Surgery presentations for knots coloured by metabelian groups

TL;DR

This paper classifies knots with metabelian group colorings up to a specific equivalence, using analogues of Seifert matrices to understand their structure and transformations.

Contribution

It introduces a classification of G-coloured knots for certain metabelian groups using Seifert matrix analogues and a new local move concept.

Findings

Classification of G-coloured knots up to rho-equivalence for specific groups

Development of a Seifert matrix analogue for G-coloured knots

Establishment of a new local move as a G-coloured crossing change

Abstract

A G-coloured knot is a knot together with a representation of its knot group onto G. Two G-coloured knots are said to be rho-equivalent if they are related by surgery around unit framed unknots in the kernels of their colourings. The induced local move is a G-coloured analogue of the crossing change. For certain families of metabelian groups G, we classify G-coloured knots up to rho-equivalence. Our method involves passing to a problem about G-coloured analogues of Seifert matrices.