Rational tangle replacements and knot Floer homology

TL;DR

This paper introduces new lower bounds for the rational unknotting number of links using knot Floer homology, computes these bounds for certain torus knots, and explores their properties under knot operations.

Contribution

It defines and analyzes lower bounds for the rational unknotting number derived from link Floer complexes, including explicit calculations for specific torus knots.

Findings

Lower bounds for rational unknotting number from Floer homology

Explicit bounds for torus knots T_{p,pk+1}

Behavior of bounds under connected sum operations

Abstract

From the link Floer complex of a link $K$, we extract a lower bound $t_q'(K)$ for the rational unknotting number of $K$ (i.e. the minimum number of rational replacements required to unknot $K$). Moreover, we show that the torsion obstruction $t_q(K)=\hat{t}(K)$ from an earlier paper of Alishahi and the author is a lower bound for the proper rational unknotting number. Moreover, $t_q(K\#K')=\max\{t_q(K),t_q(K')\}$ and $t'_q(K\#K')=\max\{t'_q(K),t'_q(K')\}$. For the torus knot $K=T_{p,pk+1}$ we compute $t'_q(K)=\lfloor p/2\rfloor$ and $t_q(K)=p-1$.