Peter Feller

On cobordisms between knots, braid index, and the Upsilon-invariant

We use Ozsváth, Stipsicz, and Szabó’s Upsilon-invariant to provide bounds on cobordisms between knots that ‘contain full-twists’. In particular, we recover and generalize a classical consequence of the Morton–Franks–Williams inequality for knots: positive braids that contain a positive full-twist realize the braid index of their closure. We also establish that quasi-positive braids that are sufficiently twisted realize the minimal braid index among all knots that are concordant to their closure. Finally, we provide inductive formulas for the Upsilon-invariant of torus knots and compare it to the Levine–Tristram signature profile.

Gordian adjacency for torus knots

A knot K is called Gordian adjacent to a knot L if there exists an unknotting sequence for L containing K. We provide a sufficient condition for Gordian adjacency of torus knots via the study of knots in the thickened torus. We also completely describe Gordian adjacency for torus knots of index 2 and 3 using Levine-Tristram signatures as obstructions to Gordian adjacency. Finally, Gordian adjacency for torus knots is compared to the notion of adjacency for plane curve singularities.

On the topological 4-genus of torus knots

We prove that the topological locally flat slice genus of large torus knots takes up less than three quarters of the ordinary genus. As an application, we derive the best possible linear estimate of the topological slice genus for torus knots with non-maximal signature invariant.

The degree of the Alexander polynomial is an upper bound for the topological slice genus

We use the famous knot-theoretic consequence of Freedmans disc theorem---knots with trivial Alexander polynomial bound a locally-flat disc in the 4-ball---to prove the following generalization. The degree of the Alexander polynomial of a knot is an upper bound for twice its topological slice genus. We provide examples of knots where this determines the topological slice genus.

On 2-bridge knots with differing smooth and topological slice genera

We give infinitely many examples of 2-bridge knots for which the topological and smooth slice genera differ. The smallest of these is the 12-crossing knot

On the Upsilon invariant and satellite knots

12a255

The signature of positive braids is linearly bounded by their first Betti number

. These also provide the first known examples of alternating knots for which the smooth and topological genera differ.

On classical upper bounds for slice genera

We study the effect of satellite operations on the Upsilon invariant of Ozsváth–Stipsicz–Szabó. We obtain results concerning when a knot and its satellites are independent; for example, we show that the set

Calculating the homology and intersection form of a 4-manifold from a trisection diagram

\{D_{2^i,1}\}_{i=1}^\infty