How to Pick Out the Slicing Degree of Knots Using a Spork

The slicing degree of a knot K is the smallest integer k such that K is k-slice (i.e., bounds a disk with self-intersection number –k) in #n(-CP)^2 for some n. In this talk, we establish bounds on the slicing degrees of knots using Rasmussen’s s-invariant, knot Floer homology, and singular instanton homology.

We also introduce sporks, defined as pairs (W, f) consisting of a contractible 4-manifold W and a boundary diffeomorphism f that extends smoothly inside. Sporks appear naturally in certain k-RBG links and produce knots with the same k-trace; although too blunt to produce exotic smooth structures, they are effective in detecting slicing degree in the examples we consider.