Winding homology of knotoids

Kutluay, Deniz

doi:10.48550/arxiv.2002.07871

Cited by 3 publications

(4 citation statements)

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“…Intrinsic invariants of knotoids, as well as invariants induced from classical and virtual knot invariants have been studied by many researchers. See [3][4][5][14][15][16]19,20,25,26,29,36,39]. Turaev showed that knotoids in S 2 are in one-to-one correspondence with simple Θ-curves and multi-knotoids in S 2 , immersions of the unit interval and a finite number of circles in S 2 , are in one-to-one correspondence with simple theta-links [39].…”

Section: Introductionmentioning

confidence: 99%

Invariants of Multi-linkoids

Gabrovšek

Gügümcü

2023

Mediterr. J. Math.

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Section: Introductionmentioning

confidence: 99%

Invariants of Multi-linkoids

Gabrovšek

Gügümcü

2023

Mediterr. J. Math.

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“…For example, mentioned above Turaev's extension of the Kauffman bracket polynomial has an additional indeterminate counting intersections of an arc connecting the endpoints with the rest part of a diagram. In [5] The work is supported by the Russian Science Foundation under grant 19-11-00151.…”

Section: Introductionmentioning

confidence: 99%

“…An additional grading in winding homology corresponds with additional indeterminate in the extended bracket polynomial. These improvements makes the polynomial and winding homology more informative than direct analogues of the Kauffman bracket polynomial and Khovanov homology which are also studied in [8] and [5].…”

Section: Introductionmentioning

confidence: 99%

Homological Casson type invariant of knotoids

Tarkaev¹

2020

Preprint

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We consider an analogue of well-known Casson knot invariant for knotoids. We start with a direct analogue of the classical construction which gives two different integer-valued knotoid invariants and then focus on its homology extension. Value of the extension is a formal sum of subgroups of the first homology group H 1 (Σ) where Σ is an oriented surface with (maybe) non-empty boundary in which knotoid diagrams lie. To make the extension informative for spherical knotoids it is sufficient to transform an initial knotoid diagram in S 2 into a knotoid diagram in the annulus by removing small disks around its endpoints. As an application of the invariants we prove two theorems: a sharp lower bound of the crossing number of a knotoid (the estimate differs from its prototype for classical knots proved by M. Polyak and O. Viro in 2001) and a sufficient condition for a knotoid in S 2 to be a proper knotoid (or pure knotoid with respect to Turaev's terminology). Finally we give a table containing values of our invariants computed for all spherical prime proper knotoids having diagrams with at most 5 crossings.

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“…The height is the minimum of the number of the intersections over all representative diagrams and all such an arcs disjoint from crossings (see Section 2 for precise definition). Turaev in [13] obtained a lower bound for the height of a knotoid via the extended bracket polynomial which is a purely knotoid generalization of the Kauffman bracket polynomial (see also [11] where corresponding Khovanovtype invariant is constructed). In [5] some known polynomial invariants of virtual knots are extended to the case of classical and virtual knotoids, and, in particular, it is shown that these invariants give a lower bounds for the height of a knotoid.…”

Section: Introductionmentioning

confidence: 99%