Genus two mutant knots with the same dimension in knot Floer and Khovanov homologies

We exhibit an infinite family of knots with isomorphic knot Heegaard Floer homology. Each knot in this infinite family admits a nontrivial genus two mutant which shares the same total dimension in both knot Floer homology and Khovanov homology. Each knot is distinguished from its genus two mutant by both knot Floer homology and Khovanov homology as bigraded groups. Additionally, for both knot Heegaard Floer homology and Khovanov homology, the genus two mutation interchanges the groups in $\delta$-gradings $k$ and $-k$.Comment: Information about $\delta$-graded homology has been changed along with statement of Theorem 1 and Table 1. Significant changes to Section

Surgery on links of linking number zero and the Heegaard Floer dd-invariant

We study Heegaard Floer homology and various related invariants (such as the $h$-function) for two-component L-space links with linking number zero. For such links, we explicitly describe the relationship between the $h$-function, the Sato-Levine invariant and the Casson invariant. We give a formula for the Heegaard Floer $d$-invariants of integral surgeries on two-component L-space links of linking number zero in terms of the $h$-function, generalizing a formula of Ni and Wu. As a consequence, for such links with unknotted components, we characterize L-space surgery slopes in terms of the $\nu^{+}$-invariants of the knots obtained from blowing down the components. We give a proof of a skein inequality for the $d$-invariants of $+1$ surgeries along linking number zero links that differ by a crossing change. We also describe bounds on the smooth four-genus of links in terms of the $h$-function, expanding on previous work of the second author, and use these bounds to calculate the four-genus in several examples of links.Comment: This version accepted for publication in Quantum Topolog

Determinants of Simple Theta Curves and Symmetric Graphs

A theta curve is a spatial embedding of the $\theta$-graph in the three-sphere, taken up to ambient isotopy. We define the determinant of a theta curve as an integer-valued invariant arising from the first homology of its Klein cover. When a theta curve is simple, containing a constituent unknot, we prove that the determinant of the theta curve is the product of the determinants of the constituent knots. Our proofs are combinatorial, relying on Kirchhoff's Matrix Tree Theorem and spanning tree enumeration results for symmetric, signed, planar graphs.Comment: 12 pages, 4 figures, 1 tabl

L-space knots have no essential Conway spheres

We prove that L-space knots do not have essential Conway spheres with the technology of peculiar modules, a Floer theoretic invariant for tangles.Comment: 22 pages, 11 color figures created with PSTricks and TikZ. Comments welcome