Generalized Mazur Patterns and Immersed Heegaard Floer Homology

TL;DR

This paper introduces new examples of pattern knots that produce non-slice satellite knots in any rational homology 4-ball, using immersed curve techniques to compute concordance invariants.

Contribution

It generalizes Mazur patterns and provides a closed formula for tau and epsilon invariants of satellite knots via immersed Heegaard Floer homology.

Findings

Infinite examples of non-slice patterns in rational homology 4-balls

Closed formula for tau and epsilon invariants of satellite knots

Application of immersed curve techniques from bordered Floer homology

Abstract

Generalizing prior work of Levine, we give infinitely many examples of pattern knots P such that P(K) is not slice in any rational homology 4-ball, for any companion knot K. To show this, we establish a closed formula for the concordance invariants tau and epsilon of a family of satellite knots obtained from generalized Mazur patterns. Our main computational tool is the immersed curve technique from bordered Heegaard Floer homology arising from the work of Chen-Hanselman.