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Title: Z/2-equivariant Heegaard Floer cohomology and transverse knots
Author: Kang, Sungkyung
ISNI:       0000 0004 7654 1938
Awarding Body: University of Oxford
Current Institution: University of Oxford
Date of Award: 2018
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This thesis consists roughly of two parts. In the first part, we study various properties of Z2-equivariant Heegaard Floer cohomology. Then, in the second part, we apply the results of the first part to study transverse knot invariants in the standard contact S3. When (L,p) is a based link in S3, the branched double cover Sigma(L) is a rational homology sphere which carries a natural Z2-action. Hendricks, Lipshitz, and Sarkar studied this action on the Heegaard Floer setting to define HFZ2(Sigma(L), p), the Z2-equivariant Heegaard Floer cohomology of Sigma(L), using bridge diagrams of L drawn on a sphere. They also proved that its isomorphism class as an F2[theta]-module is an isotopy invariant of (L,p), which also implies that we can drop p from its notation when L is a knot. In this thesis, we prove that, when L=K is a knot, then HFZ2(Sigma(K)) satisfies naturality and admits cobordism maps for based knot cobordisms. Then we move on to show that HFZ2(Sigma(K)) can be calculated in terms of bridge diagrams of L, which are now drawn on any weakly admissible Heegaard diagrams representing S3. Based on this observation, we point out that, for knots K, we can consider HFZ2(Sigma(K)) as a weak Heegaard invariant, and prove that it is actually a strong Heegaard invariant. We also prove that the definition of HFZ2(Sigma(K)) in terms of bridge diagrams and its alternative definition as a strong Heegaard diagram are equivalent, by constructing an invertible natural transformation between the two TQFT functors induced by those two definitions. Using that fact, we construct Z2-equivariant knot Floer cohomology HFKZ2(Sigma(K)), and show that it is a refinement of both HFK(-S3,K) and HFZ2(Sigma(K)). We then construct the Z2-equivariant contact class cZ2(xiK) of a transverse knot K inside the standard contact 3-sphere (S3,xistd) std, as a well-defined element of HFZ2(Sigma(K)), and prove that it is a transverse knot invariant which refines the contact class c(xiK) of the contact branched double cover (Sigma(K),xiK) of (S3,xistd) along K. We also construct the Z2-equivariant LOSS invariant TZ2(K) of a transverse knot K in (S3,xistd), as an element of HFKZ2(Sigma(K),K) and prove that it is also a transverse knot invariant which refines both cZ2(xiK) and the LOSS invariant T(K). Lastly, we give an interesting topological application of cZ2(xiK), by showing that it can be used to prove the nonvanishing of c(xiK) when K achieves equality in a Bennequin-type inequality.
Supervisor: Juhasz, Andras Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available