An Alternative Basis for the Kauffman Bracket Skein Module of the Solid Torus via Braids

Diamantis, Ioannis

doi:10.1007/978-3-030-16031-9_16

Cited by 10 publications

(17 citation statements)

References 8 publications

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“…It is well known that the Temperley-Lieb algebras are related to the Kauffman bracket polynomial (see [32]). In [10], and with the use of the Tempereley-Lieb algebra of type B, an alternative basis for the Kauffman bracket skein module of the solid torus is presented. It would be interesting to construct Kauffman-type invariants for pseudo links in ST using a generalized pseudo Temperley-Lieb algebra, and then extend this invariant to and invariant for pseudo links in the lens spaces L(p, q).…”

Section: Remarkmentioning

confidence: 99%

Pseudo links and singular links in the Solid Torus

Diamantis

2021

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In this paper we introduce and study the theories of pseudo links and singular links in the Solid Torus, ST. Pseudo links are links with some missing crossing information that naturally generalize the notion of knot diagrams, and that have potential use in molecular biology, while singular links are links that contain a finite number of self-intersections. We consider pseudo links and singular links in ST and we set up the appropriate topological theory in order to construct invariants for these types of links in ST. In particular, we formulate and prove the analogue of the Alexander theorem for pseudo links and for singular links in ST. We then introduce the mixed pseudo braid monoid and the mixed singular braid monoid, with the use of which, we formulate and prove the analogue of the Markov theorem for pseudo links and for singular links in ST.Moreover, we introduce the pseudo Hecke algebra of type A, P Hn, the cyclotomic and generalized pseudo Hecke algebras of type B, P H1,n, and discuss how the pseudo braid monoid (cor. the mixed pseudo braid monoid) can be represented by P Hn (cor. by P H1,n). This is the first step toward the construction of HOMFLYPT-type invariants for pseudo links in S 3 and in ST. We also introduce the cyclotomic and generalized singular Hecke algebras of type B, SH1,n, and we present two sets that we conjecture that they form linear bases for SH1,n. Finally, we generalize the bracket polynomial for pseudo links in ST.

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Section: Remarkmentioning

confidence: 99%

Pseudo links and singular links in the Solid Torus

Diamantis

2021

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“…On the level of braids the basis B H 2 is more natural since it consists of elements with no moving crossings (see Figure 15). As explained in [D,DL1,DL2,DL3,DL4], the braid band moves that reflect isotopy in a closed-connected-oriented (c.c.o.) 3-manifold M obtained by H 2 by surgery, are naturally described with the use of the new basis B H 2 .…”

Section: Discussionmentioning

confidence: 93%

“…In this paper we present two new bases for KBSM(H 2 ) with the use of braids and techniques developed in [LR1,LR2,La1,OL,DL2,D]. More precisely, we start from the Przytycki-basis B H 2 of KBSM(H 2 ) and using Alexander's theorem in H 2 and the looping generators illustrated in Figure 6 (see also Definition 2) we present elements in B H 2 to open braid form.…”

Section: Introduction and Overviewmentioning

confidence: 99%

The Kauffman bracket skein module of the handlebody of genus 2 via braids

Diamantis¹

2019

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In this paper we present two new bases, B ′ H 2 and BH 2 , for the Kauffman bracket skein module of the handlebody of genus 2 H2, KBSM(H2). We start from the well-known Przytycki-basis of KBSM(H2), BH 2 , and using the technique of parting we present elements in BH 2 in open braid form. We define an ordering relation on an augmented set L consisting of monomials of all different "loopings" in H2, that contains the sets BH 2 , B ′ H 2 and BH 2 as proper subsets. Using the Kauffman bracket skein relation we relate BH 2 to the sets B ′ H 2 and BH 2 via a lower triangular infinite matrix with invertible elements in the diagonal. The basis B ′ H 2 is an intermediate step in order to reach at elements in BH 2 that have no crossings on the level of braids, and in that sense, BH 2 is a more natural basis of KBSM(H2). Moreover, this basis is appropriate in order to compute Kauffman bracket skein modules of c.c.o. 3-manifolds M that are obtained from H2 by surgery, since isotopy moves in M are naturally described by elements in BH 2 .

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“…It is worth mentioning that in [5], a different basis for the classical Kauffman bracket skein module of ST, elements of which are presented in Figure 25 in terms of knotoids on T. Note that in this setting, the bold unknot represents the complementary ST in S 3 .…”

Section: Figure 24 the Basis Of Kbsm(t)mentioning

confidence: 99%

“…The computation of skein modules is a difficult task in general. The braid approach using mixed braids, mixed braid groups and appropriate knot algebras has allowed us to compute the Kauffman bracket skein module of the Solid Torus in [5] and of the handlebody of genus 2 in [6]. Moreover, for the case of HOMFLYPT skein modules, which are even more difficult to compute, the braid approach has been successfully applied so far for the case of the Solid Torus in [7], and significant steps toward the computation of the HOMFLYPT skein module of the lens spaces L(p, 1) have been done in [8,9,11,4].…”

Section: Figure 25 a Different Basis Of Kbsm(t)mentioning

confidence: 99%

Knotoids, pseudo knotoids, Braidoids and pseudo braidoids on the Torus

Diamantis

2021

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In this paper we study the theory of knotoids and braidoids and the theory of pseudo knotoids and pseudo braidoids on the torus T. In particular, we introduce the notion of mixed knotoids in S 2 , that generalize the notion of mixed links in S 3 , and we present an isotopy theorem for mixed knotoids. We then generalize the Kauffman bracket polynomial, <; >, for mixed knotoids and we present a state sum formula for <; >. We also introduce the notion of mixed pseudo knotoids, that is, multi-knotoids on two components with some missing crossing information. More precisely, we present an isotopy theorem for mixed pseudo knotoids and we extend the Kauffman bracket polynomial for pseudo mixed knotoids. Finally, we introduce the theories of mixed braidoids and mixed pseudo braidoids as counterpart theories of mixed knotoids and mixed pseudo knotoids respectively. With the use of the L-moves, that we also introduce here for mixed braidoid equivalence, we formulate and prove the analogue of the Alexander and the Markov theorems for mixed knotoids. We also formulate and prove the analogue of the Alexander theorem for mixed pseudo knotoids.

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An Alternative Basis for the Kauffman Bracket Skein Module of the Solid Torus via Braids

Cited by 10 publications

References 8 publications

Pseudo links and singular links in the Solid Torus

Pseudo links and singular links in the Solid Torus

The Kauffman bracket skein module of the handlebody of genus 2 via braids

Knotoids, pseudo knotoids, Braidoids and pseudo braidoids on the Torus

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