Knot Floer homology of satellite knots with (1,1)-patterns

Abstract

For pattern knots admitting genus-one bordered Heegaard diagrams, we show the knot Floer chain complexes of the corresponding satellite knots can be computed using immersed curves. This, in particular, gives a convenient way to compute the τ -invariant. For patterns P obtained from two-bridge links b(p, q), we derive a formula for the τ -invariant of P (T 2,3 ) and P (−T 2,3 ) in terms of (p, q), and use this formula to study whether such patterns induce homomorphisms on the concordance group, providing a glimpse at a conjecture due to Hedden.

An infinite-rank summand from iterated Mazur pattern satellite knots

“…We show that a 3), we have a 3 −1 or a 3 = 1. However, a 3 = −1 by Lemma 3.5 (1). Therefore, we have a 3 = 1.…”

“…Let C R denote the reduced R-complex obtained by edge-reduction of CF DA(S 1 × D 2 \nb(M )) ⊠ CF D(C n−1 ), and let A denote the R-complex corresponding to C R can be computed in terms immersed curves by an approach given in [1]: First represent CF D(C n−1 ) as an immersed curve on the punctured torus using the algorithm given in [9], and denote this curve by α. Then let (Σ, β, α a 1 , α a 2 , w, z) be a genus-one doubly pointed bordered Heegaard diagram for the Mazur pattern.…”

“…We move to determine the absolute gradings. By symmetry of the knot Floer homology group, one can see the Alexander grading of c is 0 (see Page 31 of [1]). This fact together with the relative Alexander grading determine the absolute Alexander grading.…”

“…However, the bimodule of S 1 ×D 2 −M is quite big and hence makes this approach very involved; see Theorem 3.4 of [16] for the bimodule of S 1 × D 2 − M . Instead, we partially compute the hat-version knot Floer chain complex of M n (D 2 ) using the immersed-curve techniques developed in [1], and use it to constrain possible local equivalence classes for CF K R (M n (D 2 )), and finally, we determine the local equivalence classes using obstructions for knot Floer chain complexes to be realized by knots. This in turn determines the ϕ i -invariants.…”

“…To see this, note the only intersection point of Alexander grading 2 and Maslov grading 0 is a 1 14 . The minimal subcomplex of C R which contain differentials initiating from a 1 14 is shown in Figure 6. Claim (4) can be seen after a filtered change of basis.…”