The Braid Approach to the HOMFLYPT Skein Module of the Lens Spaces L(p, 1)

Abstract

Abstract. In this paper we present recent results toward the computation of the HOMFLYPT skein module of the lens spaces L(p, 1), S (L(p, 1)), via braids. Our starting point is the knot theory of the solid torus ST and the Lambropoulou invariant, X, for knots and links in ST, the universal analogue of the HOMFLYPT polynomial in ST. The relation between S (L(p, 1)) and S(ST) is established in [DLP] and it is shown that in order to compute S (L(p, 1)), it suffices to solve an infinite system of equations obtained by performing all possible braid band moves on elements in the basis of S(ST), Λ, presented in [DL2]. The solution of this infinite system of equations is very technical and is the subject of a sequel paper [DL3]. IntroductionSkein modules were independently introduced in 1987 by Przytycki [P] and Turaev [Tu]. They generalize knot polynomials in S 3 to knot polynomials in arbitrary 3-manifolds. The essence is that skein modules are quotients of free modules over ambient isotopy classes of links in 3-manifolds by properly chosen local (skein) relations.Let M be an oriented 3-manifold, R = Z[u ±1 , z ±1 ], L the set of all oriented links in M up to ambient isotopy in M and let S be the submodule of RL generated by the skein expressions For convenience we allow the empty knot, ∅, and add the relation u −1 ∅ − u∅ = zT 1 , where T 1 denotes the trivial knot. Then the HOMFLYPT skein module of M is defined to be:

Tied Pseudo Links & Pseudo Knotoids

“…3-manifolds is currently being done [12]. Finally, it is worth mentioning that this braid approach in defining invariants for knots in various 3-manifolds has been implemented for the construction of HOMFLYPT type invariants for knots in the Solid Torus in [14,34] and work toward the construction of such invariants for knots in lens spaces L(p, 1) has been done in [11,13,15,16,18]. For the case of Kauffman bracket type invariants via braids the reader is referred to [9] for knots in the Solid Torus and in [10] for knots in the genus 2 handlebody.…”

HOMFLYPT skein sub-modules of the lens spaces L(p,1)

“…The reader is now referred to [15,16,[5][6][7]9] for other "symmetric" relations, and also for more details on the techniques applied in order to obtain the infinite change of basis matrix relating the sets Λ and Λ .…”

“…For the braiding "tail" occurring after applying Theorem 7, we apply Theorem 6 again, and this procedure will eventually stop and the result will be a sum of elements in Λ aug of lower order than the initial element. For more details of how this procedure terminates the reader is referred to [5] and [6].…”

“…The paper is organized as follows: In §1 we discuss isotopy & braid equivalence for knots and links in L(p, 1) ( [4,17]) and in §2 we recall the setting and the essential techniques and results from [15][16][17][18][4][5][6][7]9] in order to describe the HOMFLYPT skein module of ST via braids. In particular, we present the generalized Hecke algebra of type B, and through a unique trace defined on this algebra, we present the Lambropoulou invariant for knots and links in ST that captures S(ST).…”

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

“…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. 3-manifolds (see also [13] for the case of the HOMFLYPT skein module of the lens spaces L(p, 1)).…”