Quandle coloring and cocycle invariants of composite knots and abelian extensions

Quandle colorings and cocycle invariants are studied for composite knots, and applied to chirality and abelian extensions. The square and granny knots, for example, can be distinguished by quandle colorings, so that a trefoil and its mirror can be distinguished by quandle invariants of composite kno...

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Bibliographic Details
Published in:arXiv.org 2016-01
Main Authors: Clark, W Edwin, Saito, M, Vendramin, L
Format: Article
Language:English
Subjects:
Chirality
Coloring
Invariants
Knots
Online Access:Get full text
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Summary:Quandle colorings and cocycle invariants are studied for composite knots, and applied to chirality and abelian extensions. The square and granny knots, for example, can be distinguished by quandle colorings, so that a trefoil and its mirror can be distinguished by quandle invariants of composite knots. We investigate this and related phenomena. Quandle cocycle invariants are studied in relation to the connected sum, and formulas are given for computing the cocycle invariant from the number of colorings of composite knots. Relations to corresponding abelian extensions of quandles are studied, and extensions are examined for the table of small connected quandles, called Rig quandles. Computer calculations are presented, and summaries of outputs are discussed.
ISSN:2331-8422
DOI:10.48550/arxiv.1407.5803