The Chord Index, its Definitions, Applications, and Generalizations

Cheng, Zhongtao

doi:10.4153/s0008414x20000061

Cited by 9 publications

(7 citation statements)

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“…The notion of index was axiomatized by Z. Cheng [6] in the spirit of Manturov's parity axioms [31]. M. Xu [43] found the general conditions for an index to define an invariant polynomial and introduced the notion a weak chord index.…”

Section: Indicesmentioning

confidence: 99%

“…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%

“…After the paper of A. Henrich [15], the index appeared in works of several authors [3,20,28] and others. Z. Cheng [6] gave an axiomatic description of the index. M. Xu [43] found the most general formulation of index which he called weak chord index.…”

Section: Introductionmentioning

confidence: 99%

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Crossing indices, traits and the principle of indistinguishability

Nikonov¹

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.

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Section: Indicesmentioning

confidence: 99%

Section: Indicesmentioning

confidence: 99%

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Crossing indices, traits and the principle of indistinguishability

Nikonov¹

2021

Preprint

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“…Other types of indices can be found in [7,13,3,16,5]. The notion of index was axiomatized by Z. Cheng [4], and finally, M. Xu [25] found the general conditions for an index to define an invariant polynomial.…”

Section: Introductionmentioning

confidence: 99%

Crossing tribes of tangles in a thickened surface

Nikonov¹

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.

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“…Folwaczny [11], J. Kim [27], M.J. Jeong [21], H. Gao, M. Xu [12], N. Petit [42], K. Kaur, M. Prabhakar, A. Vesnin [26] and others. The first axiomatization of index properties is due to Z. Cheng [5]. M. Xu [44] found the general conditions for an index to define an invariant polynomial and defined weak chord index.…”

Section: Introductionmentioning

confidence: 99%