On Symmetries of Knots and Their Surgeries

Abstract

We investigate relationships between some knot invariants and symmetries of knots. In the first chapter, we recall the definitions of knots, the symmetries we will investigate, and some classical knot invariants including the signature, the genus, and the Alexander polynomial.

In the second chapter we investigate the relation between the knot Floer homology of a periodic knot and the knot Floer homology of its quotient knot. Specifically, we prove a rank inequality between them using a spectral sequence of Hendricks, Lipshitz, and Sarkar. We further conjecture a filtration on this inequality for which we provide evidence and consequences including a signature inequality for alternating periodic knots.

In the third chapter we define Dehn surgery, and discuss covering maps between Dehn surgeries on the same knot. We classify such covers for torus knots, and conjecture some strong restrictions on when such a covering can occur for hyperbolic knots. We check this conjecture for knots with 8 or fewer crossings.

In the final chapter we prove that the quotient of a definite knot is definite.