to Table 1 is two-fold. Firstly we need to show that there is no extraneous entry, i.e. that all entries in the table

U.1

represent non-split handlebody links,
U.2

represent irreducible handlebody links,
U.3

are mutually inequivalent, up to mirror image,
U.4

attain minimal crossing numbers.

Secondly we have to prove that the table is complete; namely, there is no missing handlebody link with $6$ crossings or less.

In Section 3 we prove U.1-U.3, making use of invariants such as the linking number [16], irreducibility criteria [2], and the Kitano-Suzuki invariant [13] (Theorems 3.2, 3.7, and 3.1, respectively). We prove the completeness of Table 1 by exhausting all—except for those obviously non-minimal—diagrams of non-split, irreducible $(n,1)$ -handlebody links up to six crossings (Section 4).

We first observe that the underlying plane graph of a diagram of a non-split, irreducible $(n,1)$ -handlebody link necessarily has edge connectivity equal to $2$ or $3$ ; for the sake of simplicity, such a diagram is said to have $2$ - or $3$ -connectivity, respectively. Diagrams with $3$ -connectivity up to six crossings are generated by a computer code, whereas to recover handlebody links represented by diagrams with $2$ -connectivity, we employ the knot sum—the order- $2$ vertex connected sum—of spatial graphs [17]. In more detail, a minimal diagram $D$ with $2$ -connectivity can be decomposed by decomposing circles³³3a circle that intersects $D$ at two different arcs. into simpler tangle diagrams, each of which induces a spatial graph that admits a minimal diagram with $3$ - or $4$ -connectivity, as illustrated in Fig. 1.1. This decomposition allows us to recover the handlebody link represented by $D$ by performing the knot sum between prime links and a spatial graph that admits a minimal diagram with $3$ -connectivity.

Refer to caption — Figure 1.1. Decomposing a minimal diagram with $2$ -connectivity.

Once a list containing all possible minimal diagrams of non-split, irreducible handlebody links is produced, we examine each entry on the list manually (Appendix A), and show that either it is non-minimal or it represents a handlebody link ambient isotopic to one in Table 1, up to mirror image. This proves the completeness, and also implies U.4, given U.1-U.3.

Theorem 1.2.

All but $5_{1}$ , $6_{3}$ , $6_{6}$ , $6_{7}$ , $6_{8}$ , $6_{10}$ in Table 1 are achiral.

The main tool used to inspect chirality is Theorem 5.3, where we prove a uniqueness result for the decomposition of non-split, irreducible handlebody links in terms of order- $2$ connected sum of handlebody-link-disk pairs (Definition 5.1).

Theorem 1.3.

Table 5 enumerates all non-split, reducible $(n,1)$ -handlebody links up to $6$ crossings, up to mirror image.

Theorem 1.3 follows from the irreducibility of handlebody links in Table 1 and a uniqueness factorization theorem (Theorem 6.1) for non-split, reducible $(n,1)$ -handlebody links in terms of order- $1$ connected sum (Definition 6.1).

The structure of the paper is the following. Basic properties of handlebody links are reviewed in Section 2; uniqueness, unsplittability, and irreducibility of handlebody links in Table 1 are examined in Section 3. The completeness of the table is discussed in Section 4. Section 5 introduces the notion of decomposable handlebody links, which is used for examining the chirality of handlebody links in Table 1. Lastly, a complete classification of non-split, reducible handlebody links, up to $6$ crossings, is given in Section 6. In the appendix we include an analysis on the output of the code, available at http://dmf.unicatt.it/paolini/handlebodylinks/.

Throughout the paper we work in the $\operatorname{PL}$ category; for the illustrative purposes, the drawings often appear smooth. In the case of $3$ -dimensional submanifolds in $\mathbb{S}^{3}$ , the $\operatorname{PL}$ category is equivalent to the smooth category due to [4, Theorem $5$ ], [7, Theorems $7.1$ , $7.4$ ], [20, Theorem $8.8$ , $9.6$ , $10.9$ ].

Table 1. Non-split, irreducible handlebody links up to six crossings.

$4_{1}$ [Uncaptioned image] $5_{1}$
$6_{1}$ $6_{2}$ $6_{3}$ $6_{4}$
$6_{5}$ $6_{6}$ $6_{7}$ $6_{8}$
$6_{9}$ $6_{10}$ $6_{11}$ $6_{12}$
$6_{13}$ $6_{14}$ $6_{15}$

2. Preliminaries

2.1. Handlebody links and spatial graphs

Definition 2.1 (Embeddings in $\mathbb{S}^{3}$ ).

A handlebody link ${\rm HL}$ (resp. a spatial graph $G$ ) is an embedding of finitely many handlebodies of positive genus (resp. a finite graph⁴⁴4A finite graph is a graph with finitely many vertices and edges; in addition, we require that no component has a positive Euler characteristic, to avoid trivial objects. A circle is regarded as a graph without vertices as in [9].) in the oriented $3$ -sphere $\mathbb{S}^{3}$ .

The genus of a handlebody link is the sum of the genera of its components; a spatial graph is trivalent if the underlying graph is trivalent (each node has degree 3). By a slight abuse of notation, we also use ${\rm HL}$ (resp. $G$ ) to denote the image of the embedding in $\mathbb{S}^{3}$ . Let $r{\rm HL}$ (resp. $rG$ ) be the mirror image of ${\rm HL}$ (resp. $G$ ).

Definition 2.2 (Equivalence).

Two handlebody links ${\rm HL}$ , ${\rm HL}^{\prime}$ (resp. spatial graphs $G,G^{\prime}$ ) are equivalent if they are ambient isotopic; they are equivalent up to mirror image if ${\rm HL}$ (resp. $G$ ) is equivalent to ${\rm HL}^{\prime}$ or $r{\rm HL}^{\prime}$ (resp. $G^{\prime}$ or $rG^{\prime}$ ).

A regular neighborhood of a spatial graph defines a handlebody link, up to equivalence [22, $3.24$ ], and a spine of a handlebody link ${\rm HL}$ is a spatial graph $G$ with ${\rm HL}$ a regular neighborhood of $G$ [9]. In practice, it is more convenient to consider spines that are trivalent.

Lemma 2.1.

Every handlebody link admits a (trivalent) spine.

Proof.

It suffices to prove the connected case. We suppose $\operatorname{HK}$ is a handlebody knot of genus $g$ , and $\mathbf{D}=\{D_{1},\cdots,D_{3g-3}\}$ is a set of disjoint incompressible disks in $\operatorname{HK}$ such that the complement $\operatorname{HK}\setminus\bigcup_{i}N(D_{i})$ of their tubular neighborhoods $N(D_{i})$ in $\operatorname{HK}$ consists of $2(g-1)$ $3$ -balls, each of which has non-trivial intersection with exactly three components of $\coprod_{i=1}^{3g-3}\partial\overline{N(D_{i})}$ . Such a disk system always exists.

Let disks $D_{i1},D_{i2},D_{i3}$ be components of $B_{i}\cap\big{(}\bigcup_{k=1}^{3g-3}\overline{N(D_{k})}\big{)}$ , and choose points $v_{i1},v_{i2},v_{i3}$ in the interior of $D_{i1},D_{i2},D_{i3}$ and a point $v_{i}$ in the interior of $B_{i}$ . Then, join $v_{i}$ to $v_{ij}$ by a path for each $j$ ; this gives us a trivalent vertex. Repeat the construction for every $i$ , and then glue the $v_{ij}$ together so that the vertices $v_{ij}$ and $v_{i^{\prime}j^{\prime}}$ are identified if they are in the same $\overline{N(D_{k})}$ , for some $k$ . This way, we obtain a connected trivalent spine of $\operatorname{HK}$ with $2(g-1)$ trivalent vertices. ∎

In general, a trivalent spine of a $n$ -component handlebody link of genus $g$ has $2(g-n)=2t$ trivalent vertices, and we call such a handlebody link a $(n,t)$ -handlebody link. This paper is primarily concerned with the case $t=1$ .

2.2. Diagrams

Let $\mathbb{S}^{k}=\mathbb{R}^{k}\cup\infty$ , and without loss of generality, it may be assumed handlebody links or spatial graphs are away from $\infty$ .

Definition 2.3 (Regular projection).

A regular projection of a spatial graph $G$ is a projection $\pi:\mathbb{S}^{3}\setminus\infty\rightarrow\mathbb{S}^{2}\setminus\infty$ such that the set $\pi^{-1}(x)\cap G$ is finite with its cardinality $\#(\pi^{-1}(x)\cap G)\leq 2$ for any $x\in\mathbb{S}^{2}\setminus\infty$ , and no $0$ -simplex of the polygonal subset $G$ of $\mathbb{S}^{3}$ is in the preimage of a double point, a double point being a point $x\in\mathbb{S}^{2}\setminus\infty$ with $\#(\pi^{-1}(x)\cap G)=2$ .

As with the case of knots, up to ambient isotopy, every spatial graph admits a regular projection: the idea is to choose a vector $v$ neither parallel to a $1$ -simplex in the polygonal subset $G\subset\mathbb{S}^{3}\setminus\infty=\mathbb{R}^{3}$ nor in a plane containing a $0$ -simplex and a $1$ -simplex or two $1$ -simplices; then isotopy $G$ slightly to remove those points $x$ with $\#\pi_{v}^{-1}(x)\cap G>2$ , where $\pi_{v}$ is the projection onto the plane normal to $v$ .

Definition 2.4 (Diagram of a spatial graph).

A diagram of a spatial graph $G$ is the image of a regular projection of $G$ with relative height information added to each double point.

The convention is to make breaks in the line corresponding to the strand passing underneath; thus each double point becomes a crossing of the diagram.

Definition 2.5 (Diagram of a handlebody link).

A diagram of a handlebody link ${\rm HL}$ is a diagram of a spine of ${\rm HL}$ .

A diagram of $G$ (resp. ${\rm HL}$ ) is trivalent if it is obtained from a regular projection of a trivalent spatial graph (resp. spine).

Definition 2.6 (Crossing numbers).

The crossing number $c(D)$ of a diagram $D$ of a handlebody link ${\rm HL}$ (resp. of a spatial graph $G$ ) is the number of crossings in $D$ . The crossing number $c({\rm HL})$ of ${\rm HL}$ (resp. $c(G)$ of $G$ ) is the minimum of the set

\{c(D)\mid\text{$D$ a diagram of ${\rm HL}$ $($resp. $G)$}\}.

Definition 2.7 (Minimal diagram).

A minimal diagram $D$ of a handlebody link ${\rm HL}$ (resp. of a spatial graph $G$ ) is a diagram of ${\rm HL}$ (resp. $G$ ) with $c(D)=c({\rm HL})$ (resp. $c(D)=c(G)$ ).

Every multi-valent vertex in a minimal diagram $D$ can be replaced with some trivalent vertices by the inverse of the contraction move [9, Fig. $1$ ] without changing the crossing number, so for a handlebody link (resp. a spatial graph) there always exists a trivalent minimal diagram $D$ . From now on, we shall use the term “a diagram” to refer to a trivalent diagram of either a spatial graph or a handlebody link.

Observe that, regarding each crossing as a quadrivalent vertex, we obtain a plane graph, a finite graph embedded in the $2$ -sphere. In practice, we work backward and start with a plane graph having only trivalent and quadrivalent vertices, and produce diagrams by replacing quadrivalent vertices with under- or over-crossings. If the plane graph has $2t$ trivalent vertices and $c$ quadrivalent vertices, then we can recover $2^{c-1}$ diagrams from it, up to mirror image. In particular, a $c$ -crossing $(n,t)$ -handlebody link can be recovered from one of these plane graphs. Therefore if one can enumerate all plane graphs with $2t$ trivalent vertices and up to $c$ quadrivalent vertices, then one can find all $(n,t)$ -handlebody links up to $c$ crossings.

2.3. Moves

Definition 2.8 (Moves).

Local changes in a diagram depicted in Fig. 2.1 and Fig. 2.2 are called generalized Reidemeister moves, and the local change in Fig. 2.3 is called an IH-move.

Note that spines of equivalent handlebody links might be inequivalent as spatial graphs; indeed, the following holds.

Theorem 2.2 ([12, Theorem $2.1$ ], [26]).

Two trivalent spatial graphs are equivalent if and only if their diagrams are related by a finite sequence of generalized Reidemeister moves.

Theorem 2.3 ([9, Corollary $2$ ]).

Two handlebody links are equivalent if and only if their trivalent diagrams are related by a finite sequence of generalized Reidemeister moves and IH-moves.

When analyzing the data from the code (Appendix A), it is more convenient to call a diagram IH-minimal if the number of crossings cannot be reduced by generalized Reidemeister moves and IH moves, that is, “minimal” as a diagram of a handlebody link, and to call a diagram R-minimal if the number of crossings cannot be reduced by generalized Reidemeister moves, that is, “minimal” as a diagram of a spatial graph.

2.4. Non-split, irreducible handlebody links

Definition 2.9 (Edge connectivity of a graph).

The edge-connectivity of a graph is the minimum number of edges whose deletion disconnects the graph.

Definition 2.10 (Connectivity of a diagram).

A diagram has $e$ -connectivity if its underlying plane graph has edge-connectivity $e$ .

Definition 2.11 (Split handlebody link).

A handlebody link ${\rm HL}$ is split if there exists a $2$ -sphere $\mathfrak{S}\subset\mathbb{S}^{3}$ such that $\mathfrak{S}\cap{\rm HL}=\emptyset$ and both components of the complement $\overline{\mathbb{S}^{3}\setminus\mathfrak{S}}$ have non-trivial intersection with ${\rm HL}$ .

Definition 2.12 (Reducible handlebody link).

A handlebody link ${\rm HL}$ is reducible if its complement admits a $2$ -sphere $\mathfrak{S}$ such that $\mathfrak{S}\cap{\rm HL}$ is an incompressible disk in ${\rm HL}$ ; otherwise it is irreducible.

Note that $\mathfrak{S}$ in Definition 2.12 factorizes ${\rm HL}$ into two handlebody links, each called a factor of the factorization of ${\rm HL}$ .

A diagram with $0$ -connectivity (resp. $1$ -connectivity) represents a split (resp. reducible) handlebody link, so only diagrams with connectivity greater than $1$ are of interest to us; on the other hand, the connectivity of a diagram of a $(n,t)$ -handlebody link with $t>0$ cannot exceed $3$ .

Now, we recall the order- $2$ vertex connected sum between spatial graphs [17], which is used to produce handlebody links represented by minimal diagrams with $2$ -connectivity. A trivial ball-arc pair of a spatial graph $G$ is a $3$ -ball $B$ with $G\cap B$ a trivial tangle in $B$ ; it is oriented if an orientation of $G\cap B$ is given.

Definition 2.13 (Knot sum).

Given two spatial graphs $G_{1},G_{2}$ with oriented trivial ball-arc pairs $B_{1},B_{2}$ of $G_{1},G_{2}$ , respectively, their order- $2$ vertex connected sum $(G_{1},B_{1})\#(G_{2},B_{2})$ is a spatial graph obtained by removing the interiors of $B_{1},B_{2}$ and gluing the resulting manifolds $\overline{\mathbb{S}^{3}\setminus B_{1}}$ and $\overline{\mathbb{S}^{3}\setminus B_{2}}$ by an orientation-reserving homoemorphism

h:\big{(}\partial(\overline{\mathbb{S}^{3}\setminus B_{1}}\big{)},\partial(G_{1}\cap B_{1}))\rightarrow(\partial\big{(}\overline{\mathbb{S}^{3}\setminus B_{2}}\big{)},\partial(G_{2}\cap B_{2})).

The notation $G_{1}\#G_{2}$ denotes the set of order- $2$ vertex connected sums of $G_{1},G_{2}$ given by all possible trivial ball-arc pairs.

Since an order- $2$ vertex connected sum depends only on the edges of $G_{1},G_{2}$ intersecting with $B_{1},B_{2}$ and their orientations, $G_{1}\#G_{2}$ is a finite set.

3. Uniqueness, non-splittability, and irreducibility

Recall that, given a finite group $\mathsf{G}$ , the Kitano-Suzuki invariant $ks_{\mathsf{G}}({\rm HL})$ of a handlebody link ${\rm HL}$ is the number of conjugate classes of homomorphisms from $\pi_{1}(\overline{\mathbb{S}^{3}\setminus{\rm HL}})$ to $\mathsf{G}$ [13]. Table 2 lists the invariants $ks_{\mathsf{A}_{4}}({\rm HL})$ and $ks_{\mathsf{A}_{5}}({\rm HL})$ of each handlebody link ${\rm HL}$ in Table 1, $\mathsf{A}_{k}$ being the alternating group on $k$ letters, as well as an upper bound of the rank of $\pi_{1}(\overline{\mathbb{S}^{3}\setminus{\rm HL}})$ computed by Appcontour [21].

The entry “split” refers to the split handlebody link $\operatorname{HL}$ given by a trivial handlebody knot and an unknotted solid torus; the entry “fake $6_{5}$ ” is the split handlebody link consisting of the handlebody knot $\operatorname{HK}4_{1}$ , Ishii-Kishimoto-Suzuki-Moriuchi’s $4_{1}$ in [10], and an unknotted solid torus; the entry “fake $6_{11}$ ” is $6_{11}$ in Table 1 with one of the bottom crossings reversed, thus making the lower solid torus component split off.

Table 2. Kitano-Suzuki invariant for entries in Table 1.

handlebody link	components	$ks_{\mathsf{A}_{4}}$	$ks_{\mathsf{A}_{5}}$	rank
split	trivial + unknot	178	3675	3
$4_{1}$	trivial + unknot	114	600	3
$5_{1}$	trivial + unknot	98	660	$\leq$ 4
$6_{1}$	trivial + unknot	90	600	3
$6_{2}$	trivial + unknot	106	689	3
$6_{3}$	trivial + unknot	90	469	3
$6_{4}$	HK $4_{1}$ + unknot	106	689	3
$6_{5}$	HK $4_{1}$ + unknot	210		$\leq$ 4
fake $6_{5}$	HK $4_{1}$ + unknot	274
$6_{6}$	trivial + unknot	130	1380	3
$6_{7}$	trivial + unknot	98	597	$\leq$ 4
$6_{8}$	trivial + unknot	114	1401	3
$6_{9}$	trivial + 2 unknots	310	1841	4
$6_{10}$	trivial + 2 unknots	326		4
$6_{11}$	trivial + 2 unknots	486	5876	4
fake $6_{11}$	trivial + 2 unknots	694
$6_{12}$	trivial + 2 unknots	502	5883	4
$6_{13}$	trivial + 2 unknots	822		4
$6_{14}$	trivial + 2 unknots	486	5876	4
$6_{15}$	trivial + 3 unknots	1242		5

Theorem 3.1 (Uniqueness).

Entries in Table 1 are all inequivalent.

Proof.

All entries in Table 1 except for the pairs $(6_{2},6_{4})$ and $(6_{11},6_{14})$ are distinguished by comparing their $ks_{\mathsf{A}_{4}}$ and $ks_{\mathsf{A}_{5}}$ invariants (shown in Table 2). On the other hand, $6_{2}$ and $6_{4}$ cannot be equivalent because the removal of the “unknot” component produces inequivalent handlebody knots: one being trivial, the other being $\operatorname{HK}4_{1}$ . Similarly, one can distinguish $6_{11}$ and $6_{14}$ by removing the solid torus component having a non-trivial linking number with the genus $2$ handlebody component [16] in each of them, and observing that, for $6_{14}$ , the resulting handlebody link is $4_{1}$ , whereas for $6_{11}$ , we get the trivial split handlebody link. ∎

Remark 3.1.

The pairs $(6_{2},6_{4})$ and $(6_{11},6_{14})$ in fact have homeomorphic complements, and hence the fundamental group cannot discriminate. Fig. 3.1 and 3.2 illustrate how to obtain the complements of $6_{2}$ and $6_{11}$ from $6_{4}$ and $6_{14}$ , respectively, via $3$ -dimensional Dehn-twists (indicated by arrows).

Remark 3.2.

$6_{5}$ viewed as a diagram of a spatial graph is the notorious figure eight puzzle devised by Steward Coffin [3]. The goal of the puzzle is to free the circle component from the knotted handcuff graph, i.e. to obtain the fake $6_{5}$ as a spatial graph. The impossibility of solving the puzzle then follows from computing $ks_{\mathsf{A}_{4}}(\bullet)$ of $6_{5}$ and fake $6_{5}$ (Table 2). See [1], [15] for other proofs of this.

Theorem 3.2 (Unsplittability).

Entries in Table 1 are all unsplittable.

Proof.

In most cases $(5_{1},6_{1},6_{2},6_{3},6_{4},6_{7},6_{9},6_{10})$ unsplittability follows by computing the linking number [16] between pairs of components of a handlebody link. There are a few cases where the linking number vanishes, and we deal with these cases by computing the $ks_{\mathsf{A}_{4}}$ - and $ks_{\mathsf{A}_{5}}$ -invariants of the corresponding split handlebody links (Table 2).

If $6_{5}$ were split, then $6_{5}$ would be equivalent to the fake $6_{5}$ but this is not possible by Table 2. In the case of $4_{1}$ , $6_{6}$ , $6_{8}$ , if any of them were split, than it would be equivalent to “split” in Table 2, but that is not the case. A similar argument can be applied to $6_{11}$ and $6_{14}$ : if one of them were split, it would be equivalent to the fake $6_{11}$ , in contradiction to Table 2. Lastly, we observe that $6_{12}$ and $6_{13}$ are non-split, for otherwise $4_{1}$ would be split. ∎

Below we recall the irreducibility test developed in [2]. A $r$ -generator link is a link whose knot group, the fundamental group of its complement, is of rank $r$ .

Lemma 3.3.

If the trivial knot is a factor of some factorization of a reducible $(n,1)$ -handlebody link ${\rm HL}$ , then

12\mid ks_{\mathsf{A}_{4}}({\rm HL})+6\cdot 3^{n}+2\cdot 4^{n}\quad\textbf{and}\quad 60\mid ks_{\mathsf{A}_{5}}({\rm HL})+14\cdot 4^{n}+19\cdot 3^{n}+22\cdot 5^{n}.

(3.1)

Lemma 3.4.

If a $2$ -generator knot is factor of some factorization of a reducible $(n,1)$ -handlebody link ${\rm HL}$ , then

12+24p\mid ks_{\mathsf{A}_{4}}({\rm HL})+(6+16p)\cdot 3^{n}+(2+6p)\cdot 4^{n},\hskip 3.00003pt\textbf{where $p=0$ or $1$}.

(3.2)

Lemma 3.5.

If a $2$ -component, $2$ -generator link is a factor of some factorization of a reducible $(n,1)$ -handlebody link ${\rm HL}$ , then

48+24p\mid ks_{\mathsf{A}_{4}}({\rm HL})+(26+16p)\cdot 3^{n-1}+(8+6p)\cdot 4^{n-1},\hskip 3.00003pt\textbf{where $p=0,1,2,3$ or $4$.}

(3.3)

From the above lemmas, one derives the following irreducibility test (see [2] for more details), making use of the Grushko theorem [6].

Corollary 3.6 (Irreducibility test).

A $3$ -generator $(2,1)$ -handlebody link is irreducible if it fails to satisfy (3.1); a $4$ -generator $(2,1)$ -handlebody link is irreducible if it fails to satisfy (3.2); a $4$ -generator $(3,1)$ -handlebody link or a $5$ -generator $(4,1)$ -handlebody link is irreducible if it fails to satisfy (3.1) and (3.3).

Theorem 3.7 (Irreducibility).

Entries in Table1 are irreducible.

Proof.

Corollary 3.6, together with Table 2, shows that all but $6_{9}$ , $6_{12}$ are irreducible. The irreducibility of $6_{12}$ and $6_{9}$ follows from computing the linking number between each pair of components in each of them. Specifically, if $6_{12}$ (resp. $6_{9}$ ) is reducible, then either the trivial knot or a $2$ -generator $2$ -component link is a factor of some factorization of $6_{12}$ (resp. $6_{9}$ ). For $6_{12}$ , the former case is not possible by (3.1); the latter impossible too, for otherwise the two solid torus components would have a trivial linking number. The same argument implies that $6_{9}$ cannot have a $2$ -generator $2$ -component link as a factor, and the trivial knot cannot be its factor either, since the homomorphism of integral homology

H_{1}(V_{1})\oplus H_{1}(V_{2})\rightarrow H_{1}(\overline{\mathbb{S}^{3}\setminus W})

is onto, where $V_{1},V_{2}$ are the solid torus components, and $W$ the genus $2$ component. ∎

Remark 3.3.

The complement of $6_{9}$ is in fact $\partial$ -reducible; one can see this by performing the twist operation, indicated by the arrow in Fig. 3.3, where it shows that its complement is homeomorphic to the complement of the order- $1$ connected sum (Definition 6.1) between two Hopf links (Fig. 3.3, right). In the connected case, no irreducible handlebody knot of genus $2$ admits a $\partial$ -reducible complement [24, Theorem $1$ ], but when the genus is larger than $2$ , such handlebody knots exist [23, Example $5.5$ ], [24, Section $5$ ]. For a $(n,1)$ -handlebody link, we suspect that $6_{9}$ attains the lowest possible $n$ for such a phenomenon to happen.

4. Completeness

This section discusses completeness of Table 1. Recall first that a minimal diagram of a non-split, irreducible handlebody link has either $2$ - or $3$ -connectivity. IH-minimal diagrams with $3$ -connectivity are obtained from a software code, and IH-minimal diagrams with $2$ -connectivity are recovered by knot sum of spatial graphs.

4.1. Minimal diagrams with 3-connectivity

We consider plane graphs with two trivalent vertices and up to six quadrivalent vertices satisfying the properties:

(1)

each of them has edge-connectivity $3$ as an abstract graph,
(2)

their double arcs can only connect two quadrivalent vertices as abstract graphs, and
(3)

their double arcs only form a “bigon” (a polygon with two sides; the case ‘i’ in Fig. 4.1) as plane graphs.

The reason of considering only double arcs connecting two quadrivalent vertices with a bigon configuration is because all the other cases lead to either non-R-minimal diagrams or diagrams with connectivity less than $3$ (see Fig. 4.1, where “d, e, f, g, h” illustrate those double arcs connecting at least one trivalent vertex and “j, k, l” those connecting two quadrivalent vertices with a non-bigon configuration.)

We enumerate such plane graphs by the software code, and then recover diagrams from these plane graphs by adding an over- or under-crossing to each quadrivalent vertex. Note that the number ( $n$ in Table 3) of components of the associated spatial graphs is independent of how over/under-crossings are chosen. To provide a glimpse of how the code works, we record in Table 3 the number of such plane graphs with $c$ quadrivalent vertices for each $n$ . To recover $(n,1)$ -handlebody links with $n>1$ represented by IH-minimal diagrams with $3$ -connectivity, we need to consider the cases with $n>1$ in Table 3. On the other hand, to produce $(n,1)$ -handlebody links represented by IH-minimal diagrams with $2$ -connectivity, spatial graphs admitting an R-minimal diagram with $3$ -connectivity up to $4$ crossings are required; thus all cases with $c\leq 4$ have to be examined.

Table 3. Plane graphs given by the code.

$c$	$n=1$	$n=2$	$n=3$	total
2	1			1
3	2	1		3
4	8	2		10
5	29	8		37
6	144	34	3	181

IH-minimal diagrams. We examine IH-minimality of diagrams produced by plane graphs with $n\geq 2$ , and discard those obviously not IH-minimal. This excludes all diagrams produced by the code up to $5$ crossings (Table 7), but for diagrams with $6$ crossings, some diagrams are potentially IH-minimal: they represent handlebody links $6_{1}$ , $6_{2}$ , $6_{3}$ or $6_{9}$ in Table 1.

Lemma 4.1.

An IH-minimal diagram with $3$ -connectivity has crossing number $c\geq 6$ , and if $c=6$ , it represents a handlebody link equivalent to $6_{1}$ , $6_{2}$ , $6_{3}$ or $6_{9}$ , up to mirror image.

Note that we cannot conclude diagrams of $6_{1},6_{2},6_{3}$ and $6_{9}$ in Table 1 are IH-minimal yet, as they might admit diagrams with $2$ -connectivity and fewer crossings.

R-minimal diagrams. To produce minimal diagrams with $2$ -connectivity up to $6$ crossings, we need R-minimal diagrams up to $4$ crossings. Inspecting R-minimality of diagrams produced by the code (Table 6) gives us the following lemma.

Lemma 4.2.

An R-minimal diagram with $3$ -connectivity and crossing number less than $5$ represents one of the spatial graphs in Table 4, up to mirror image.

[Uncaptioned image] — Table 4. Spatial graphs up to four crossings.

4.2. Minimal diagrams with 2-connectivity

Recall a diagram $D$ with $2$ -connectivity can be decomposed into finitely many simpler tangle diagrams such that each associated diagram of spatial graphs has $3$ - or $4$ -connectivity (Fig. 1.1). Furthermore, if $D$ is R-minimal, each induced spatial graph diagram is also R-minimal. In particular, an IH-minimal diagram with $2$ -connectivity can be recovered by performing the order-2 vertex connected sum between spatial graphs admitting a minimal diagram with $k$ -connectivity, $k>2$ . Since we are interested in $(n,1)$ -handlebody links, only one summand is a spatial graph with two trivalent vertices, and the rest are links admitting a minimal diagram with $4$ -connectivity. Note that the simplest minimal diagram with $4$ -connectivity represents the Hopf link, and since we only consider minimal diagrams up to $6$ crossings, there are at most three link summands. Thus, IH-minimal diagrams with $2$ -connectivity can be recovered by considering the seven possible configurations below:

(1)

$G\#L_{1}$ ,
(2)

( $G\#L_{1})\#L_{2}$ ,
(3)

$G\#(L_{1}\#L_{2})$ ,
(4)

(( $G\#L_{1})\#L_{2})\#L_{3}$ ,
(5)

( $G\#L_{1})\#(L_{2}\#L_{3})$ ,
(6)

( $G\#(L_{1}\#L_{2}))\#L_{3}$ ,
(7)

$G\#((L_{1}\#L_{2})\#L_{3})$ ,

where $G$ is a spatial graph admitting a minimal diagram with $3$ -connectivity, and $L_{i}$ is a link admitting a minimal diagram with $4$ -connectivity. In general it is not known if a minimal diagram with $4$ -connectivity always represents a prime link; it is the case, however, when the crossing number is less than $5$ . In fact, there are only four minimal diagrams with $4$ -connectivity up to $4$ crossings, and they represent the Hopf link, the trefoil knot, the figure eight, and Solomon’s knot (L4a1), respectively.

Cases $4$ through $7$ are easily dealt with since $G$ must have no crossings, and hence it is the trivial theta curve $\operatorname{G0_{1}}$ , and thus each $L_{i}$ is necessarily the Hopf link, so the knot sums actually consist in ‘inserting a ring’ somewhere to the result of the previous knot sums. To produce irreducible handlebody links there is only one possibility, that is, adding one Hopf link to each of the three arcs of the trivial theta curve, and this gives us entry $6_{15}$ in Table 1.

Cases $2$ and $3$ forces $G$ to have $2$ crossings at most. It cannot have zero crossings (trivial theta curve), for otherwise, we could only get diagrams representing reducible handlebody links. Note also that there is no $R$ -minimal diagram with $1$ crossing. Now, there is only one $R$ -minimal diagram with two crossings, this is, entry G $2_{1}$ in Table 4 (Moriuchi’s $2_{1}$ in [18]).

Now, to add two Hopf links to it, i.e. to place two rings successively, we observe that one of them must be placed around the connecting arc of the handcuff graph by irreducibility. The second ring can be placed in three inequivalent ways, which yield entries $6_{12}$ , $6_{13}$ and $6_{14}$ of Table 1.

Case $1$ is more complicated, and we divided it into subcases based on the crossing number $c:=c(G)$ . The case $c=0$ is immediately excluded by irreducibility, so three possibilities remain: $c\in\{2,3,4\}$ .

Subcase $c(G)=2$ . $G$ is necessarily G $2_{1}$ in Table 4, and $L$ cannot be a knot. Since the crossing number of $L$ cannot exceed $4$ , $L$ is either L2a1 (Hopf link) or L4a1 (Solomon’s knot). In either case, $L$ is to be added to the connecting arc of the handcuff graph to produce irreducible handlebody links, yielding entries $4_{1}$ and $6_{8}$ in Table 1.

Subcase $c(G)=3$ . $G$ is necessarily G $3_{1}$ (Moriuchi’s theta curve $3_{1}$ in [17]), so $L$ cannot be a knot, and hence is the Hopf link. There is only one place to add $L$ by irreducibility, and this leads to entry $5_{1}$ in Table 1.

Subcase $c(G)=4$ . In this case, $L$ can only be the Hopf link; and there are five possible spatial graphs for $G$ , namely G $4_{1}$ , G $4_{2}$ , G $4_{3}$ , G $4_{4}$ , and G $4_{5}$ :

•

For G $4_{1}$ in Table 4 (Moriuchi’s non-prime handcuff graph $2_{1}\#_{3}2_{1}$ [19]), there are two inequivalent ways to add $L$ which produce entries $6_{4}$ and $6_{5}$ .
•

For G $4_{2}$ and G $4_{3}$ in Table 4 (Moriuchi’s prime handcuff graph $4_{1}$ [18] and prime theta-curve $4_{1}$ [17], respectively), there is only one way to add the Hopf link in each case by irreducibility, and this gives $6_{6}$ , $6_{7}$ in Table 1, respectively.
•

For G $4_{4}$ and G $4_{5}$ in Table 4, again by irreducibility, there is only one way to add the Hopf link in each case, which gives us $6_{10}$ and $6_{11}$ in Table 1, respectively.

We summarize the discussion above in the following:

Lemma 4.3.

A non-split, irreducible handlebody link admitting an IH-minimal diagram with $2$ -connectivity and crossing number $\leq 6$ is equivalent, up to mirror image, to one of the following handlebody links:

4_{1},5_{2},6_{4},6_{5},6_{6},6_{7},6_{8},6_{10},6_{11},6_{12},6_{13},6_{14}.

(4.1)

By Lemma 4.1, if any of (4.1) admits an IH-minimal diagram with $3$ -connectivity, it is equivalent to one of $6_{1},6_{2},6_{3}$ , $6_{9}$ , while by Lemma 4.1 if $6_{1},6_{2},6_{3}$ or $6_{9}$ admits an IH-minimal diagram with $2$ -connectivity and less than $6$ crossings, it is equivalent to $4_{1}$ or $5_{1}$ , but neither situation can happen by Theorem 3.1.

Corollary 4.4.

Diagrams in Table 1 are all IH-minimal.

5. Chirality

5.1. Decomposable links

We consider order- $2$ connected sum (compare with Definition 6.1) for handlebody-link-disk pairs. A handlebod-link-disk pair is a handlebody link ${\rm HL}$ with an oriented incompressible disk $D\subset{\rm HL}$ . A trivial knot with a meridian disk is considered as the trivial handlebody-link-disk pair.

Definition 5.1 (Order-2 connected sum).

Given two handlebody-link-disk pairs $({\rm HL}_{1},D_{1})$ , $({\rm HL}_{2},D_{2})$ the order-2 connected sum $({\rm HL}_{1},D_{1})\#({\rm HL}_{2},D_{2})$ is obtained as follows: first choose a $3$ -ball $B_{i}$ of $D_{i}$ in $\mathbb{S}^{3}$ for each $i$ with $\mathring{B_{i}}\cap{\rm HL}_{i}$ a tubular neighborhood $N(D_{i})$ of $D_{i}$ in ${\rm HL}_{i}$ ; next, identify $\overline{N(D_{i})}$ with $D_{i}\times[0,1]$ via the orientation of $D_{i}$ . Then $({\rm HL}_{1},D_{1})\#({\rm HL}_{2},D_{2})$ is given by removing $\mathring{B_{i}}$ and gluing the resulting manifolds via an orientation-reversing homeomorphism:

h:\partial(\overline{\mathbb{S}^{3}\setminus B_{1}})\rightarrow\partial(\overline{\mathbb{S}^{3}\setminus B_{2}})\quad\textbf{with $h(D_{1}\times\{j\})=D_{2}\times\{k\}$, $k\equiv j+1$ mod $2$}.

A non-split, irreducible handlebody link is decomposable if it is equivalent to an order-2 connected sum of non-trivial handlebody-link-disk pairs.

Decomposability is reflected in minimal diagrams in most examples here, thus we state the following conjecture.

Conjecture 5.1.

Minimal diagrams of a decomposable handlebody link have $2$ -connectivity.

Lemma 5.2.

For a non-split, irreducible handlebody link ${\rm HL}$ , the following statements are equivalent:
• ${\rm HL}$ is decomposable;
• there exists a $2$ -sphere $\mathfrak{S}$ in $\mathbb{S}^{3}$ such that $\mathfrak{S}$ and ${\rm HL}$ intersect at two incompressible disks in ${\rm HL}$ and neither of $\overline{B_{i}\setminus{\rm HL}}$ , $i=1,2$ , is a solid torus, where $B_{1},B_{2}$ are components of $\overline{\mathbb{S}^{3}\setminus\mathfrak{S}}$ ;
• $\overline{\mathbb{S}^{3}\setminus{\rm HL}}$ admits an incompressible, non-boundary parallel (or $\partial$ -incompressible) annulus $A$ with $\partial A$ inessential in ${\rm HL}$ .

Proof.

This follows from the definition of ( $\partial$ -) incompressibility. ∎

The annulus $A$ in Lemma 5.2 is called a decomposing annulus of ${\rm HL}$ . Similarly, a decomposiing annulus $A$ of a handlebody-link-disk pair $({\rm HL},D)$ is an incompressible, non-boundary parallel annulus $A$ with $\partial A$ inessential in ${\rm HL}$ and disjoint from $\partial D$ ; note that ${\rm HL}$ in $({\rm HL},D)$ could be reducible.

We now prove a unique decomposition theorem for handlebody links with no genus $g>2$ component; a unique decomposition theorem for handlebody knots of arbitrary genus is given [14, Appendix $B$ ] (see also [11]).

Theorem 5.3.

Given a non-split, irreducible handlebody link ${\rm HL}$ , suppose no component of $\operatorname{HL}$ has genus greater than 2, and $A,A^{\prime}$ are decomposing annuli inducing

{\rm HL}\simeq({\rm HL}_{1},D_{1})\#({\rm HL}_{2},D_{2}),\;{\rm HL}\simeq({\rm HL}_{1}^{\prime},D_{1}^{\prime})\#({\rm HL}_{2}^{\prime},D_{2}^{\prime}),\textbf{ respectively}.

(5.1)

If $(\overline{\mathbb{S}^{3}\setminus{\rm HL}_{i}},D_{i}),i=1,2$ , admits no decomposing annulus. Then $A,A^{\prime}$ are isotopic, in the sense that there exists an ambient isotopy $f_{t}:\mathbb{S}^{3}\rightarrow\mathbb{S}^{3}$ fixing $\operatorname{HL}$ with $f_{1}(A)=A^{\prime}$ .

Proof.

Note first that if $A$ , $A^{\prime}$ are disjoint, then the assumption implies that they must be parallel and hence isotopic. Suppose $A\cap A^{\prime}\neq\emptyset$ . Then we isotopy $A$ such that the number of components of $A\cap A^{\prime}$ is minimized.

Claim: any circle or arc in $A\cap A^{\prime}$ is essential in both $A$ and $A^{\prime}$ . Observe first that a circle component $C$ of $A\cap A^{\prime}$ is either essential or inessential in both $A$ and $A^{\prime}$ , for assuming otherwise would contradict the incompressibility of $A,A^{\prime}$ . Suppose $C$ is inessential in both $A$ and $A^{\prime}$ , and is innermost in $A^{\prime}$ . Then $C$ bounds disks $D,D^{\prime}$ in $A,A^{\prime}$ , respectively. Since ${\rm HL}$ is non-split, $D\cup D^{\prime}$ bounds a $3$ -ball $B$ in $\overline{\mathbb{S}^{3}\setminus{\rm HL}}$ . Isotopy $D$ across $B$ to $D^{\prime}$ induces a new annulus $A$ isotopic to the original one with $A\cap A^{\prime}$ having less components, contradicting the minimality.

Similarly, if $l$ is an arc component of $A\cap A^{\prime}$ , then it is either essential or inessential in both $A$ and $A^{\prime}$ , for assuming otherwise would contradict the $\partial$ -incompressibility of $A,A^{\prime}$ . Suppose $l$ is inessential in both $A$ and $A^{\prime}$ and an innermost arc in $A^{\prime}$ . Then $l$ cuts off a disk $D^{\prime}$ from $A^{\prime}$ and a disk $D$ from $A$ . Let $\hat{D}:=D\cup D^{\prime}$ . If $\partial\hat{D}$ is inessential, then we can remove the intersection $l$ by isotopying $A$ across the ball bounded by $\hat{D}$ and the disk bounded by $\partial\hat{D}$ in $\partial\operatorname{HL}$ , contradicting the minimality. If $\partial\hat{D}$ is essential, then isotopying $\hat{D}$ , we can disjoin $\hat{D}$ from $A$ .

Now, it may be assumed that $D_{1}$ is in a genus one component of $\operatorname{HL}_{1}$ , and hence $D_{2}$ is in a component of $\operatorname{HL}_{2}$ with genus $\leq 2$ . Since $\partial\hat{D}$ is essential, $\hat{D}$ has to be in a genus $2$ component of $\operatorname{HL}_{2}$ containing $D_{2}$ . Because $\partial\hat{D}\cap D_{2}=\emptyset$ , if $\partial\hat{D}$ is essential on the boundary of the embedded solid torus $\big{(}{\rm HL}_{2}\setminus N(D_{2})\big{)}\subset\mathbb{S}^{3}$ , $\partial\hat{D}$ would be its longitude, where $N(D_{2})$ is a tubular neighborhood of $D_{2}$ , disjoint from $\hat{D}$ , in $\operatorname{HL}_{2}$ . Particularly, ${\rm HL}_{2}$ and therefore ${\rm HL}$ would be reducible, a contradiction. On the other hand, if $\partial\hat{D}$ bounds a disk on $\partial\big{(}{\rm HL}_{2}\setminus N(D_{2})\big{)}$ that contains some components of $\partial\overline{N(D_{2})}$ , then $\partial\hat{D}$ is inessential in ${\rm HL}_{2}$ , and hence in ${\rm HL}$ , again contradicting the irreducibility of ${\rm HL}$ .

The claim is proved, which also implies $A\cap A^{\prime}$ contains either circles or arcs.

No essential circles. Suppose $C$ is an essential circle, and a closest circle to $\partial A^{\prime}$ . Let $R^{\prime}$ be the annulus cut off by $C$ from $A^{\prime}$ with $A\cap R^{\prime}=C$ and $R$ an annulus cut off by $C$ from $A$ . We isotopy the incompressible annulus $\hat{R}:=R\cup R^{\prime}$ away from $A$ . Since components of $\partial\hat{R}$ are inessential in $\operatorname{HL}$ , by the assumption, $\hat{R}$ is either parallel to $A$ or boundary-parallel. In the former case, replacing $A$ with $R$ leads to a contradiction since $R\cap A^{\prime}$ has less components than $A\cap A^{\prime}$ . In the latter case, isotopying $R$ through the solid torus $V$ bounded by $\hat{R}$ and the part of $\partial{\rm HL}$ parallel to $\hat{R}$ gives a new $A$ isotopic to the original one but with less components in $A\cap A^{\prime}$ , contradicting the minimality.

No essential arcs. Suppose $l_{1}$ is an essential arc. Then choose the essential arc $l_{2}$ next to $l_{1}$ in $A^{\prime}$ such that the disk $D^{\prime}$ cut off by $l_{1},l_{2}$ from $A^{\prime}$ has $D^{\prime}\cap A=l_{1}\cup l_{2}$ and is on the side of $A$ containing components of $\operatorname{HL}_{1}$ . Let $D$ be a disk cut off by $l_{1},l_{2}$ from $A$ . It may be assumed, by pushing $D$ away from $A$ , that $D\cup D^{\prime}$ is disjoint from $A$ , and hence is on the genus $1$ complement of $\operatorname{HL}_{1}$ containing $D_{1}$ . Since $\hat{A}:=D\cup D^{\prime}$ is disjoint from $D_{1}$ , it is necessarily an annulus, for if it were a Möbious band, we would get a non-orientable surface embedded in $\mathbb{S}^{3}$ . Furthermore, each component of $\partial\hat{A}$ is necessarily inessential in $\operatorname{HL}_{1}$ , so it either bounds a meridian disk or is inessential in $\partial\operatorname{HL}_{1}$ . Note also it cannot be the case that one component of $\partial\hat{A}$ is essential in $\partial\operatorname{HL}_{1}$ and the other inessential by the irreducibility of $\operatorname{HL}$ . Suppose both components are inessential in $\partial\operatorname{HL}_{1}$ . Then $\hat{A}$ , together with disks on $\partial\operatorname{HL}_{1}$ bounded by $\partial\hat{A}$ , bounds a $3$ -ball, with which we can isotopy $A$ to remove the intersection $l_{1},l_{2}$ , contradicting the minimality. Suppose both components bound meridian disks in $\operatorname{HL}_{1}$ . Then $\tilde{A}=D^{c}\cup D^{\prime}$ has $\partial\tilde{A}$ inessential in $\partial\operatorname{HL}_{1}$ , where $D^{c}=\overline{A\setminus D}$ . Thus we reduce it to the previous case. ∎

5.2. Chirality

We divide the proof of Theorem 1.2 into two lemmas.

Lemma 5.4.

All handlebody links except for $5_{1}$ , $6_{3}$ , $6_{6}$ , $6_{7}$ , $6_{8}$ , $6_{10}$ in Table 1 are achiral.

Proof.

Except $6_{2}$ , equivalences between these handlebody links and their mirror images are easy to construct; an equivalence between $6_{2}$ and r $6_{2}$ is depicted in Fig. 5.2

∎

Lemma 5.5.

$5_{1}$ , $6_{3}$ , $6_{6}$ , $6_{7}$ , $6_{8}$ , $6_{10}$ in Table 1 are chiral.

Proof.

Recall that, given a handlebody link ${\rm HL}$ , if ${\rm HL}$ and $r{\rm HL}$ are equivalent, then there is an orientation-reversing self-homeomorphism of $\mathbb{S}^{3}$ sending ${\rm HL}$ to ${\rm HL}$ .

Observe that each of $5_{1},6_{6},6_{8},6_{10}$ admits an obvious decomposing annulus satisfying conditions in Theorem 5.3; particularly the annulus in each of them is unique. Their chirality then follows readily from the fact that torus links are chiral.

To see chirality of $6_{3}$ , we observe that, given a $(2,1)$ -handlebody link ${\rm HL}$ , any self-homeomorphism of $\mathbb{S}^{3}$ preserving ${\rm HL}$ sends the meridian $m$ and the preferred longitude $l$ of the circle component to $m^{\pm 1}$ and $l^{\pm 1}$ , respectively. In particular, any isomorphism on knot groups induced by such a homeomorphism sends the conjugacy class of $m\cdot l$ in $\pi_{1}(\overline{\mathbb{S}^{3}\setminus{\rm HL}})$ to the conjugacy class of $m\cdot l$ , $m^{-1}\cdot l$ , $m\cdot l^{-1}$ or $m^{-1}\cdot l^{-1}$ , depending on whether the homeomorphism is orientation-preserving.

Let $N$ be the number of conjugacy classes of homomorphisms from $\pi_{1}(\overline{\mathbb{S}^{3}\setminus{\rm HL}})$ to a finite group $\mathsf{G}$ that sends $m\cdot l$ (and hence $m^{-1}\cdot l^{-1}$ ) to $1$ , and $rN$ the number of conjugacy classes of homomorphisms from $\pi_{1}(\overline{\mathbb{S}^{3}\setminus{\rm HL}})$ to $\mathsf{G}$ that sends $m\cdot l^{-1}$ (and hence $m^{-1}\cdot l$ ) to $1$ . Now, if ${\rm HL}$ and its mirror image $r{\rm HL}$ are equivalent, then $N=rN$ . This is however not the case with $6_{3}$ ; when $\mathsf{G}=A_{5}$ , we have $(N,rN)=(77,111)$ as computed by [21].

In the case of $6_{7}$ by Theorem 5.3, every self-homeomorphism $f$ of $\mathbb{S}^{3}$ sending $6_{7}$ to itself induces a self-homeomorphism $f$ sending the handlebody-knot-disk pair $(T,D)$ in Fig. 3(a) to itself, or alternatively the (fattened) figure eight with an arc $(\operatorname{K4_{1}},\alpha)$ in Fig. 3(b) to itself, where $\alpha$ is the dual one-simplex to $D$ .

Let $S$ be a minimal Seifert surface of the figure eight (Fig. 3(c)) containing the arc $\alpha$ . Then it can be further assumed that $f(S)=S$ , because the complement of the tubular neighborhood $N(\alpha)$ of $\alpha$ in $S$ is a Seifert surface of a Hopf link, and up to ambient isotopy, the Hopf link admits two minimal Seifert surfaces, and only one of them can give us $S$ after gluing $N(\alpha)$ back.

Since $f$ sends the complement $S\setminus N(\alpha)$ to $S\setminus f\big{(}N(\alpha)\big{)}$ , if $f$ is orientation-reversing, then the two oriented Hopf links in Fig. 5.4 are ambient isotopic, but that is not possible by their linking numbers. ∎

6. Reducible handlebody links

In this section, we show that Table 5 classifies, up to ambient isotopy and mirror image, all non-split, reducible $(n,1)$ -handlebody links up to six crossings (Theorem 1.3). We begin by considering the order- $1$ connected sum for handlebody links.

6.1. Order-1 connected sum

A handlebody-link-component pair $({\rm HL},h)$ is a handlebody link ${\rm HL}$ with a selected component $h$ of ${\rm HL}$ .

Definition 6.1 (Order- $1$ connected sum).

Let $({\rm HL}_{1},h_{1})$ and $({\rm HL}_{2},h_{2})$ be two handlebody-link-component pairs. Then their order- $1$ connected sum $({\rm HL}_{1},h_{1})$ – $({\rm HL}_{2},h_{2})$ is given by removing the interior of a $3$ -ball $B_{1}$ (resp. $B_{2}$ ) in $\mathbb{S}^{3}$ with $B_{1}\cap\partial{\rm HL}_{1}=B_{1}\cap\partial h_{1}$ (resp. $B_{2}\cap\partial{\rm HL}_{2}=B_{2}\cap\partial h_{2}$ ) a $2$ -disk, and then gluing the resulting $3$ -manifolds $\overline{\mathbb{S}^{3}\setminus B_{1}}$ , $\overline{\mathbb{S}^{3}\setminus B_{2}}$ via an orientation-reversing homeomorphism $f:(\partial B_{1},(\partial B_{1})\cap h_{1})\rightarrow(\partial B_{2},(\partial B_{2})\cap h_{2})$ . We use ${\rm HL}_{1}$ – ${\rm HL}_{2}$ to denote the set of order- $1$ connected sums between ${\rm HL}_{1},{\rm HL}_{2}$ with all possible selected components.

The following generalizes the case of handlebody knots in [24, Theorem $2$ ].

Theorem 6.1 (Uniqueness).

Given a non-split, reducible $(n,1)$ -handlebody link ${\rm HL}$ , if ${\rm HL}\simeq({\rm HL}_{1},h_{1})\text{-{}-}({\rm HL}_{2},h_{2})$ , and ${\rm HL}\simeq({\rm HL}_{1}^{\prime},h_{1}^{\prime})\text{-{}-}({\rm HL}_{2}^{\prime},h_{2}^{\prime})$ , then $({\rm HL}_{i},h_{i})\simeq({\rm HL}_{i}^{\prime},h_{i}^{\prime})$ , $i=1,2$ , up to reordering.

Proof.

Note first that, since ${\rm HL}$ is non-split and reducible, ${\rm HL}_{i},{\rm HL}_{i}^{\prime}$ , $i=1,2$ , are non-split, and $\pi_{1}(\overline{\mathbb{S}^{3}\setminus{\rm HL}})$ is a non-trivial free product $\mathsf{G}_{1}\ast\mathsf{G}_{2}$ , where $\mathsf{G}_{i}$ is the knot group of ${\rm HL}_{i}$ , $i=1,2$ .

Let $D$ and $D^{\prime}$ be the separating disks in $\overline{\mathbb{S}^{3}\setminus{\rm HL}}$ given by the factorizations ${\rm HL}\simeq({\rm HL}_{1},h_{1})\text{-{}-}({\rm HL}_{2},h_{2})$ and ${\rm HL}\simeq({\rm HL}_{1}^{\prime},h_{1}^{\prime})\text{-{}-}({\rm HL}_{2}^{\prime},h_{2}^{\prime})$ , respectively. Suppose neither $\mathsf{G}_{1}$ nor $\mathsf{G}_{2}$ is isomorphic to $\mathbb{Z}$ . Then, up to isotopy, $D^{\prime}\cap D=\emptyset$ by the innermost circle/arc argument.

Suppose one of $\mathsf{G}_{i},i=1,2$ , say $\mathsf{G}_{1}$ , is isomorphic to $\mathbb{Z}$ , that is, ${\rm HL}_{1}$ is a trivial solid torus in $\mathbb{S}^{3}$ . Then $\mathsf{G}_{2}$ must be non-cyclic, since $n>1$ . Let $D_{l}$ be the disk bounded by the longitude of ${\rm HL}_{1}$ , and isotopy $D,D_{l}$ such that the number $n$ (resp. $n_{l}$ ) of components of $D^{\prime}\cap D$ (resp. $D^{\prime}\cap D_{l}$ ) is minimized.

Claim: $n_{l}=0$ . Note first that the minimality implies that $D^{\prime}\cap D_{l}$ contains no circle components. Now, consider a tubular neighborhood $N(D_{l})$ of $D_{l}$ in $\overline{\mathbb{S}^{3}\setminus\operatorname{HL}}$ small enough such that $\overline{N(D_{l})}\cap D=\emptyset$ and $\overline{N(D_{l})}\cap D^{\prime}$ are some disks, each of which intersects $D_{l}^{+}$ (resp. $D_{l}^{-}$ ) at exactly one arc on its boundary, where $D_{l}^{\pm}\subset\partial\overline{N(D_{l})}$ are proper disks in $\overline{\mathbb{S}^{3}\setminus\operatorname{HL}}$ parallel to $D_{l}$ . The claim then follows once we have shown that $N(D_{l})$ can be isotopied away from $D^{\prime}$ .

To see this, we construct a labeled tree $\Upsilon$ from the complement of the intersection $D^{\prime}\cap D_{l}^{\pm}$ in $D^{\prime}$ , where $D^{\pm}:=D_{l}^{+}\cup D_{l}^{-}$ . Regard each component of $D^{\prime}\setminus\big{(}D^{\prime}\cap D^{\pm}\big{)}$ as a node in $\Upsilon$ , and each arc in $D^{\prime}\cap D_{l}^{\pm}$ as an edge in $\Upsilon$ connecting the two nodes representing the components of $D^{\prime}\setminus\big{(}D^{\prime}\cap D^{\pm}\big{)}$ whose closures intersect at the arc. Since each arc in $D^{\prime}\cap D^{\pm}$ cuts $D^{\prime}$ into two, $\Upsilon$ is a tree. The first two figures from the left in Fig. 6.2 illustrate the construction.

We label nodes and edges of $\Upsilon$ as follows: A node is labeled with $I$ if the corresponding component of $D^{\prime}\setminus\big{(}D^{\prime}\cap D^{\pm}\big{)}$ is inside $N(D_{l})$ ; otherwise the node is labeled with $O$ . An edge of $\Upsilon$ is labeled with $+$ if the corresponding component of $D^{\prime}\cap D^{\pm}_{l}$ is in $D_{l}^{+}$ ; otherwise, it is labeled with $-$ .

The labeling on $\Upsilon$ has the following properties: (a) adjacent nodes have different labels; (b) a node with label $I$ is bivalent, and the two adjacent edges are labeled with $+$ and $-$ , respectively, whereas a node labeled with $O$ could be multi-valent; (c) a one-valent node corresponds to an innermost arc in $D^{\prime}$ , and always has label $O$ .

Consider a maximal path $\Gamma\subset\Upsilon$ starting from a one-valent node and with the property that adjacent edges of $\Gamma$ have different labels. Then the other end point of the path must be labeled with $O$ and it is either a one-valent node of $\Upsilon$ or a multi-valent node with all adjacent edges having the same label; the two figures from the right in Fig. 6.2 illustrate two possible maximal paths.

Without loss of generality, we may assume that the adjacent edge of the starting one-valent node of $\Gamma$ is labeled with $+$ . Denote the closure of the corresponding component of $D^{\prime}\setminus\big{(}D^{\prime}\cap D^{\pm}\big{)}$ by $D_{\Gamma}^{s}$ . Then $\partial D_{\Gamma}^{s}$ bounds a disk $T$ on $\partial({\rm HL}\cup\overline{N(D_{l})})$ . If $T\cap D_{l}^{-}=\emptyset$ , then $D_{\Gamma}^{s}\cap D=\emptyset$ and hence $T\cap D=\emptyset$ by the minimality of $n$ ; however, if it were the case, one could reduce $n_{l}$ by isotopying $D_{l}$ across the $3$ -ball bounded by $D_{\Gamma}^{s}$ and $T$ . Hence $T$ must contain $D_{l}^{-}$ . Since the adjacent edge of the starting node is labeled with $+$ , adjacent edges of the end node of $\Gamma$ in $\Upsilon$ are labeled with $-$ . Denote by $D_{\Gamma}^{e}$ the closure of the component corresponding to the end node. Then $\partial D_{\Gamma}^{e}$ bounds a disk in $\partial({\rm HL}\cup\overline{N(D_{l})})$ that is contained in $T$ and has no intersection with $D_{l}^{+}$ . Particularly, $D_{\Gamma}^{e}\cap D=\emptyset$ by the minimality of $n$ , and there is an arc in $\partial D_{\Gamma}^{e}$ cutting a disk $D^{\prime\prime}$ off $\overline{T\setminus D_{l}^{-}}$ with $\mathring{D}^{\prime\prime}\cap D^{\prime}=D^{\prime\prime}\cap D=\emptyset$ , so one can slide $N(D_{l})$ over $D^{\prime\prime}$ (Fig. 6.3) to decrease $n_{l}$ , a contradiction.

Consequently, such a path $\Gamma$ cannot exist, but this can happen only if $\Upsilon$ is empty. The claim is thus proved. It implies that ${\rm HL}_{1},{\rm HL}_{1}^{\prime}$ are trivial solid tori in $\mathbb{S}^{3}$ , and ${\rm HL}_{2}$ , ${\rm HL}_{2}^{\prime}$ are equivalent to $\overline{N(D_{l})}\cup{\rm HL}$ . ∎

6.2. Non-split, reducible handlebody links

Table 5. Reducible links with up to six crossings.

crossings	$c(L_{1})$ + $c(L_{2})$	description	$\|L_{1}$ – $L_{2}\|$
2 (1)	0 + 2	unknot – Hopf	1
4 (4)	0 + 4	unknot – L4a1	1
		unknot – Hopf#Hopf	2
	2 + 2	Hopf – Hopf	1
5 (4)	0 + 5	unknot – Whitehead	1
		unknot – Trefoil#Hopf	2
	3 + 2	trefoil – Hopf	1
6 (17)	0 + 6	unknot – L6a $i$ , $i=1,\dots,5$	1
		unknot – L6n1	1
		unknot – L4a1#Hopf	3
		unknot – (Hopf#Hopf)#Hopf	4
	2 + 4	Hopf – L4a1	1
		Hopf – Hopf#Hopf	2
	4 + 2	K4a1 – Hopf	1

Table 5 lists all non-split, reducible $(n,1)$ -handlebody links obtained by performing order- $1$ connected sum on pairs of links $(L_{1},L_{2})$ with crossing numbers $(c_{1},c_{2})$ and $c_{1}+c_{2}\leq 6$ . Since $n>1$ , one of $L_{1}$ , $L_{2}$ , say $L_{2}$ , is a link with more than one component. The number in parentheses indicates the total number of inequivalent reducible handlebody links of the given crossing number. By Theorem 6.1, isotopy types of $L_{1}$ and $L_{2}$ with selected components determine the isotopy type of the resulting handlebody link $L_{1}$ – $L_{2}$ . Thus there are no duplicates in Table 5.

On the other hand, by Lemmas 4.1 and 4.3 and Theorem 3.7, minimal diagrams of non-split, reducible $(n,1)$ -handlebody links up to $6$ crossings cannot have $k$ -connectivity, $k>1$ . This shows the completeness of Table 5.

In particular, every non-split, reducible $(n,1)$ -handlebody link in Table 5 admits a minimal diagram with $1$ -connectivity; thus we postulate the following conjecture.

Conjecture 6.2.

Every non-split, reducible handlebody link admits a minimal diagram with $1$ -connectivity.

Not every minimal diagram of a reducible handlebody link has $1$ -connectivity. By Theorem 6.1, Conjecture 6.2 implies the additivity of the crossing number (Conjecture 6.3), a reminiscence of a one-hundred years old problem in knot theory.

Conjecture 6.3.

If $({\rm HL}_{1},h_{1})\text{-{}-}({\rm HL}_{2},h_{2})$ is a $(n,1)$ -handlebody link, then

c\big{(}({\rm HL}_{1},h_{1})\text{-{}-}({\rm HL}_{2},h_{2})\big{)}=c({\rm HL}_{1})+c({\rm HL}_{2}).

(6.1)

Acknowledgements

The first author benefits from the support of the GNAMPA (Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni) of INdAM (Istituto Nazionale di Alta Matematica). The second author benefits from the support of the Swiss National Science Foundation Professorship grant PP00P2_179110/1. The fourth author is supported by National Center of Theoretical Sciences.

References

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[4] M. Brown, Locally Flat Imbeddings of Topological Manifolds Ann. Math., Second Series, 75, (1962), 331–341.
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[6] I. A. Grushko, On the bases of a free product of groups, Matematicheskii Sbornik, 8 (1940), 169–182.
[7] M. W. Hirsch, B. Mazur, Smoothings of Piecewise Linear Manifolds, (AM-80), Princeton University Press, Princeton, NJ, (1974).
[8] J. Hoste, M. Thistlethwaite, J. Weeks The first 1,701,936 knots, Math. Intelligencer 20 (1998), 33–48.
[9] A. Ishii, Moves and invariants for knotted handlebodies, Algebr. Geom. Topol. 8 (2008), 1403–1418.
[10] A. Ishii, K. Kishimoto, H. Moriuchi, M. Suzuki, A table of genus two handlebody-knots up to six crossings, J. Knot Theory Ramifications 21 (2012).
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[14] Y. Koda, M. Ozawa, with an appendix by C. Gordon, Essential surfaces of non-negative Euler characteristic in genus two handlebody exteriors, Trans. Amer. Math. Soc. 367 (2015), no. 4, 2875–2904.
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Appendix A Output of the code

A.1. Minimal diagrams from the code

The software code used in the paper exhaustively enumerates $3$ -edge-connected plane graphs with two trivalent vertices and $q$ quadrivalent vertices, $0<q\leq 6$ , without double arcs that form a non-bigon. Note that the trivial theta curve is the only $3$ -edge-connected plane graph without quadrivalent vertices. The output of the code is examined and summarized in Table 3, while the detailed list is available on http://dmf.unicatt.it/paolini/handlebodylinks/, where each plane graph is described by its adjacent matrix together with a fixed ordering (clockwise or counterclockwise) of the edges adjacent to every vertex, as determined by the planar embedding.

A.1.1. Four crossings or less

In Table 6, we analyze the output of the code up to $4$ quadrivalent vertices, where the column “quad. v.” lists the number of quadrivalent vertices and “ref. no.” the reference number of each plane graph in the output of the code. The column “induced diagrams” describes minimality of diagrams induced by each plane graph. Most induced diagrams are not minimal, and we record those that are and their isotopy types as special graphs or handlebody links, up to mirror image. Up to $4$ -crossings, no IH-minimal diagram with more than one component is found.

Table 6. Diagrams with up to

4

crossings.

quad. v.	ref. no.	induced diagrams
1	none	none
2	#1	R-minimal; G $2_{1}$ in Table 4; not IH-minimal
3	#1,	not R-minimal
3	#2,#3	R-minimal; G $3_{2}$ in Table 4; not IH-minimal
4	#1,#2	IH-minimal; G $4_{1}$ in Table 4
	#3	R-minimal; G $3_{2}$ in Table 4; not IH-minimal
	#4, #8	R-minimal; G $4_{2}$ in Table 4; not IH-minimal
	#5, #6, #7	R-minimal; G $4_{3}$ in Table 4; not IH-minimal
	#9	R-minimal; G $4_{4}$ in Table 4; not IH-minimal
	#10	R-minimal; G $4_{5}$ in Table 4; not IH-minimal

A.1.2. Five and six crossing cases

Table 7. Diagrams with

5

crossings.

ref. no.	description
#6, #11, #14	not R-minimal
#22, #26, #35	not IH-minimal
#36	not IH-minimal
#37	not R-minimal

In the $5$ crossings case, the code finds $8$ plane graphs with more than one components, out of a total of $37$ planar embeddings. Table 7 records the analysis for their induced diagrams; none of them gives IH-minimal diagrams. In the $6$ crossing case, out of $181$ plane graphs, $37$ induces diagrams with more than one components. Table 8 records the minimality of their induced diagrams.

Table 8. Diagrams with

6

crossings.

ref. no.	description
#5	$6_{1}$ in Table 1
#15, #22, #34, #45, #54	not R-minimal
#56	$6_{1}$ in Table 1
#60	$6_{2}$ in Table 1
#70	$6_{3}$ in Table 1
#73	not R-minimal
#83	$6_{2}$ in Table 1
#84	$6_{1}$ in Table 1
#86, #91, #92, #93	not R-minimal
#104, #105, #114, #117, #123	not IH-minimal
#134, #135, #137, #144	not IH-minimal
#161, #165	$6_{1}$ in Table 1
#168, #169, #170,#171	not IH-minimal
#175	$6_{9}$ in Table 1
#176	not IH-minimal
#177	not R-minimal
#179, #180	not IH-minimal
#181	$6_{9}$ in Table 1

Fig. A.1 exemplifies how the analysis is done. Fig. 1(a) shows how the diagrams induced by Plane Graph #5 are equivalent to those by #161 and #165 in the case of 6 crossings, and Fig. 1(b) explains non-minimality of diagrams induced by Plane Graphs #168, #169, #170, #171.

A.1.3. Inequivalent planar embeddings

As a side remark, Fig. A.2 illustrates two examples of abstract graphs with inequivant planar embeddings: one with five quadrivalent vertices and the other with six. Note that the abstract graphs have $2$ -vertex-connectivity, consistent with the Whitney uniqueness theorem [25].

A table of nn-component handlebody links of genus n+1n+1 up to six crossings

Abstract.

1. Introduction

Theorem 1.1.

Theorem 1.2.

Theorem 1.3.

2. Preliminaries

2.1. Handlebody links and spatial graphs

Definition 2.1 (Embeddings in 𝕊3\mathbb{S}^{3}).

Definition 2.2 (Equivalence).

Lemma 2.1.

Proof.

2.2. Diagrams

Definition 2.3 (Regular projection).

Definition 2.4 (Diagram of a spatial graph).

Definition 2.5 (Diagram of a handlebody link).

Definition 2.6 (Crossing numbers).

Definition 2.7 (Minimal diagram).

2.3. Moves

Definition 2.8 (Moves).

Theorem 2.2 ([12, Theorem 2.12.1], [26]).

Theorem 2.3 ([9, Corollary 22]).

2.4. Non-split, irreducible handlebody links

Definition 2.9 (Edge connectivity of a graph).

Definition 2.10 (Connectivity of a diagram).

Definition 2.11 (Split handlebody link).

Definition 2.12 (Reducible handlebody link).

Definition 2.13 (Knot sum).

3. Uniqueness, non-splittability, and irreducibility

Theorem 3.1 (Uniqueness).

Proof.

Remark 3.1.

Remark 3.2.

Theorem 3.2 (Unsplittability).

Proof.

Lemma 3.3.

Lemma 3.4.

Lemma 3.5.

Corollary 3.6 (Irreducibility test).

Theorem 3.7 (Irreducibility).

Proof.

Remark 3.3.

4. Completeness

4.1. Minimal diagrams with 3-connectivity

Lemma 4.1.

Lemma 4.2.

4.2. Minimal diagrams with 2-connectivity

Lemma 4.3.

Corollary 4.4.

5. Chirality

5.1. Decomposable links

Definition 5.1 (Order-2 connected sum).

Conjecture 5.1.

Lemma 5.2.

Proof.

Theorem 5.3.

Proof.

5.2. Chirality

Lemma 5.4.

Proof.

Lemma 5.5.

Proof.

6. Reducible handlebody links

6.1. Order-1 connected sum

Definition 6.1 (Order-11 connected sum).

Theorem 6.1 (Uniqueness).

Proof.

6.2. Non-split, reducible handlebody links

Conjecture 6.2.

Conjecture 6.3.

Acknowledgements

References

Appendix A Output of the code

A.1. Minimal diagrams from the code

A.1.1. Four crossings or less

A.1.2. Five and six crossing cases

A.1.3. Inequivalent planar embeddings

A table of $n$ -component handlebody links of genus $n+1$ up to six crossings

Definition 2.1 (Embeddings in $\mathbb{S}^{3}$ ).

Theorem 2.2 ([12, Theorem $2.1$ ], [26]).

Theorem 2.3 ([9, Corollary $2$ ]).

Definition 6.1 (Order- $1$ connected sum).