An algorithm for computing the $\Upsilon$-invariant and the $d$-invariants of Dehn surgeries

TL;DR

This paper presents an explicit algorithm leveraging grid homology to compute the $0$-invariant and $d$-invariant for Dehn surgeries on knots, enabling calculations for all prime knots up to 11 crossings.

Contribution

It introduces a novel algorithm that explicitly computes these invariants using grid homology, expanding computational capabilities in knot theory.

Findings

Computed invariants for all prime knots with up to 11 crossings

Provided a practical method for calculating $0$ and $d$-invariants

Enhanced understanding of Dehn surgeries on small knots

Abstract

By using grid homology theory, we give an explicit algorithm for computing Ozsv\'ath-Stipsicz-Szab\'o's $\Upsilon$-invariant and the $d$-invariant of Dehn surgeries along knots in $S^3$. As its application, we compute the two invariants for all prime knots with up to 11 crossings.