Torus surguries on knot traces

Abstract

We initiate the study of torus surgeries on knot traces. Our key technical insight is realizing the annulus twisting construction of Osoinach as a torus surgery on a knot trace. We present several applications of this idea. We find exotic elliptic surfaces that can be realized as surgery on null-homologously embedded traces in a manner similar to that proposed by Manolescu and Piccirillo. Then we exhibit exotic traces with novel properties and improve upon the known geography for exotic Stein fillings. Finally, we construct new potential counterexamples to the smooth 4-dimensional Poincar\'e Conjecture.