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

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 $K\subset S^3$.

Figures (9)

• Figure 1: The data contained in a genus-one doubly-pointed bordered Heegaard diagram (on the left) of the Mazur pattern can be equivalently understood as a 5-tuple (on the right). This convention comes from chen2019knot.
• Figure 2: Examples of paring diagrams. The curves $\beta(P)$ and $\alpha(K)$ are drawn in blue and red, respectively.
• Figure 3: The immersed curve for $-T_{2,3}$ in an infinite cylinder (left) and the $\alpha$-curves for $-T_{2,3}$ in a torus (right), where the circled numbers can be used to follow the construction.
• Figure 4: A choice of $\beta_0$ (highlighted in cyan) that corresponds to the Mazur pattern. (With a slight abuse of notation, we use the same symbols for the lifts of the two basepoints, respectively.)
• Figure 5: A choice of $\alpha_0$ (highlighted in magenta) that corresponds to $T_{2,3}$.

Theorems & Definitions (12)

• Conjecture 1.1
• Conjecture 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.7: chen2019knot, Theorem 1.2
• Lemma 2.1: whitney1937regular, Theorem 1; see also geiges2009contact, Theorem 1
• proof : Proof of Theorem \ref{['thm:second']}
• Remark 3.1
• proof : Proof of Theorem \ref{['thm:main']}
• Remark 4.1