From chord parity to chord index

2021

Preprint

Section: Parity Cycle Induced By a Biquandle 1-cocyclementioning

confidence: 71%

Section: Parity Cycle Induced By a Biquandle 1-cocyclementioning

confidence: 99%

See 1 more Smart Citation

Intersection formulas for parities on virtual knots

2021

Preprint

“…M. Xu [43] found the general conditions for an index to define an invariant polynomial and introduced the notion a weak chord index. See also [6,7,9] for other index axiomatics and examples of chord indices.…”

Section: Indicesmentioning

confidence: 99%

Crossing indices, traits and the principle of indistinguishability

2021

Preprint

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.

“…Using technics of [19], in Section 2 we prove the main theorem of the paper which shows that a tribe of crossings is determined by the component, order and homotopy types. This result allows to give the negative answer to the question [5,6] whether nontrivial indices for classical knots exist (we postpone this conclusion to Corollary 5 in Section 4.4). Section 3 is devoted to description of tribes for flat tangles in the surface.…”

Section: Introductionmentioning

confidence: 96%

Crossing tribes of tangles in a thickened surface

2021

Preprint

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.