Computing rotation and self-linking numbers in contact surgery diagrams

TL;DR

This paper provides explicit formulas for calculating the rotation number, self-linking number, and d3-invariant of knots in contact surgery diagrams, enhancing computational tools in contact topology.

Contribution

It introduces new explicit formulas for invariants of Legendrian and transverse knots in contact (1/n)-surgeries, extending previous methods.

Findings

Explicit formula for rotation number in contact (1/n)-surgery diagrams

Formula for self-linking number of transverse knots

Extension of Ding-Geiges-Stipsicz formula for d3-invariant

Abstract

We give an explicit formula to compute the rotation number of a nullhomologous Legendrian knot in contact (1/n)-surgery diagrams along Legendrian links and obtain a corresponding result for the self-linking number of transverse knots. Moreover, we extend the formula by Ding-Geiges-Stipsicz for computing the d3-invariant to (1/n)-surgeries.