Alternating Surgeries

TL;DR

This thesis investigates when the double branched cover of an alternating knot can be obtained via Dehn surgery on S^3, using a surgery obstruction based on lattice structures and Heegaard Floer homology, with applications to unknotting and characterizing slopes.

Contribution

It introduces a new lattice-based obstruction for surgeries producing double branched covers of alternating knots and characterizes unknotting number one alternating knots through diagram analysis.

Findings

Double branched cover arises from non-integer surgery iff rational tangle replacement exists.

Unknotting number one for alternating knots iff an unknotting crossing exists in every alternating diagram.

New bounds on surgeries producing double branched covers of alternating knots.

Abstract

This thesis is concerned with the question of when the double branched cover of an alternating knot can arise by Dehn surgery on a knot in $S^3$. We approach this problem using a surgery obstruction, first developed by Greene, which combines Donaldson's Diagonalization Theorem with the $d$-invariants of Ozsv{\'a}th and Szab{\'o}'s Heegaard Floer homology. This obstruction shows that if the double branched cover of an alternating knot or link $L$ arises by surgery on $S^3$, then for any alternating diagram the lattice associated to the Goeritz matrix takes the form of a changemaker lattice. By analyzing the structure of changemaker lattices, we show that the double branched cover of $L$ arises by non-integer surgery on $S^3$ if and only if $L$ has an alternating diagram which can be obtained by rational tangle replacement on an almost-alternating diagram of the unknot. When one considers half-integer surgery the resulting tangle replacement is simply a crossing change. This allows us to show that an alternating knot has unknotting number one if and only if it has an unknotting crossing in every alternating diagram. These techniques also produce several other interesting results: they have applications to characterizing slopes of torus knots; they produce a new proof for a theorem of Tsukamoto on the structure of almost-alternating diagrams of the unknot; and they provide several bounds on surgeries producing the double branched covers of alternating knots which are direct generalizations of results previously known for lens space surgeries. The thesis begins with an exposition of the changemaker surgery obstruction, giving an amalgamation of results due to Gibbons, Greene and the author. It then gives background material on alternating knots and changemaker lattices. The latter part of the thesis is then taken up with the applications of this theory.