Tied links in the solid torus

Flores, Marcelo

doi:10.48550/arxiv.1910.10778

Cited by 2 publications

(2 citation statements)

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“…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.…”

Section: Pseudo Links and Singular Links In Stmentioning

confidence: 99%

Pseudo links and singular links in the Solid Torus

Diamantis

2021

Preprint

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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.

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Section: Pseudo Links and Singular Links In Stmentioning

confidence: 99%

Pseudo links and singular links in the Solid Torus

Diamantis

2021

Preprint

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“…Generalized Ties. Following the same ideas as in [AJ1,F], we first need to get the analogue of Propositions 1, 2 for tied links in M .…”

Section: Tied Links In Mmentioning

confidence: 99%

Tied Links in various topological settings

Diamantis¹

2020

Preprint

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Tied links in S 3 were introduced by Aicardi and Juyumaya as standard links in S 3 equipped with some non-embedded arcs, called ties, joining some components of the link. Tied links in the Solid Torus were then naturally generalized by Flores. In this paper we study this new class of links in other topological settings. More precisely, we study tied links in the lens spaces L(p, 1), in handlebodies of genus g, and in the complement of the g-component unlink. We introduce the tied braid groups T Bg,n by combining the algebraic mixed braid groups defined by Lambropoulou and the tied braid monoid, and we state and prove Alexander's and Markov's theorems for tied links in the 3-manifolds mentioned above. Finally, we emphasize on further steps needed in order to study tied links in knot complements and c.c.o. 3-manifolds, which is the subject of a sequel paper.

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Tied links in the solid torus

Cited by 2 publications

References 18 publications

Pseudo links and singular links in the Solid Torus

Pseudo links and singular links in the Solid Torus

Tied Links in various topological settings

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