Signed Heights of Knotoids

A knotoid diagram is an immersion of an interval into $S^{2}$ with only transverse double crossings, together with over/under crossing information. A knotoid is a class of knotoid diagrams up to planar isotopy and the Reidemeister moves performed away from the endpoints. (We will provide a geometric definition as well, in Section 2.6.) Knotoids are always considered to be oriented, that is, the endpoints are labelled as the tail $v_{0}$ and the head $v_{1}$.

A shortcut for a knotoid diagram $K$ is an embedded path from $v_{0}$ to $v_{1}$ that intersects $K$ transversely and does not intersect the crossings. Of course, every knotoid diagram has many shortcuts. The intersections between $K$ and a shortcut are signed; see Figure 1. (The endpoints are not considered to be intersections.)

A pair $(K,a)$, where $a$ is a shortcut for $K$, will be called a shortcut diagram. We can connect any two shortcut diagrams for a knotoid by planar isotopy, the Reidemeister moves away from the shortcut, and the three shortcut moves shown in Figure 2.

For each diagram $K$ and shortcut $a$, the height $h(K,a)$ of the pair is the number of intersection points with $a$. A sign sequence is a finite sequence with values in $\{+,-\}$, and for a shortcut diagram $(K,a)$ we define $\operatorname{Seq}(K,a)$ to be the sign sequence of length $h(K,a)$ expressing the signs of the intersections between $K$ and $a$ in the order they appear when following $K$ from $v_{0}$ to $v_{1}$. The positive (resp. negative) height $h_{\pm}(K,a)$ is the number of appearances of $+$ (resp. $-$) in $\operatorname{Seq}(K,a)$.

These values give rise to natural invariants of knotoids:

There is a simple relationship between the height of a knotoid $k$ and its signed heights.

1.3 reduces questions about the height of a knotoid to questions about its signed heights, which form a height pair $(h_{+},h_{-})$. This will make it easier to compute the heights of some knotoids. The theorem also implies that all minimal attainable sequences for a knotoid are rearrangements of each other. The next theorem provides another restriction on the set of minimal attainable sequences.

For a sign sequence $A$, a left shift move of size $n$ on $A$ is the result of deleting $n$ appearances of $(-,+)$ as a consecutive subsequence and then inserting $n$ copies of $(+,-)$. Similarly, a right shift move deletes copies $(+,-)$ and adds copies of $(-,+)$. The deletions and insertions all happen at the same time. For example, a nontrivial shift move on $(-,-,+,+)$ must be a left shift of size $1$, deleting the second and third entries. The possible results after inserting $(+,-)$ are $(+,-,-,+)$, $(-,+,-,+)$, and $(-,+,+,-)$.

The signed heights of a knotoid can be bounded by some polynomial invariants, in particular the Turaev polynomial $\langle\kern-5.16663pt~{}\langle k\rangle\kern-5.16663pt~{}\rangle_{\circ}$ of [2] and the index polynomial $F_{k}$ of [5]. For a nonzero Laurent polynomial $p(t)\in\mathbb{Z}[t^{\pm 1}]$, we will write $\deg^{+}(p)$ for $\max\{\operatorname{maxdeg}(p),0\}$ and $\deg^{-}(p)$ for $\max\{-\operatorname{mindeg}(p),0\}$. We also set $\deg^{\pm}(0)=0$. For a Laurent polynomial in multiple variables, the signed degree in a specific variable will be denoted by (for example) $\deg_{t}^{\pm}$.

The index polynomial also gives more specific information about attainable sign sequences.

In the theorem above, of course, we treat $+$ terms as $+1$ and $-$ as $-1$.

In Section 2, we give background information on knotoids, including basic operations and the relationship between knotoids and theta-curves. In Section 3, we prove 1.3 and 1.4. Section 4 addresses the behavior of height and sign sequences under knotoid operations. In Section 5, we recall background on $n$-writhes and the Turaev polynomial, and we prove Theorems 1.5 through 1.8. Section 6 contains interesting examples and applications to knotoids of low height.

I would like to thank my advisor, Ian Agol, for his thorough and careful feedback. I am also grateful to Kyle Miller for our very helpful conversations.

The first examples of knotoid invariants are the over- and underpass closures. A shortcut diagram for $k$ gives rise to an oriented knot diagram by incorporating the shortcut into the diagram to create an immersion of a circle into $S^{1}$. By taking the shortcut to pass over (resp. under) the knotoid at each crossing, we obtain a diagram of the overpass (resp. underpass) closure of $k$, denoted $k_{+}$ (resp. $k_{-}$).

Conversely, given a diagram of an oriented knot $\kappa$, and a point on an edge of the diagram, we may obtain a knotoid diagram by deleting an open interval around the chosen point. The resulting knotoid depends only on $\kappa$ and is denoted $\kappa^{\bullet}$. By construction, $\kappa^{\bullet}$ has height $0$ and $(\kappa^{\bullet})_{\pm}=\kappa$. Any knotoid of height $0$ may be obtained in this way, and so we have a natural identification of the set of height-$0$ knotoids with oriented knots. Such knotoids are called knot-type. Knotoids that are not knot-type are proper.

The set of knotoid types has a natural noncommutative product. Given knotoid diagrams $K_{1}$ and $K_{2}$, we may form a diagram $K_{1}K_{2}$ by deleting small open disks around the head of $K_{1}$ and tail of $K_{2}$, then gluing appropriately along the boundaries. The resulting knotoid depends only on the knotoids represented by $K_{1}$ and $K_{2}$. It also makes sense to refer to the product of two shortcut diagrams as another shortcut diagram.

Knotoid multiplication is associative. The trivial knotoid, the one that can be drawn without crossings, is an identity. The over/underpass closure operations and the $\kappa\mapsto\kappa^{\bullet}$ operation are monoid homomorphisms.

A prime knotoid is one that cannot be written as a nontrivial product. Every knotoid has a unique decomposition of the form

where each $k_{i}$ is a proper prime knotoid. A knot-type knotoid is prime if and only if the corresponding knot is prime. Two distinct prime knotoids commute if and only if one or both is knot-type ([2]).

For a knotoid $k$, the reverse $\operatorname{rev}(k)$ is obtained by switching the orientation on a diagram of $K$, that is, swapping the labels of the vertices. The mirror image $\operatorname{mir}(k)$ is obtained from switching the over/under information on each crossing, and the symmetry operation acts by reversing the orientation of the ambient $S^{2}$. Rotation is the composition of symmetry and mirror image reflection. See Figure 4. The basic involutions generate a group isomorphic to $(\mathbb{Z}/2\mathbb{Z})^{3}$.

Given a diagram $K$ of some knotoid $k$ and positive $n$, we may choose a lift of $K$ to the $n$-fold cover of $S^{2}$ branched over $v_{0}$ and $v_{1}$. The result is a new diagram $K/n$ with orientation and over/under information inherited from $K$, and $k/n$ is a well-defined knotoid. See Figure 5. For all $k$, the sequence $k/n$ stabilizes to a knot-type knotoid; we define $k/\infty$ to be the corresponding knot.

A similar construction is studied in [4]: The entire preimage of $K$ under the double cover of $S^{2}$ branched over the endpoints constitutes a diagram of an unoriented knot. The unoriented knot is an invariant and is called the double branched cover of $k$.

A framing of a knotoid $k$ is a class of diagrams of $k$ up to regular isotopy, that is, up to Reidemeister moves I’, II, and III. Two diagrams of $k$ are in the same framing class if and only if they have the same writhe.

Similarly, a shortcut framing for $k$ is a class of shortcut diagrams related by all moves except the Type I shortcut move. Shortcut framings are classified by algebraic intersection number between the main strand $K$ and the shortcut $a$.

The theta graph $\Theta$ is the graph with two vertices, $v_{0}$ and $v_{1}$, and three oriented edges $e_{0}$, $e_{+}$, and $e_{-}$ from $v_{0}$ to $v_{1}$. A theta-curve $\theta$ is an embedding of $\Theta$ into $S^{3}$. Such a curve is simple if (the image of) $e_{+}\cup e_{-}$ is unknotted. A spanning disk $D$ for a simple theta-curve is a choice of embedded disk with boundary $\partial D=e_{+}\cup e_{-}$ such that $D$ intersects $e_{0}$ transversely.

Like knotoids, isotopy classes of simple theta-curves form a monoid: The product of $\theta_{1}$ and $\theta_{2}$ is formed by deleting small balls around $v_{1}$ in $\theta_{1}$ and $v_{0}$ in $\theta_{2}$, and gluing the boundaries so that each edge of $\theta_{1}$ is glued to the edge of $\theta_{2}$ with the same label. The resulting theta-curve is well-defined up to isotopy, because the pure mapping class group of a thrice-punctured sphere is trivial. Similarly, we may multiply isotopy classes of pairs $(\theta,D)$.

The height $h(\theta)$ of $\theta$ is the minimal number of intersections of a spanning disk with $e_{0}$, and the positive and negative heights $h_{\pm}$ are the minimal numbers of intersections of those signs. The sign sequence associated to $(\theta,D)$ is the sequence of signs of the intersections of $e_{0}$ with $D$, in the order along $e_{0}$ from $v_{0}$ to $v_{1}$. Sequences obtained this way are attainable for $\theta$.

There is a natural map $\tau$ from the set of knotoids to the set of isotopy classes of simple theta-curves. Given a shortcut diagram $(K,a)$, for a knotoid $k$, we may form a simple theta-curve by considering the diagram as lying in a neighborhood of $S^{2}\subset S^{3}$, with $K$ lifting to an embedded path $e_{0}$, and adding edges $e_{+}$ and $e_{-}$ over and under $a$. The resulting theta-curve represents $\tau(k)$. Note that, in $\tau(k)$, the isotopy class of $e_{0}\cup e_{\pm}$ is $k_{\pm}$.

The construction above also yields a correspondence between shortcut diagrams for $k$ and spanning disks for $\tau(k)$: For each such diagram $(K,a)$, we may choose $D$ to be the “vertical” disk between $e_{+}$ and $e_{-}$ in the constructed theta-curve $\theta$. Under this construction, the sign sequence of $(\theta,D)$ is the same as that of $(K,a)$. Every isotopy class of spanned embeddings $(\theta,D)$ for $\tau(k)$ may be obtained in this way from a shortcut diagram for $k$. Thus, heights and attainable sign sequences for $k$ correspond directly with heights and attainable sequences for $\tau(k)$. Isotopy classes of spanned embeddings $(\theta,D)$ for $\tau(k)$ correspond to classes of shortcut diagrams of $k$ under planar isotopy, the Reidemeister moves away from $a$, and the Type III shortcut move.

The signed heights of knotoids behave in straightforward ways under the basic knotoid involutions.

More specifically, we can say the following.

4.1, together with 1.3, has implications for unsigned heights of knotoids, such as for rotatable knotoids, which are addressed in [4]. A knotoid $k$ is rotatable if it equals $\operatorname{rot}(k)$.

In this section we relate the set of attainable sign sequences for a product to the attainable sequences of its factors.

Note that as a particular case of the statement above, if $A$ is any attainable sequence for a knotoid $k$, then the result of appending $+$ or $-$ to either end of $A$ is also attainable for $k$, because on any shortcut diagram for $k$ we can perform a Type I shortcut move around either endpoint.

The next theorem is a converse for 4.4.

We will prove 4.5 using a modification of original argument appearing in [2] for 4.7 below, which is an immediate corollary.

We will treat knotoids as simple theta-curves, and use the following lemmas:

For a shortcut diagram $(K,a)$ of a knotoid $k$, there are $n$ lifts of $a$ to a shortcut for $K/n$. The total number of positive/negative intersections of $K/n$ with all such lifts is equal to $h_{\pm}(K,a)$. Of course, that amount must be at least $n$ times the minimal number of positive/negative intersections with each of the $n$ lifts of $a$.

Furthermore, we can obtain attainable sign sequences for $k/n$ from attainable sequences for $k$ in the following way. Given a sign sequence $A$ of length $r$, for each $i\in\{0,\ldots,r\}$ let $p_{A}(i)$ be the sum of the terms of $A$ from indices $1$ to $i$. For $i\in\{1,\ldots,r\}$, let $q_{A}(i)$ be the maximum of $p_{A}(i-1)$ and $p_{A}(i)$. Then for $x\in\mathbb{Z}/n\mathbb{Z}$, let $A^{x}$ be the subsequence of $A$ consisting of only the terms from indices $i$ with $q_{A}(i)\equiv x\pmod{n}$. Given $(K,a)$, we may label the $n$ lifts of $a$ as $a^{1},\ldots,a^{n}$ in such a way that they increment counterclockwise around the lift of $v_{0}$, and the initial direction of $K/n$ is between $a^{n}$ and $a^{1}$. Then the $i$^th intersection of $K$ with $a$ lifts to an intersection of $K/n$ with $a^{q(i)}$, so for each $x$, $\operatorname{Seq}(K/n,a^{x})=\operatorname{Seq}(K,a)^{x}$. This implies the following.

Given a crossing $c$ in a knotoid diagram $K$, there is a unique resolution of $c$ that respects orientation. This resolution creates an oriented diagram with two components, a loop $L$ and an interval $K^{\prime}$ with the same endpoints as $K$. The winding number of $L$ around the twice-punctured sphere is called the intersection index of $c$, denoted $\operatorname{Ind}(c)$. The index is equal to the intersection number of $L$ with any shortcut, or with $K^{\prime}$. Note that the index of a crossing doesn’t depend on any over/under information. If a crossing has index $n$, it will be called an $n$-crossing.

The following are immediate consequences of the definition of $n$-writhe:

The $n$-writhes of a knotoid can be encoded in the coefficients of a polynomial. The index polynomial for $k$ is

This is closely related to its similarly-named predecessor, the affine index polynomial of [3], defined by

where $w(c)$ is $\operatorname{sign}(c)\operatorname{ssgn}(c)\operatorname{Ind}(c)$, and $\operatorname{ssgn}(c)$ is as shown in Figure 9. Note that $w(c)$ differs from $\operatorname{Ind}(c)$ only by sign. The affine index polynomial satisfies $P_{k}(t)=P_{k}(t^{-1})$ for all $k$ ([3]), so it is related to the index polynomial by the formula

The $t^{\pm n}$ coefficient $J_{n}(k)+J_{-n}(k)$ of the affine index polynomial of $k$ equals the natural linking number of consecutive components in a periodic diagram for $k/n$ (see Figure 5).

The degree of the affine index polynomial was shown to be a lower bound for the height of a knotoid in [3]. Because of the relationship in eq. 3, this is equivalent to Proposition 3.12 of [5]. 1.5, together with 1.3, is an improvement on this bound in the case that $\deg^{+}(F_{k})$ and $\deg^{-}(F_{k})$ are both positive.

We now prove Theorems 1.5, 1.8, and 1.7.