Cobordisms of graphs: A sliceness criterion for stably odd free knots and related results on cobordisms

Fedoseev, D. A.; Manturov, Vassily Olegovich

doi:10.1142/s0218216518420117

Cited by 4 publications

(15 citation statements)

References 11 publications

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“…In another direction, Manturov and Fedoseev have produced slice obstructions for free knots [24,10,11]. A free knot is an equivalence class of 4-valent graphs, and a Gauss code representing a virtual knot may be projected to a code representing a free knot by forgetting the signs and directions of its chords.…”

Section: Statement Of Resultsmentioning

confidence: 99%

Computations of the slice genus of virtual knots

Rushworth¹

2019

Topology and its Applications

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This paper subsumes the (now withdrawn) arXiv submission On the virtual Rasmussen invariant.Abstract. A virtual knot is an equivalence class of embeddings of S 1 into thickened (closed oriented) surfaces, up to self-diffeomorphism of the surface and certain handle stabilisations. The slice genus of a virtual knot is defined diagrammatically, in direct analogy to that of a classical knot. However, it may be defined, equivalently, as follows: a representative of a virtual knot is an embedding of S 1 into a thickened surface Σg ×I; what is the minimal genus of oriented surfaces S → M × I with the embedded S 1 as boundary, where M is an oriented 3-manifold with ∂M = Σg?We compute and estimate the slice genus of all virtual knots of 4 classical crossings or less. We also compute or estimate the slice genus of 46 virtual knots of 5 and 6 classical crossings whose slice status is not determined in the work of Boden, Chrisman, and Gaudreau. The computations are made using two distinct virtual extensions of the Rasmussen invariant, one due to Dye, Kaestner, and Kauffman, the other due to the author. Specifically, the computations are made using bounds on the two extensions of the Ramussen invariant which we construct and investigate. The bounds are themselves generalisations of those on the classical Rasmussen invariant due, independently, to Kawamura and Lobb. The bounds allow for the computation of the extensions of the Rasmussen invariant in particular cases. As asides we identify a class of virtual knots for which the two extensions of the Rasmussen invariant agree, and show that the extension due to Dye, Kaestner, and Kauffman is additive with respect to the connect sum.

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Section: Statement Of Resultsmentioning

confidence: 99%

Computations of the slice genus of virtual knots

Rushworth¹

2019

Topology and its Applications

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“…We can assume that the initial state coincide with the configuration considered above. Then by Theorem 6 there is a homomorphism φ n : P B n → G 3 n and we need only describe explicitly the images of the generators of the group P B n .…”

Section: Homomorphism Of Pure Braids Into G 3 Nmentioning

confidence: 99%

“…For any i < j the pure braid b ij can be presented as the following dynamical system: As we check all the situations in the dynamical systems where three points lie on the same line, and write down these situations as letters in a word of the group G 3 n we get exactly the element…”

Section: Homomorphism Of Pure Braids Into G 3 Nmentioning

confidence: 99%

On Groups $G_{n}^{k}$ and $Γ_{n}^{k}$: A Study of Manifolds, Dynamics, and Invariants

Manturov,

Fedoseev,

Kim

et al. 2019

Preprint

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Recently the first named author defined a 2-parametric family of groups G k n [19]. Those groups may be regarded as a certain generalisation of braid groups. Study of the connection between the groups G k n and dynamical systems led to the discovery of the following fundamental principle: "If dynamical systems describing the motion of n particles possess a nice codimension one property governed by exactly k particles, then these dynamical systems admit a topological invariant valued in G k n ". The G k n groups have connections to different algebraic structures, Coxeter groups and Kirillov-Fomin algebras, to name just a few. Study of the G k n groups led to, in particular, the construction of invariants, valued in free products of cyclic groups.Later the first and the fourth named authors introduced and studied the second family of groups, denoted by Γ k n , which are closely related to triangulations of manifolds. The spaces of triangulations of a given manifolds have been widely studied. The celebrated theorem of Pachner [98] says that any two triangulations of a given manifold can be connected by a sequence of bistellar moves or Pachner moves. See also [62,91]; the Γ k n naturally appear when considering the set of triangulations with the fixed number of points.There are two ways of introducing this groups: the geometrical one, which depends on the metric, and the topological one. The second one can be thought of as a "braid group" of the manifold and, by definition, is an invariant of the topological type of manifold; in a similar way, one can construct the smooth version.In the present paper we give a survey of the ideas lying in the foundation of the G k n and Γ k n theories and give an overview of recent results in the study of those groups, manifolds, dynamical systems, knot and braid theories.

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“…In the classical case, for stubborn knots such as 8 18 with vanishing signature, signature function, and Rasmussen invariant, the slice genus can be investigated by other techniques, such as the T -genus and the triple point method [MS84]. It would be interesting to generalize these results to virtual knots, and in their recent paper [FM17], Fedoseev and Manturov define a slice criterion for free knots using triple points of free knot cobordisms. Their work represents a promising development toward realizing this goal.…”

Section: Applicationsmentioning

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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Cobordisms of graphs: A sliceness criterion for stably odd free knots and related results on cobordisms

Cited by 4 publications

References 11 publications

Computations of the slice genus of virtual knots

Computations of the slice genus of virtual knots

On Groups $G_{n}^{k}$ and $Γ_{n}^{k}$: A Study of Manifolds, Dynamics, and Invariants

Concordance group of virtual knots

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