Kauffman type invariants for tied links

Abstract

Abstract. We define two new invariants for tied links. One of them can be thought as an extension of the Kauffman polynomial and the other one as an extension of the Jones polynomial which is constructed via a bracket polynomial for tied links. These invariants are more powerful than both the Kauffman and the bracket polynomials when evaluated on classical links. Further, the extension of the Kauffman polynomial is more powerful of the Homflypt polynomial, as well as of certain new invariants introduced recently. Also we propose a new algebra which plays in the case of tied links the same role as the BMW algebra for the Kauffman polynomial in the classical case. Moreover, we prove that the Markov trace on this new algebra can be recovered from the extension of the Kauffman polynomial defined here.

New skein invariants of links

“…[38,39]. See also [3]. (3) The categorification of the new skein invariants is another interesting problem and, for the invariant θ(q, E), which generalizes the Jones polynomial and is a specialization of Θ(q, z, E) [27], it is the object of the ongoing research project [8].…”

“…(3) The categorification of the new skein invariants is another interesting problem and, for the invariant θ(q, E), which generalizes the Jones polynomial and is a specialization of Θ(q, z, E) [27], it is the object of the ongoing research project [8]. (4) The invariants in the present paper can be formulated in terms of the invariants of tied links, using the methods of Aicardi and Juyumaya [2,3]. This will be investigated in a subsequent paper.…”

Truncations and extensions of the Brauer-Chen algebra

“…Here we prove that the same phenomenon occurs for the Brauer-Chen algebra. In particular, there is a natural KZ-type connection on Br(W ) ⋉ L that 'covers' in some sense these two different constructions, and which should be related, when W = S n , with the tied-BWM algebra introduced by Aicardi and Juyumaya in [1]. In the framework of links invariants, this tied-BMW algebra supports the Markov trace responsible for the Kauffman and HOMFLY as well as their 'tied' variants.…”

“…Then the quotient Br Q (W, L) of Br 0 (W, L) by the relations e H Q H (e H ) = 0 has finite rank over k.The existence of the flat connection of Theorem 2.6 raises the following question : Question 2.8. When W = S n and L is the lattice of all reflection subgroups of W , does this monodromy representation of the braid group over Br(W, L) provide the braid group representations factoring through the tied-BMW algebra of Aicardi and Juyumaya (see[1]) ?…”

Brauer and Jones tied monoids

“…In [4] is defined a Kauffman type invariant for tied links that is more powerful than the Kauffman polynomial when restricting to classical links, and also a tied BMW algebra denoted by tBMW. This algebra is defined by a presentation that consists of four types of generators: braid generators, tangle generators, tie generators and tied tangle generators.…”