Date

2014

Department or Program

Mathematics

Additional Department or Program (if any)

Mathematics

Title of Approved Individual Major

Mathematics

Primary Wellesley Thesis Advisor

Ismar Volic

Abstract

A central question in knot theory is the classification of knots. Given two knots, how can we determine if they are different or the same? To answer this question, we develop and study knot invariants which are properties of knots that remain unchanged under isotopy. Khovanov homology is a powerful knot invariant that is able to distinguish many knots. However, because it is constructed in a combinatorial and algebraic manner, Khovanov homology lacks any geometric or topological motivation. Since Khovanov homology encodes information about topological objects, it would be ideal if it could be interpreted from a topological perspective. One way to approach this is through the lens of manifold calculus of functors and more specifically, the Taylor tower for spaces of long knots. Recent developments have shown that the Jones polynomial, another knot invariant, is encoded in the Taylor tower for knots. Since the Jones polynomial can be extracted from Khovanov homology, it is natural to ask if the Taylor tower can provide a space level realization of Khovanov homology. This paper provides an introduction to Khovanov homology and calculus of functors and offers conjectures that relate the two notions.

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