Representations of Composite Braids and Invariants for Mutant Knots and Links in Chern-Simons Field Theories

TL;DR

This paper investigates the limitations of Chern-Simons knot invariants in distinguishing mutant knots and links, introduces composite braid invariants, and explores their effectiveness in identifying mutant links.

Contribution

It develops a new framework for composite braid invariants using Wess-Zumino conformal field theory, enhancing the understanding of knot invariants in Chern-Simons theory.

Findings

Composite invariants can distinguish some mutant links.

Mutant knots remain indistinguishable by these invariants.

Framework applies broadly to non-abelian gauge groups.

Abstract

We show that any of the new knot invariants obtained from Chern-Simons theory based on an arbitrary non-abelian gauge group do not distinguish isotopically inequivalent mutant knots and links. In an attempt to distinguish these knots and links, we study Murakami (symmetrized version) $r$-strand composite braids. Salient features of the theory of such composite braids are presented. Representations of generators for these braids are obtained by exploiting properties of Hilbert spaces associated with the correlators of Wess-Zumino conformal field theories. The $r$-composite invariants for the knots are given by the sum of elementary Chern-Simons invariants associated with the irreducible representations in the product of $r$ representations (allowed by the fusion rules of the corresponding Wess-Zumino conformal field theory) placed on the $r$ individual strands of the composite braid. On the other hand, composite invariants for links are given by a weighted sum of elementary multicoloured Chern-Simons invariants. Some mutant links can be distinguished through the composite invariants, but mutant knots do not share this property. The results, though developed in detail within the framework of $SU(2)$ Chern-Simons theory are valid for any other non-abelian gauge group.