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Gabrovšek, Boštjan; Mroczkowski, Maciej

doi:10.1016/j.topol.2014.07.003

Cited by 21 publications

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“…We define the product t k 1 t k 2 · · · t ks , s ∈ N, as the links t k i placed consecutively along U as illustrated in Figure 7 Theorem 4 (Turaev [32]). S3(T ) is a free R-module generated by 14]). S3(L(p, 1)) is a free R-module generated by…”

Section: The Kauffman Bracket Skein Modulementioning

confidence: 99%

Knot Invariants in Lens Spaces

Gabrovšek

Horvat

2019

Knots, Low-Dimensional Topology and Applications

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In this survey we summarize results regarding the Kauffman bracket, HOMFLYPT, Kauffman 2-variable and Dubrovnik skein modules, and the Alexander polynomial of links in lens spaces, which we represent as mixed link diagrams. These invariants generalize the corresponding knot polynomials in the classical case. We compare the invariants by means of the ability to distinguish between some difficult cases of knots with certain symmetries.

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Section: The Kauffman Bracket Skein Modulementioning

confidence: 99%

Knot Invariants in Lens Spaces

Gabrovšek

Horvat

2019

Knots, Low-Dimensional Topology and Applications

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“…We equip these diagrams with an additional isotopy move SL p,q also known as the slide move [16,6] or in some literature as the band move [21,22,19,7]. This move arises from the gluing map h p,q and is presented in Figure 5.…”

Section: ←→mentioning

confidence: 99%

“…In contrast to the KBSM , see for example [35,16,31,27,25,26], the HSM is not a widely studied knot invariant. Uncoincidentally, the HSM is a much stronger invariant and is much more difficult to compute, see for example [35,14,18,6]. The HSM has been calculated for the solid torus using diagrammatic methods [25] as well as algebraic methods [35,14,18,8], for S 1 × S 2 [17,25], and recently for the family of lens spaces of type L(p, 1) [6], see also [9].…”

Section: ←→mentioning

confidence: 99%

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Tabulation of Prime Knots in Lens Spaces

Gabrovšek

2017

Mediterr. J. Math.

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Using computational techniques we tabulate prime knots up to five crossings in the solid torus and the infinite family of lens spaces L(p, q). For these knots we calculate the second and third skein module and establish which prime knots in the solid torus are amphichiral. Most knots are distinguished by the skein modules. For the handful of cases where the skein modules fail to detect inequivalent knots, we calculate and compare the hyperbolic structures of the knot complements. We were unable to resolve a handful of 5-crossing cases for p ≥ 13.

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“…So, in the last decades, the research focused on the study of links, i.e. closed 1-submanifold, embedded in these manifolds: the techniques of representation developed for lens spaces produced different notions of diagrams for the links inside them, establishing a connection with the widespread theory of links in S 3 and determining the possibility of extending a lot of classical invariants (see for example [2,7,10,11,14,16,18,19,23]).…”

Section: Introductionmentioning

confidence: 99%

Diffeomorphic vs Isotopic Links in Lens Spaces

Cattabriga

Manfredi

2018

Mediterr. J. Math.

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Links in lens spaces may be defined to be equivalent by ambient isotopy or by diffeomorphism of pairs. In the first case, for all the combinatorial representations of links, there is a set of Reidemeistertype moves on diagrams connecting isotopy equivalent links. In this paper we provide a set of moves on disk, band and grid diagrams that connects diffeo equivalent links: there are up to four isotopy equivalent links in each diffeo equivalence class. Moreover, we investigate how the diffeo equivalence relates to the lift of the link in the 3-sphere: in the particular case of oriented primitive-homologous knots, the lift completely determines the knot class in L(p, q) up to diffeo equivalence, and thus only four possible knots up to isotopy equivalence can have the same lift. Mathematics Subject Classification 2010: Primary 57M27, 57M10; Secondary 57M25.the same family, but in a general lens space: using results of [4,27] and the diffeo moves, we completely characterize, up to isotopy, which knots among the family have the same lift, in terms of the parameters of the lens space and of the amphicheirality of the lift (see Theorem 8).Further development In [13] a tabulations of knots, represented via band diagrams and up to isotopy equivalence is given. Diffeo-moves can be used to detect diffeo equivalent knots in the tabulation. For example, the two knots 5 26 and 5 27 , which are non-isotopic in L(p, q) with p ≤ 12 and conjecturally non-isotopic in the other lens spaces, are related by a τ move.Another interesting development could be to study the diffeo moves in more general manifolds such Seifert one, for which isotopy equivalence moves are already known (see [15]).

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The HOMFLYPT skein module of the lens spacesLp,1

Abstract: We compute the HOMFLYPT skein module of the lens spaces L p,1 and present a free basis of this module for each p.

Cited by 21 publications

References 7 publications

Knot Invariants in Lens Spaces

Knot Invariants in Lens Spaces

Tabulation of Prime Knots in Lens Spaces

Diffeomorphic vs Isotopic Links in Lens Spaces

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