Localization for Khovanov homologies:
Thesis advisor: Julia Elisenda Grigsby === Thesis advisor: David Treumann === In 2010, Seidel and Smith used their localization framework for Floer homologies to prove a Smith-type rank inequality for the symplectic Khovanov homology of 2-periodic links in the 3-sphere. Hendricks later used similar...
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ndltd-BOSTON-oai-dlib.bc.edu-bc-ir_1084702019-05-24T03:02:08Z Localization for Khovanov homologies: Zhang, Melissa Thesis advisor: Julia Elisenda Grigsby Thesis advisor: David Treumann Text thesis 2019 Boston College English electronic application/pdf In 2010, Seidel and Smith used their localization framework for Floer homologies to prove a Smith-type rank inequality for the symplectic Khovanov homology of 2-periodic links in the 3-sphere. Hendricks later used similar geometric techniques to prove analogous rank inequalities for the knot Floer homology of 2-periodic links. We use combinatorial and space-level techniques to prove analogous Smith-type inequalities for various flavors of Khovanov homology for periodic links in the 3-sphere of any prime periodicity. First, we prove a graded rank inequality for the annular Khovanov homology of 2-periodic links by showing grading obstructions to longer differentials in a localization spectral sequence. We remark that the same method can be extended to p-periodic links. Second, in joint work with Matthew Stoffregen, we construct a Z/p-equivariant stable homotopy type for odd and even, annular and non-annular Khovanov homologies, using Lawson, Lipshitz, and Sarkar's Burnside functor construction of a Khovanov stable homotopy type. Then, we identify the fixed-point sets and apply a version of the classical Smith inequality to obtain spectral sequences and rank inequalities relating the Khovanov homology of a periodic link with the annular Khovanov homology of the quotient link. As a corollary, we recover a rank inequality for Khovanov homology conjectured by Seidel and Smith's work on localization and symplectic Khovanov homology. Khovanov homology Khovanov homotopy type link invariants localization Smith theory Copyright is held by the author, with all rights reserved, unless otherwise noted. Thesis (PhD) — Boston College, 2019. Submitted to: Boston College. Graduate School of Arts and Sciences. Discipline: Mathematics. http://hdl.handle.net/2345/bc-ir:108470 |
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Khovanov homology Khovanov homotopy type link invariants localization Smith theory |
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Khovanov homology Khovanov homotopy type link invariants localization Smith theory Zhang, Melissa Localization for Khovanov homologies: |
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Thesis advisor: Julia Elisenda Grigsby === Thesis advisor: David Treumann === In 2010, Seidel and Smith used their localization framework for Floer homologies to prove a Smith-type rank inequality for the symplectic Khovanov homology of 2-periodic links in the 3-sphere. Hendricks later used similar geometric techniques to prove analogous rank inequalities for the knot Floer homology of 2-periodic links. We use combinatorial and space-level techniques to prove analogous Smith-type inequalities for various flavors of Khovanov homology for periodic links in the 3-sphere of any prime periodicity. First, we prove a graded rank inequality for the annular Khovanov homology of 2-periodic links by showing grading obstructions to longer differentials in a localization spectral sequence. We remark that the same method can be extended to p-periodic links. Second, in joint work with Matthew Stoffregen, we construct a Z/p-equivariant stable homotopy type for odd and even, annular and non-annular Khovanov homologies, using Lawson, Lipshitz, and Sarkar's Burnside functor construction of a Khovanov stable homotopy type. Then, we identify the fixed-point sets and apply a version of the classical Smith inequality to obtain spectral sequences and rank inequalities relating the Khovanov homology of a periodic link with the annular Khovanov homology of the quotient link. As a corollary, we recover a rank inequality for Khovanov homology conjectured by Seidel and Smith's work on localization and symplectic Khovanov homology. === Thesis (PhD) — Boston College, 2019. === Submitted to: Boston College. Graduate School of Arts and Sciences. === Discipline: Mathematics. |
author |
Zhang, Melissa |
author_facet |
Zhang, Melissa |
author_sort |
Zhang, Melissa |
title |
Localization for Khovanov homologies: |
title_short |
Localization for Khovanov homologies: |
title_full |
Localization for Khovanov homologies: |
title_fullStr |
Localization for Khovanov homologies: |
title_full_unstemmed |
Localization for Khovanov homologies: |
title_sort |
localization for khovanov homologies: |
publisher |
Boston College |
publishDate |
2019 |
url |
http://hdl.handle.net/2345/bc-ir:108470 |
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AT zhangmelissa localizationforkhovanovhomologies |
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1719191840619495424 |