Pseudo links and singular links in the Solid Torus

Abstract

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.

Pseudo links in handlebodies

“…In this paper we extend the results of [9] for pseudo links in the genus g handlebody, H g . More precisely, we translate isotopy moves for pseudo links in H g to isotopy moves for mixed links in S 3 and we generalize the pseudo bracket polynomial for pseudo links in H g , by first extending it to the genus two handlebody, H 2 .…”

“…In [9] the theory of pseudo links is generalized for the case of the Solid Torus, ST. More precisely, pseudo links in ST are presented as mixed links in S 3 , and the pseudo bracket polynomial for pseudo links in S 3 is generalized for the case pseudo links in ST. Moreover, the mixed pseudo braid monoid of type B is introduced and studied ( [9]), with the use of which, the analogues of the Alexander and the Markov theorems for pseudo links in ST are obtained. The theory of pseudo links in ST plays an important role for studying pseudo links in a genus g handlebody, H g , since H g is an oriented 3-manifold which is homeomorphic to a connected sum of g-solid tori.…”

“…The paper is organized as follows: In § 1 we recall results on pseudo links in S 3 and in the Solid Torus from [3,19,4,9,22,24]. More precisely, in § 1.1 we recall the definition of pseudo links in S 3 and we present the pseudo bracket polynomial for pseudo links in S 3 .…”

“…1.2. Pseudo links in ST. We now view ST as the complement of a solid torus in S 3 and we present results from [9]. An oriented link L in ST can be viewed as an oriented mixed link in S 3 , that is, a link in S 3 consisting of the unknotted fixed part I that represents the complementary solid torus in S 3 , and the moving part L that links with I.…”

“…By the analogue of the Alexander theorem for pseudo links in the Solid Torus ST ( [9]), a mixed pseudo link diagram I ∪ L of I ∪ L may be turned into a mixed pseudo braid I ∪ β with isotopic closure, where the closure of a mixed pseudo braid is defined as in the case of classical braids in S 3 . Mixed pseudo braids are pseudo braids in S 3 where, without loss of generality, their first strand represents I, the fixed part, and the other strands, β, represent the moving part L. The subbraid β shall be called the moving part of I ∪ β.…”