Pseudo links and singular links in the Solid Torus

In this subsection we recall results on pseudo links and on singular links in $S^{3}$. In particular, we recall the pseudo braid monoid $PM_{n}$, introduced in [BJW], that is related to the singular braid monoid $SM_{n}$ ([Ba], [Bi]), and we recall all necessary results in order to present the analogues of the Alexander and the Markov theorems for pseudo links and singular links in $S^{3}$. A pseudo diagram of a knot consists of a regular knot diagram with some missing crossing information, that is, there is no information about which strand passes over and which strand passes under the other. These undetermined crossings are called pre-crossings (for an illustration see Figure 1).

As explained in [BJW], pseudo knots are closed related to singular knots, that is, knots that contain a finite number of self-intersections. In particular, there exists a bijection $f$ from the singular knot diagrams to the set of pseudo knot diagrams where singular crossings are mapped to pre-crossings. In that way we may also recover all of the pseudo knot Reidemeister moves, with the exception of the pseudo-Reidemeister I (PR1) move (see Figure 2). Moreover, $f$ induces an onto map from singular knots to pseudo knots, since the image of two isotopic singular knot diagrams are also isotopic pseudo knot diagrams with exactly the same sequence of Reidemeister moves. We now introduce the pseudo braid monoid, $PM_{n}$, following [BJW].

Note that $p_{i}$ corresponds to a standard crossing and not to a singular crossing. We denote a singular crossing by $\tau_{i}$ and we have that if we replace the pre-crossings $p_{i}$ of Definition  2.2 by singular crossings $\tau_{i}$, we obtain the singular braid monoid, $SM_{n}$, defined in [Ba] and [Bi]. Thus, we obtain the following result:

Define now the closure of a pseudo braid (cor. of a singular braid) as in the standard case (for an illustration of the closure of a pseudo braid see Figure 8). By considering $PM_{n}\subset PM_{n+1}$, we can consider the inductive limit $PM_{\infty}$. Using the analogue of the Alexander theorem for singular knots ([Bi]), in [BJW] the analogue of the Alexander theorem for (oriented) pseudo links is presented. In particular:

We now recall the notion of $L$-moves for pseudo knots and results from [D] and [La]. $L$-moves make up an important tool for braid equivalence in any topological setting and they allow us to formulate sharpened versions of the analogues of the Markov theorems. For more details the reader is referred to [La] and references therein.

$L$-moves for pseudo braids are defined in the same way as in the singular case, that is, the two strands that appear after the performance of an $L$-move should cross the rest of the braid only with real crossings (all over in the case of an $L_{o}$-move or all under in the case of an $L_{u}$-move). For an illustration see Figure 4 for the case of an $L$-move performed on a pseudo braid.

As explained in [D], $L$-moves on pseudo knots can be used in order to obtain the analogue of the Alexander theorem for pseudo knots following the braiding algorithm in [Bi], [A] and [LR1]. The main idea of this algorithm is to keep the arcs of the oriented links diagram that go downwards with respect to the height function unaffected and replace arcs that go upwards with braid strands. We first isotope the pre-crossings in such a way that the braiding algorithm will not affect them (see Figure 10), and we may then apply the braiding algorithm of [Bi] or [LR1] by ignoring the pre-crossings. The same is true for the case of singular links in $S^{3}$ (for a detailed proof for the case of singular links in $S^{3}$ see [PR] Theorem 2.3). In § 3.2 we recall the braiding algorithm from [LR1] and we modify it in order to obtain the analogue of the Alexander theorem for pseudo links and singular links in ST (see also [La2]). We now state the analogue of the Markov theorem for oriented pseudo links and for oriented singular links in $S^{3}$ ([G], [La], [BJW] and [D]).

We now view ST as the complement of a solid torus in $S^{3}$ and we present results from [LR1]. An oriented link $L$ in ST can be represented by an oriented mixed link in $S^{3}$, that is, a link in $S^{3}$ consisting of the unknotted fixed part $\widehat{I}$ representing the complementary solid torus in $S^{3}$ and the moving part $L$ that links with $\widehat{I}$. A mixed link diagram is a diagram $\widehat{I}\cup\widetilde{L}$ of $\widehat{I}\cup L$ on the plane of $\widehat{I}$, where this plane is equipped with the top-to-bottom direction of $I$. For an illustration see Figure 6 by ignoring the pre-crossings. Consider now an isotopy of an oriented link $L$ in ST. As the link moves in ST, its corresponding mixed link will change in $S^{3}$ by a sequence of moves that keep the oriented $\widehat{I}$ pointwise fixed. This sequence of moves consists in isotopy in $S^{3}$ and the mixed Reidemeister moves. In terms of diagrams we have the following result for isotopy in ST: The mixed link equivalence in $S^{3}$ includes the classical Reidemeister moves and the mixed Reidemeister moves, which involve the fixed and the standard part of the mixed link, keeping $\widehat{I}$ pointwise fixed. By the Alexander theorem for knots in solid torus ([La2]), a mixed link diagram $\widehat{I}\cup\widetilde{L}$ of $\widehat{I}\cup L$ may be turned into a mixed braid $I\cup\beta$ with isotopic closure. This is a braid in $S^{3}$ where, without loss of generality, its first strand represents $\widehat{I}$, the fixed part, and the other strands, $\beta$, represent the moving part $L$. The subbraid $\beta$ is called the moving part of $I\cup\beta$. For an illustration see Figure 8 by ignoring the pre-crossings. The sets of braids related to ST form groups, the Artin braid groups type B, denoted $B_{1,n}$, with presentation:

where the generators $\sigma_{i}$ and $t$ are illustrated in Figure 5.

We finally have that isotopy in ST is translated on the level of mixed braids by means of the following theorem.

As explained in § 2.2 we view ST as the complement of another ST in $S^{3}$ and thus, pseudo links in ST can be seen as mixed pseudo links in $S^{3}$ containing the complementary ST. Similarly, singular links in ST can be viewed as mixed singular links in $S^{3}$. For an illustration of a mixed pseudo link see Figure 6, where by changing pre-crossings to singular crossings, one obtains a singular mixed link.

Isotopy for pseudo links in ST is now translated on the level of mixed pseudo links in $S^{3}$ by means of the following theorem:

It is worth mentioning that we do not allow special crossings, i.e. pre-crossings for the case of pseudo links and singular crossings for the case of singular links, between the fixed and the moving part of the mixed pseudo (cor. singular) braid.

We now define the mixed pseudo braids, which are related to mixed pseudo links, and the mixed singular braids, related to the mixed singular links.

Without loss of generality, we assume that the first strand of the mixed pseudo braid represents the complementary solid torus in $S^{3}$. This can be realized by performing the technique of (standard) parting, in order to separate the endpoints of the mixed braids into two different sets, the first represents the complementary solid torus in $S^{3}$ (i.e. the fixed part $I$), and the last $n$ strands represent the moving part of the mixed braid, i.e. the link in ST, and so that the resulting braids have isotopic closures. This can be realized by pulling each pair of corresponding moving strands to the right and over or under the strand of $I$ that lies on their right according to its label. We start from the rightmost pair respecting the position of the endpoints. For more details the reader is referred to [LR1] and [DL1]. We also define the closure of a mixed pseudo (cor. singular) braid as follows:

We now present a braiding algorithm for mixed pseudo links in ST and for mixed singular links in ST. By “mixed links” in the braiding algorithm that follows, we shall mean both mixed pseudo links and mixed singular links, and by special crossings we shall mean pre-crossings or singular crossings. The main idea of the braiding algorithm is to keep the arcs of the oriented mixed link diagrams that go downwards with respect to the height function unaffected, and replace arcs that go upwards with braid strands. These arcs are called up-arcs (see Figure 9).

Note that the braiding algorithm should keep the fixed part of the mixed link unaffected and should also ”take care“ of the special crossings in the mixed diagram which contain at least one up-arc. For this we apply the idea used in [Bi] for the case of singular knots, (see also [KL] for the case of virtual knots). Namely, before we apply the braiding algorithm we have to isotope the mixed link in such a way that the special crossings will only contain down-arcs, so that the braiding algorithm will not affect them. This is achieved by rotating all special crossings that contain at least one up-arc, so that the two arcs are now directed downward. This is illustrated in Figure 10 for the case of pre-crossings. Then we may apply the braiding algorithm of [La2], which is a modification of the braiding algorithm for oriented mixed links presented in [LR1], for the mixed link (ignoring the special crossings). In particular:

• •

  We first ”take care“ of the special crossings by isotoping the mixed link in such a way that the braiding algorithm will not affect them. For an illustration see Figure 10.

  

  We now recall the braiding algorithm of [La2] (for more details the reader is referred to [La2] and references therein).

• •

  Consider now a vertical line $l$ that passes through the maximum and minimum of $\hat{I}$ as illustrated in Figure 11. By small perturbations we can assume that $l$ does not pass through any crossings of the link.

  

• •

  We now apply the following braiding algorithm to the part of the link that lies to the left of the vertical line $l$, keeping $\hat{I}$ fixed:

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    Chose a base-point and we run along the diagram of the link according to its orientation.

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    When/If we run along an opposite arc, we subdivide it into smaller arcs, each containing crossings of one type only as shown in Figure 9.

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    Label now every up-arc with an “o”or a “u”, according to the crossings it contains. If it contains no crossings, then the choice is arbitrary.

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    Perform an $o$-braiding moves on all up-arcs which were labeled with an “o” and $u$-braiding moves on all up-arcs which were labeled with an “u” (see Figure 12).

    

• •

  Close the braided part of the link that lies to the left of $l$ and consider this operation to be enclosed in a tube $T_{1}$ (see left hand side of Figure 13).

• •

  Apply once again the braiding algorithm on the right hand side of $l$ keeping $\hat{I}$ fixed, and we close now this braided portion of the link and enclose the strings participating in this operation in a tube $T_{2}$ (see right hand side of Figure 13).

• •

  Rotate now around the back of the diagram in order to bring the tube $T_{1}$ to the very right of the diagram and $T_{2}$ to the very left of the diagram, and so that the resulting diagram goes around a central point, say $P_{0}$ on $l$.

• •

  Local maxima and minima in the diagram, if any, would lie on the vertical line $l$. We isolate each one in neighborhoods that do not contain other parts of the diagram, and we stretch the arcs above and below a point of symmetry $P_{0}$, so that the extrema would lie on $l$ again but in reverse order. Note also that we numerate the extrema with integers with respect to the point $P_{0}$. For an illustration see right hand side of Figure 13.

  

• •

  Open the braided diagram by cutting through a half-line starting from $P$ and we isotope in ST.

• •

  The result is a mixed pseudo braid (cor. a mixed singular braid) whose closure is isotopic to the initial mixed link.

This braiding algorithm provides a proof of the following theorem:

We now study algebraic structures related to mixed pseudo braids and mixed singular braids. Our aim is to formulate the analogue of the Markov theorem algebraically. We start by defining the mixed pseudo braid monoid $PM_{1,n}$, the counterpart of the Artin’s braid group of type B for pseudo links, and also derive the mixed singular braid monoid. In order to define the mixed pseudo braid monoid $PM_{1,n}$, we need to find the defining relations of this monoid. We analyze the moves of Theorem 3.1 and this leads to the following definition:

Note that every isotopy can be decomposed in a sequence of elementary isotopies which correspond to the relations in Definition 3.7, and that means that we have obtained a complete set of relations. It is also worth mentioning that by replacing the pre-crossings $p_{i}$ in Definition 3.7, we obtain the singular braid monoid of type B, $SM_{1,n}$, presented in Theorem 4.4 [V]. Note also that in [V], the singular braid monoid of type B is called singular braid monoid of the Annulus. Thus, we have the following result:

Moreover, by considering the natural inclusion $PM_{1,n}\subset PM_{1,n+1}$ (see Figure 14), we can define the inductive limit $PM_{1,\infty}$, and similarly, by considering the natural inclusion $SM_{1,n}\subset SM_{1,n+1}$, we may define the inductive limit $SM_{1,\infty}$.

Using the braiding algorithm presented in § 3.2 for pseudo links in $S^{3}$, we have a stronger version of the analogue of the Markov theorem for pseudo knots, similar to Theorem 2 in [La2]. In particular, we have the following:

Since all pseudo links related to the solid torus contain the same braided part $\hat{I}$, we have that corresponding pseudo braids contain the same braided part $I$. Therefore, we obtain the following:

By Corollary 3.10 and the discussion in § 3.3 we obtain the following result:

Note that from the discussion in § 2.1 and Theorem 2.8 it follows that conjugation and stabilization moves can be replaced by the $L$-moves. Hence, we obtained a sharpened version of Theorem 3.11. More precisely, we have the following:

Similarly, by Theorem 3.8, we have the following result for mixed singular braids:

The technique of [PR] can be applied for the case of pseudo knots in $S^{3}$, since, as mentioned before, pseudo knot theory of $S^{3}$ can be realized as the quotient of the theory of singular knots in $S^{3}$, modulo the pseudo-Reidemeister move 1. In that sense, we present here results from [PR] which are adapted accordingly. Recall that the Hecke algebra of type A, $\mathcal{H}_{n}$, has presentation obtained from the classical braid group $B_{n}$ by corresponding the braiding generators $\sigma_{i}$ to $g_{i}$ and by adding the quadratic relations:

That is, $\mathcal{H}_{n}\ =\ \frac{\mathbb{C}[B_{n}]}{\langle g_{i}^{2}-(q-1)g_{i}-q\rangle}$. We define the pseudo Hecke algebra of type A in a similar way. More precisely:

Note that the embedding $PM_{n}\hookrightarrow PM_{n+1}$ induces a homomorphism $\iota:P\mathcal{H}_{n}\to P\mathcal{H}_{n+1}$. Moreover, each $PM_{n}$ has a natural grading with respect to the pre-crossings, namely, $PM_{n}=\bigoplus_{d=0}^{\infty}P_{d}M_{n}$, where $P_{d}M_{n}$ denote the set of mixed pseudo braids with $d$ pre-crossings. This grading induces a grading on $P\mathcal{H}_{n}$, since the quadratic relations affect only the $g_{i}$’s. In particular, we have that:

Recall now the following linear basis for the Hecke algebra of type A [Jo]:

for $1\leq i_{1}<\ldots<i_{p}\leq n-1{\rm\;}$. The basis $S$ yields directly an inductive basis for $\mathcal{H}_{n}$, which is used in the construction of the Ocneanu trace, leading to the HOMFLYPT or $2$-variable Jones polynomial. Similarly, we want to find a basis for $P\mathcal{H}_{n}$. Although $P\mathcal{H}_{n}$ is not of finite dimension, each subspace $P_{d}\mathcal{H}_{n}$ of the graduation is of finite dimension. Indeed we have the following (compare to Proposition 3.1 [PR]):

The proof is similar to that of Proposition 3.1 [PR] by replacing the singular crossings $\tau_{i}$’s by the pre-crossings $p_{i}$’s. In order now to define HOMFLYPT-type invariants for pseudo links in $S^{3}$, one should construct a family of Markov traces on $\{P_{d}\mathcal{H}_{n}\}_{n=1}^{\infty}$ and $P\mathcal{H}_{n}$. Note that the difference between the family of traces we have to define in $\{P_{d}\mathcal{H}_{n}\}_{n=1}^{\infty}$ to that defined in [PR], is the relation that comes from the pseudo-Reidemeister move 1, and which is not allowed in the theory of singular knots. In a sequel paper we shall work toward the construction of such Markov traces. Note that the construction of a Markov trace leads naturally to a HOMFLYPT-type invariant for pseudo links in $S^{3}$. As shown in [PR], one would have to fix a Markov trace on $\{P_{d}\mathcal{H}\}_{n=1}^{\infty}$, and then follow [Jo] in order to define an invariant $X$ for pseudo knots in $S^{3}$.

In this subsection we define the generalized pseudo Hecke algebra of type B and the cyclotomic pseudo Hecke algebras of type B, related to pseudo links in ST, and similarly, we define the cyclotomic and the generalized singular Hecke algebra of type B, related to the theory of singular links in ST, following [La1]. Our aim is to apply similar techniques from [La1], [PR] and [Jo] toward the construction of HOMFLYPT-type invariants for pseudo links and singular links in the Solid Torus, ST, which is the subject of a sequel paper. It has been established that Hecke algebras of type B form a tower of B-type algebras that are related to the knot theory of ST. A presentation for the basic one is obtained from the presentation of the Artin group $B_{1,n}$ by adding the quadratic relation (2) and $t^{2}=\left(Q-1\right)t+Q$, where $q,Q\in{\mathbb{C}}\backslash\{0\}$ are seen as fixed variables. The middle B-type algebras are the cyclotomic Hecke algebras of type B, whose presentations are obtained by the quadratic relation (2) and $t^{d}=(t-u_{1})(t-u_{2})\ldots(t-u_{d})$. The topmost Hecke-like algebra in the tower is the generalized Hecke algebra of type B, $\textrm{H}_{1,n}$, which has the following presentation:

That is:

Note that in $\textrm{H}_{1,n}(q)$ the generator $t$ satisfies no polynomial relation, making the algebra $\textrm{H}_{1,n}(q)$ infinite dimensional. Also that in [La1] the algebra $\textrm{H}_{1,n}(q)$ is denoted as $\textrm{H}_{n}(q,\infty)$. Following [La1] and [PR], we now define the generalized pseudo Hecke algebra of type B and the cyclotomic pseudo Hecke algebras of type B.

As in the case of pseudo Hecke algebras of type A, the embedding $PM_{1,n}\hookrightarrow PM_{1,n+1}$ induces a homomorphism $\iota:P\mathcal{H}_{1,n}\rightarrow P\mathcal{H}_{1,n+1}$, and moreover, $PM_{1,n}$ also has a natural grading with respect to the pre-crossings, i.e. $PM_{1,n}=\bigoplus_{d=0}^{\infty}P_{d}M_{1,n}$, where $P_{d}M_{1,n}$ denotes the set of mixed pseudo braids with $d$ pre-crossings. This grading induces a grading on $P\mathcal{H}_{1,n}$:

Consider now the elements illustrated in Figure 15 and recall the following linear bases for the generalized Hecke algebra of type B [La1]:

where $k_{1},\ldots,k_{r}\in{\mathbb{Z}}$ and $\sigma$ a basic element in ${\rm H}_{n}(q)$.

The basis $\Sigma^{\prime}_{n}$ yields an inductive basis for the generalized Hecke algebra of type B, which is used in the construction of the Ocneanu trace, leading to the HOMFLYPT or $2$-variable Jones polynomial (the reader is referred to [La1] for more details). Similarly, we want to find a basis for $P\mathcal{H}_{1,n}$, with the use of which we will construct HOMFLYPT-type invariants for pseudo links in ST. This is more complicated (compared to the type A case) and further research is required. We conclude this discussion with a conjecture:

In [HD] the pseudo bracket polynomial, $\langle;\rangle$, is defined for pseudo links in $S^{3}$ extending the Kauffman bracket polynomial for classical knots presented in [LK]. Note that the orientation of a diagram in the case of pseudo links is needed in order to define a skein relation on pre-crossings. More precisely, we have the following:

It can be easily seen that the pseudo bracket is invariant under Reidemeister moves 2 and 3 and the pseudo moves PR1, PR2 and PR3 ([HD] Theorem 1). As in [LK], by normalizing the pseudo bracket polynomial using the writhe, we obtain the normalized pseudo bracket polynomial, which is an invariant of pseudo knots in $S^{3}$ ([HD], Corollary 2). In particular:

We now generalize the pseudo bracket polynomial for pseudo links in ST. Note that we now view ST as a punctured disc (for an illustration of the looping generator $t$ in this set up see Figure 16).

We have the following:

An immediate result of Theorem 1 [HD] is the following:

As in the case of pseudo links in $S^{3}$ we obtain the following result: