Intersection formulas for parities on virtual knots

2021

Preprint

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Crossings of knot diagrams can be divided into classes (tribes) compatible with Reidemeister moves. Tribes can be considered as localization of the notion of weak chord index introduced by M. Xu. In the article we describe tribes of crossings for tangles in a fixed surface and show that crossings are differentiated by their component, order and homotopy types. As a consequence, we conclude that there are no nontrivial indices on diagrams of classical knots.

Section: Tribes and Indicesmentioning

confidence: 99%

“…In [21,22] we localized the notion of index and defined tribes as classes of crossings compatible in some sense with the Reidemeister moves (see Definition 2). It follows from the definition that crossings which belong to one tribe have the same value for any (weak chord) index.…”

Section: Introductionmentioning

confidence: 99%

Crossing tribes of tangles in a thickened surface

2021

Preprint

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“…A series of signed indices (in fact, oriented parities) on virtual knot diagrams was constructed in [39].…”

Section: A6 Derived Paritiesmentioning

confidence: 99%

“…Remark A.4. The reduced stable parity on based matrices in [38] is obtained by applying the intersection formula [39] to an index which can be deduced from the index σbm (v).…”

Section: A Examples Of Indices A1 Basic Indicesmentioning

confidence: 99%

“…For virtual knots, this idea was excellently realized by V.O. Manturov who created the parity theory [31] and then applied it to strengthen knot invariants, to prove minimality theorem and to construct (counter)examples [32,33,34,35], see also [17,18,19,36,39]. Despite all achievements of the parity theory in the theory of virtual knots, it has one flow: the parity theory gives nothing to classical knots [18,Corollary 4.3].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Crossing indices, traits and the principle of indistinguishability

2021

Preprint

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A (weak chord) index is a function on the crossings of knot diagrams such that: 1) the index of a crossing does not change under Reidemeister moves; 2) crossings which can be paired by a second Reidemeister move have the same index. We show that one can omit the second condition in the case of the universal index. As a consequence, we get the following principle of indistinguishability for classical knots: crossings of the same sign in a classical knot diagram can not be distinguished by any inherent property.

Local transformations and functorial maps

2023

Preprint

Picture-valued invariants are the main achievement of parity theory by V.O. Manturov. In the paper we give a general description of such invariants which can be assigned to a parity (in general, a trait) on diagram crossings. We distinguish two types of picture-valued invariants: derivations (Turaev bracker, index polynomial etc.) and functorial maps (Kauffman bracket, parity bracket, parity projection etc.). We consider some examples of binary functorial maps.Besides known cases of functorial maps, we present two new examples. The order functorial map is closely connected with (pre)orderings of surface groups and leads to the notion of sibling knots, i.e. knots such that any diagram of one knot can be transformed to a diagram of the other by crossing switching. The other is the lifting map which is inverse to forgetting of under-overcrossings information which turns virtual knots into flat knots. We give some examples of liftable flat knots and flattable virtual ones.An appendix of the paper contains description of some smoothing skein modules. In particular, we show that ∆-equivalence of tangles in a fixed surface is classified by the extended homotopy index polynomial.