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Khovanov homology and the unknotting number


Khovanov homology and the unknotting number

Marino, Laura (2020) Khovanov homology and the unknotting number. Masters thesis, Concordia University.

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The aim of this thesis is to describe a new lower bound λ for the unknotting number.
The unknotting number u is a classical knot invariant, defined as the minimum number of crossing changes that are needed in order to turn a knot into the Unknot (where a knot is the image of a smooth embedding S^1 → R^3, and the Unknot is the "unknotted" knot). We
have that u is hard to compute, thus one of the goals of knot theory is to find lower bounds for it.
Among the tools that have recently been used to describe lower bounds for u there is Khovanov homology. It is a link invariant, constructed from algebraic structures called Frobenius systems in the following way: given a link diagram D, we associate a cube of (1+1)-cobordisms to it. Then every Frobenius system F of rank two generates a functor, called TQFT, that associates a chain complex C_F (D) to this cube. Khovanov homology is the homology of C_F (D).
Thus different Frobenius systems F generate different homology theories H_F . Among Frobenius systems, F_Univ is particularly interesting because H_F_Univ determines every other Khovanov homology H_F .
Alishahi and Dowlin (2017) defined two lower bounds λ_BN and λ_Lee for the unknotting number using the Khovanov homology theories relative to Frobenius systems F_BN and F_Lee.
Other than giving a bound for u these bounds have several interesting applications related to the convergence of some spectral sequences and to the Knight Move Conjecture. Using the structures and tools introduced by Alishahi and Dowlin, in this thesis we find a new bound that subsumes λ_BN and λ_Lee, using the Khovanov homology theory relative to F_Univ.

Divisions:Concordia University > Faculty of Arts and Science > Mathematics and Statistics
Item Type:Thesis (Masters)
Authors:Marino, Laura
Institution:Concordia University
Degree Name:M. Sc.
Date:1 September 2020
Thesis Supervisor(s):Iovita, Adrian
ID Code:987469
Deposited By: Laura Marino
Deposited On:25 Nov 2020 16:19
Last Modified:25 Nov 2020 16:19
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