Chirality of Knots $9_{42}$ and $10_{71}$ and Chern-Simons Theory

TL;DR

This paper demonstrates that advanced Chern-Simons knot invariants can detect the chirality of specific knots ($9_{42}$ and $10_{71}$) that are indistinguishable by traditional polynomial invariants.

Contribution

It shows that $SU(2)$ Chern-Simons topological field theory invariants can distinguish knot chirality where known polynomials fail.

Findings

Chern-Simons invariants detect chirality of $9_{42}$ and $10_{71}$ knots.

Traditional polynomial invariants are insensitive to these knots' chirality.

The approach extends the capability of knot invariants in topological studies.

Abstract

Upto ten crossing number, there are two knots, $9_{42}$ and $10_{71}$ whose chirality is not detected by any of the known polynomials, namely, Jones invariants and their two variable generalisations, HOMFLY and Kauffman invariants. We show that the generalised knot invariants, obtained through $SU(2)$ Chern-Simons topological field theory, which give the known polynomials as special cases, are indeed sensitive to the chirality of these knots.