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The Chord Index, its Definitions, Applications, and Generalizations

Part of: Low-dimensional topology

Published online by Cambridge University Press:  30 January 2020

Zhiyun Cheng
Zhiyun Cheng*
Affiliation:
School of Mathematical Sciences, Laboratory of Mathematics and Complex Systems, Beijing Normal University, Beijing100875, China
*
e-mail: czy@bnu.edu.cn
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Abstract

In this paper, we study the chord index of virtual knots, which can be thought of as an extension of the chord parity. We show how to use the chord index to enhance the quandle coloring invariants. The notion of indexed quandle is introduced, which generalizes the quandle idea. Some applications of this new invariant is discussed. We also study how to define a generalized chord index via a fixed finite biquandle. Finally, the chord index and its applications in twisted knot theory are discussed.

Keywords

virtual knotchord indexwrithe polynomialindexed quandletwisted knot

MSC classification

Primary: 57M25: Knots and links in $S^3$
Secondary: 57M27: Invariants of knots and 3-manifolds
Type
Article
Information
Canadian Journal of Mathematics , Volume 73 , Issue 3 , June 2021 , pp. 597 - 621
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Copyright
© Canadian Mathematical Society 2020

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Footnotes

The author is supported by NSFC 11771042, NSFC 11571038 and China Scholarship Council.

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