# Longitudinal Mapping Knot Invariant for SU(2)

## TL;DR

This paper introduces a new knot invariant called the longitudinal mapping invariant for SU(2), generalizing previous polynomial invariants and quandle cocycle invariants, with explicit calculations for specific knots.

## Contribution

It extends the knot coloring polynomial to infinite pointed groups and provides a topological interpretation, broadening the scope of knot invariants.

## Key findings

- Invariant evaluated for torus knots T(2,n)
- Invariant computed for the figure eight knot
- Provides a new perspective on quandle colorings

## Abstract

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 $SU(2)$.