The Kauffman bracket skein module of the complement of (2,2p + 1)-torus knots via braids

Diamantis, Ioannis

doi:10.1016/j.topol.2023.108433

Cited by 1 publication

(6 citation statements)

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“…In this way we obtain a closed formula for the torsion part of the module and our result agrees with that of Hoste and Przytycki [20]. The diagrammatic method via braids has been successfully applied in [5] for the case of the Kauffman bracket skein module of the complement of (2, 2p + 1)-torus knots and in [1] for the case of the lens spaces L(p, q), p = 0. The importance of the braid approach lies in the fact that it can shed light to the problem of computing (various) skein modules of arbitrary c.c.o.…”

Section: Introduction and Overviewsupporting

confidence: 84%

“…It is worth mentioning that S 1 × S 2 is the first 3-manifold, where torsion on its Kauffman bracket skein module is detected via the 'braid technique'. Finally, we compute KBSM(S 1 × S 2 ) via a different diagrammatic method based on braids following [5,3]. In this way we obtain a closed formula for the torsion part of the module and our result agrees with that of Hoste and Przytycki [20].…”

Section: Introduction and Overviewsupporting

confidence: 73%

“…It was then extended for the computation of Kauffman bracket skein modules of 3-manifolds via the generalized Temperley-Lieb algebra of type B (see [1]). The diagrammatic method for computing Kauffman bracket skein modules has been successfully implemented for the case of the handlebody of genus two ( [3]), for the complement of (2, 2p + 1)-torus knots ( [5]) and for the case of the lens spaces L(p, q) ([1]). It is also worth mentioning that these techniques have been applied toward the computation of various skein modules of different families of 'knotted objects' in (various) 3-manifolds.…”

Section: Discussionmentioning

confidence: 99%

“…Unoriented braids were defined in [1] as standard braids by ignoring the natural top-to-bottom orientation (for an illustration see Figure 16). These tools seem promising in computing Kauffman bracket skein modules of arbitrary 3-manifolds (see for example [3,5]).…”

Section: A Diagrammatic Approach Via Braidsmentioning

confidence: 99%

“…For more details the interested reader should refer to the technical lemmas [11,Lemmas 3,4,5,6,9 & 11].…”

Section: A New Basis For Hom(st)mentioning

confidence: 99%

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The Kauffman bracket skein module of the lens spaces via unoriented braids

Diamantis

2022

Commun. Contemp. Math.

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In this paper, we develop a braid theoretic approach for computing the Kauffman bracket skein module of the lens spaces [Formula: see text], KBSM([Formula: see text]), for [Formula: see text]. For doing this, we introduce a new concept, that of an unoriented braid. Unoriented braids are obtained from standard braids by ignoring the natural top-to-bottom orientation of the strands. We first define the generalized Temperley–Lieb algebra of type B, [Formula: see text], which is related to the knot theory of the solid torus ST, and we obtain the universal Kauffman bracket-type invariant, [Formula: see text], for knots and links in ST, via a unique Markov trace constructed on [Formula: see text]. The universal invariant [Formula: see text] is equivalent to the KBSM(ST). For passing now to the KBSM([Formula: see text]), we impose on [Formula: see text] relations coming from the band moves (or slide moves), that is, moves that reflect isotopy in [Formula: see text] but not in ST, and which reflect the surgery description of [Formula: see text], obtaining thus, an infinite system of equations. By construction, solving this infinite system of equations is equivalent to computing KBSM([Formula: see text]). We first present the solution for the case [Formula: see text], which corresponds to obtaining a new basis, [Formula: see text], for KBSM([Formula: see text]) with [Formula: see text] elements. We note that the basis [Formula: see text] is different from the one obtained by Hoste and Przytycki. For dealing with the complexity of the infinite system for the case [Formula: see text], we first show how the new basis [Formula: see text] of KBSM([Formula: see text]) can be obtained using a diagrammatic approach based on unoriented braids, and we finally extend our result to the case [Formula: see text]. The advantage of the braid theoretic approach that we propose for computing skein modules of c.c.o. 3-manifolds, is that the use of braids provides more control on the isotopies of knots and links in the manifolds, and much of the diagrammatic complexity is absorbed into the proofs of the algebraic statements.

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Section: Introduction and Overviewsupporting

confidence: 84%