Khovanov Homology and Some of Its Applications in Knot Theory

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Geevechi, Amir Masoud | 2016

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Type of Document: M.Sc. Thesis
Language: Farsi
Document No: 48685 (02)
University: Sharif University of Technology
Department: Mathematical Sciences
Advisor(s): Razvan, Mohammad Reza; Eftekhary, Eaman
Abstract:
In this thesis, we study a homological invariant in Knot theory, called Khovanov homology. The main property of this invariants is that it gives us the Jones polynomial, as its graded Euler characteristic. Besides, the functor (1+1) TQFT, from the category of closed one-manifolds to the category of vector spaces is employed in its construction. By making some changes to this functor and defining another functor and some other steps, the so-called Lee spectral sequence is derived which starts from Khovanov homology and converges to another homological invariant of links, called Lee-Khovanov homology. Computation of this homology is very simple. By using this spectral sequence, a numerical invariant of knots, named s-invariant is defined. s-invariant has a good behaviour under the cobordism of knots and provides a lower bound for the four-ball genus. In this manner, a combinatorial proof for the celebrated conjecture of Milnor about the four-ball genus of torus knots is obtained
Keywords:
Khovanov Homology ; Knot Theory ; Khovanov Homology ; Lee Spectral Sequence ; Four-ball Genus ; S-Invariant ; Milnor Conjecture