Non-isotopic Seifert surfaces in the 4-ball

Abstract

We give families of knots and links with pairs of Seifert surfaces that are topologically non-isotopic in $D^4$. This generalizes the main example of Hayden-Kim-Miller-Park-Sundberg and the proof is similarly based on the double branched cover and the Seifert form. Moreover, using Conway's theory of topographs, we implement an algorithm that decides whether two integral quadratic forms in two variables are isomorphic over $\mathbb Z$.