What is it about?
We consider equivalence classes of 3-coloured knots and of 5-coloured knots under colour -preserving Dehn surgery. We show that the equivalence classes are in bijective correspondence with elements of an order 3 and an order 5 cyclic group, correspondingly. The equivalence classes are represented by coloured connect-sums of (3,2) and of (5,2) torus knots.
Featured Image
Why is it important?
The idea is to develop a quantum topology for coloured knots. A coloured knot represents a branched covering space and a link inside it, which can serve as a source for coloured knot invariants. These equivalence classes are the type 0 invariants.
Perspectives
The results were extended to n-coloured knots in joint work with Kricker, and later also to knots coloured by sufficiently "nice" metabelian groups in a preprint. The "coloured untying invariant" was indepedently discovered by Fenn.
Dr Daniel Moskovich
Ben-Gurion University of the Negev
Read the Original
This page is a summary of: Surgery untying of coloured knots, Algebraic & Geometric Topology, May 2006, Mathematical Sciences Publishers,
DOI: 10.2140/agt.2006.6.673.
You can read the full text:
Resources
Contributors
The following have contributed to this page
