A note on knot Floer homology of satellite knots with (1,1)-patterns

TL;DR

This paper establishes inequalities relating the knot Floer homology dimensions of satellite knots with (1,1)-patterns to those of their companion and unknot, providing new insights into their homological properties.

Contribution

It proves inequalities for the dimensions of knot Floer homology of satellite knots with (1,1)-patterns, linking them to the unknot and the companion knot.

Findings

Dimension of K of P(K) is at least that of P(U).

Dimension of K of P(K) is at least that of K.

Results hold for any companion knot K.

Abstract

We prove that if $P$ is a $(1,1)$-pattern knot, the two inequalities $\dim \widehat{HFK} (P(K)) \geqslant \dim \widehat{HFK} (P(U))$ and $\dim \widehat{HFK} (P(K)) \geqslant \dim \widehat{HFK} (K)$ hold for the unknot $U\subset S^3$ and any companion knot $K\subset S^3$.