Weak Parities and Functorial Maps

Nikonov, Igor Mikhailovich

doi:10.1007/s10958-016-2807-0

Cited by 13 publications

(12 citation statements)

References 10 publications

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“…The Gaussian parity is a winding parity with an abelian group Z 2 and the fixed element a = 0. Definition 3.4 ( [11]). An oriented parity p is a family of maps p D : V(D) → G defined for each diagram D of the knot K, that possesses the properties described in Fig.…”

Section: Winding Parity For Knots In S G × Smentioning

confidence: 99%

Knots in $S_{g} \times S^{1}$ and winding parities

Kim¹

2021

Preprint

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A virtual knot, which is one of generalizations of knots in R 3 (or S 3 ), is, roughly speaking, an embedded circle in thickened surface Sg × I. In this talk we will discuss about knots in 3 dimensional Sg × S 1 . We introduce basic notions for knots in Sg × S 1 , for example, diagrams, moves for diagrams and so on. For knots in Sg × S 1 technically we lose over/under information, but we have information "how many times a half of the crossing of the knot in Sg × S 1 rotates along S 1 ", we call it labels of crossings. In this paper we extend this notion more generally and discuss its geometrical meaning. This paper follows from [1].

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Section: Winding Parity For Knots In S G × Smentioning

confidence: 99%

Knots in $S_{g} \times S^{1}$ and winding parities

Kim¹

2021

Preprint

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“…Note that this is the definition of "parity in the weak sense," cf. Manturov [Man13] and Nikonov [Nik16].…”

Section: Preliminariesmentioning

confidence: 99%

Alexander invariants of periodic virtual knots

Bodén

Nicas

White

2018

Dissertationes Math.

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We show that every periodic virtual knot can be realized as the closure of a periodic virtual braid and use this to study the Alexander invariants of periodic virtual knots. If K is a q-periodic and almost classical knot, we show that its quotient knot K * is also almost classical, and in the case q = p r is a prime power, we establish an analogue of Murasugi's congruence relating the Alexander polynomials of K and K * over the integers modulo p. This result is applied to the problem of determining the possible periods of a virtual knot K. One consequence is that if K is an almost classical knot with a nontrivial Alexander polynomial, then it is p-periodic for only finitely many primes p. Combined with parity and Manturov projection, our methods provide conditions that a general virtual knot must satisfy in order to be q-periodic.

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“…In [Man10], this is referred to as "parity in the weak sense". Readers interested in more details on parity are referred to [Man10], [IMN11] and [Nik13].…”

Section: Parity Projection and Concordancementioning

confidence: 99%

Concordance group of virtual knots

Bodén

Nagel²

2017

Proc. Amer. Math. Soc.

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We introduce Tristram-Levine signatures of virtual knots and use them to investigate virtual knot concordance. The signatures are defined first for almost classical knots, which are virtual knots admitting homologically trivial representations. The signatures and ω-signatures are shown to give bounds on the topological slice genus of almost classical knots, and they are applied to address a recent question of Dye, Kaestner, and Kauffman on the virtual slice genus of classical knots. A conjecture on the topological slice genus is formulated and confirmed for all classical knots with up to 11 crossings and for 2150 out of 2175 of the 12 crossing knots.The Seifert pairing is used to define directed Alexander polynomials, which we show satisfy a Fox-Milnor criterion when the almost classical knot is slice. We introduce virtual disk-band surfaces and use them to establish realization theorems for Seifert matrices of almost classical knots. As a consequence, we deduce that any integral polynomial ∆(t) satisfying ∆(1) = 1 occurs as the Alexander polynomial of an almost classical knot.In the last section, we use parity projection and Turaev's coverings of knots to extend the Tristram-Levine signatures to all virtual knots. A key step is a theorem saying that parity projection preserves concordance of virtual knots. This theorem implies that the signatures, ω-signatures, and Fox-Milnor criterion can be lifted to give slice obstructions for all virtual knots. There are 76 almost classical knots with up to six crossings, and we use our invariants to determine the slice status for all of them and the slice genus for all but four.

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Weak Parities and Functorial Maps

Cited by 13 publications

References 10 publications

Knots in $S_{g} \times S^{1}$ and winding parities

Knots in $S_{g} \times S^{1}$ and winding parities

Alexander invariants of periodic virtual knots

Concordance group of virtual knots

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