Rational tangle replacements and knot Floer homology

Abstract

From the link Floer complex of a link $K$, we extract a lower bound $t_q^{\prime }(K)$ 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, $t_q(K\#K^{\prime })=\max \{t_q(K),t_q(K^{\prime })\}$ and $t_q^{\prime }(K\#K^{\prime })=\max \{t_q^{\prime }(K),t_q^{\prime }(K^{\prime })\}$. For the torus knot $K=T_{p,pk+1}$ we compute $t_q^{\prime }(K)=\lfloor p/2\rfloor$ and $t_q(K)=p-1$.

17 pages, 6 figures