On slice knots and patterns in knot homology

Abstract

In this thesis, we consider two distinct research problems in knot theory: how knot invariants detect knot sliceness (Chapter 1) and how hyperbolic and homological measures of knot complexity are related (Chapter 2). In Chapter 1, we review a number of the most widely used knot invariants and determine their ability to detect slice knots.

In Chapter 2, we establish three new conjectures focused on the hyperbolic volume of the knot complement and knot cohomology. Our computational experiments support these findings based on the data of 12-17-crossing knots. The first conjecture states that there exists a constant a ∈ R such that

log r(K) < a · Vol(K)

for all knots K where r(K) is the total rank of knot Floer homology of K and Vol(K) is the hyperbolic volume of K.

The second conjecture states that if we fix a small cut-off value d of the total rank of knot Floer homology and let f(x) be defined as the fraction of knots whose total rank of knot Floer homology is less than d among the knots whose hyperbolic volume is less than x, then for sufficiently large crossing numbers, the following inequality must hold

f(x) < L/(1 + exp(−k · (x − x_0))) + b

where L, x_0, k, b are constants.

Further, our computational results support the following conjecture relating the knot determinant and the hyperbolic volume: there exist constants a, b ∈ R such that

log det(K) < a · Vol(K) + b

for all knots K where det(K) is the knot determinant of K and Vol(K) is the hyperbolic volume of K.