Longitudinal Mapping Knot Invariant for SU(2)

The knot coloring polynomial defined by Eisermann for a finite pointed group is generalized to an infinite pointed group as the longitudinal mapping invariant of a knot. In turn this can be thought of as a generalization of the quandle 2-cocycle invariant for finite quandles. If the group is a topol...

Full description

Saved in:
Bibliographic Details
Published in:arXiv.org 2018-02
Main Authors: Clark, W Edwin, Saito, Masahico
Format: Article
Language:English
Subjects:
Coloring
Invariants
Knots
Longitude
Mapping
Polynomials
Topology
Toruses
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary:The knot coloring polynomial defined by Eisermann for a finite pointed group is generalized to an infinite pointed group as the longitudinal mapping invariant of a knot. In turn this can be thought of as a generalization of the quandle 2-cocycle invariant for finite quandles. If the group is a topological group then this invariant can be thought of a topological generalization of the 2-cocycle invariant. The longitudinal mapping invariant is based on a meridian-longitude pair in the knot group. We also give an interpretation of the invariant in terms of quandle colorings of a 1-tangle for generalized Alexander quandles without use of a meridian-longitude pair in the knot group. The invariant values are concretely evaluated for the torus knots \(T(2,n)\), their mirror images, and the figure eight knot for the group \(SU(2)\).
ISSN:2331-8422