
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 . We approach this problem using a surgery obstruction, first developed by Greene, which combines Donaldson's Diagonalization Theorem with the -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 arises by surgery on , 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 arises by non-integer surgery on if and only if has an alternating diagram which can be obtained by rational tangle replacement on an almost-alternating diagram of the unknot. When one considers…
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsGeometric and Algebraic Topology · Homotopy and Cohomology in Algebraic Topology · Advanced Combinatorial Mathematics
