A transverse knot invariant from Z2-equivariant Heegaard Floer cohomology

Abstract

We define an invariant of based transverse links, as a well-defined element inside the equivariant Heegaard Floer cohomology of its branched double cover, defined by Lipschitz, Hendricks, and Sarkar. We prove the naturality and functoriality of equivariant Heegaard Floer cohomology for branched double covers of $S^3$ along based knots, and then prove that our transverse link invariant $c_{{\mathbb{Z}}_2}({\xi }_K)$ is an well-defined element which is always nonvanishing and functorial under certain classes of symplectic cobordisms, and describe its behavior under negative stabilization. It follows that we can use properties of $c_{{\mathbb{Z}}_2}({\xi }_K)$ to give a condition on transverse knots K which implies the vanishing/nonvanishing of the contact class $c({\xi }_K)$.

43 pages, 33 figures; new section on vanishing/nonvanishing conditions of contact classes added. comments welcome!

References in corpus (1)

• A contact invariant in sutured monopole homology

Cited by in corpus (2)

• Rank inequalities for the Heegaard Floer homology of branched covers
• An odd analog of Plamenevskaya's invariant of transverse knots