Tied Pseudo Links & Pseudo Knotoids

Abstract

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Pseudo links and singular links in the Solid Torus

“…The braiding algorithm described above can be also applied for the case of tied pseudo links in ST, that is, pseudo links in ST equipped with some non-embedded arcs called ties. The reader is referred to [8] for a treatment of tied pseudo links in S 3 and to [9] for the case of tied links in various 3-manifolds. Note that tied links in S 3 were introduced in [2] and that in [19] the author generalized the notion of tied links in ST.…”

“…From now on and throughout the paper, by pseudo links we shall mean both pseudo links and pseudo knots. Moreover, it is worth mentioning that in [8], the author introduces the L-moves for pseudo links and formulates and proves a sharpened version of the analogue of the Markov theorem for pseudo links, as well as an alternative proof of the analogue of the Alexander theorem for pseudo links in S 3 . In this paper we extend these results for pseudo links in the solid torus ST. Our aim is the construction of pseudo knot invariants in S 3 and ST, and toward that end we present the pseudo bracket polynomial for pseudo links in ST and discuss HOMFLYPT-type invariants for pseudo links in S 3 and ST via appropriate Hecke type algebras, following [25,28,33].…”

“…The paper is organized as follows: In § 1 we recall results on pseudo links, on singular links and on knots in the solid torus from [6,23,8,27,29]. In § 2 we present a braiding algorithm for pseudo and singular links in ST and we introduce the mixed pseudo braid monoid and the mixed singular braid monoid.…”

“…We now recall the notion of L-moves for pseudo knots and results from [8,27]. L-moves make up an important tool for braid equivalence in any topological setting and they allow us to formulate sharpened versions of the analogues of the Markov theorems.…”

“…As explained in [8], L-moves on pseudo knots can be used in order to obtain the analogue of the Alexander theorem for pseudo knots following the braiding algorithm in [4], [1] and [30]. The main idea of this algorithm is to keep the arcs of the oriented links diagram that go downwards with respect to the height function unaffected and replace arcs that go upwards with braid strands.…”