Twisting, mutation and knot Floer homology
This article concerns the interesting problem of detecting knot mutations by means of knot invariants such as knot polynomials and homology theories. While many knot invariants do not detect mutations, the bigraded knot Floer homology groups can detect Conway mutation and genus $2$ mutation. The author shows that the knot Floer homology $\widehat{HFK}(\mathcal{L}_{n})$ stabilizes as $n$ goes to infinity, where $\mathcal{L}_{n}$ is a knot obtained from a given knot $\mathcal{L}$ with a fixed positive crossing by replacing this crossing with $n$ positive twists. This stabilization phenomenon categorifies a similar phenomenon of the Alexander polynomial. The author uses this property to construct an infinite family of prime, positive mutant knots with isomorphic bigraded knot Floer homology groups. This family is obtained from the Kinoshita-Terasaka knot ($11n42$) and the Conway knot ($11n34$), which are positive mutants, by adding full twists to these knots just outside the mutation sphere.
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