Non-orientable 4-genus of torus knots

TL;DR

This paper investigates the non-orientable 4-genus of certain torus knots, providing bounds and a new formula for related invariants, advancing understanding of knot surfaces in four-dimensional topology.

Contribution

It extends bounds for the non-orientable 4-genus of specific torus knots and introduces a generalized formula for the $d$-invariant of -1-surgery on these knots.

Findings

Bounds for $ onorientable 4$-genus of $T_{5,q}$ and $T_{6,q}$

A non-recursive formula for $d$-invariant of $-1$ surgery on torus knots

Extension of previous research on non-orientable surfaces in 4-manifolds

Abstract

The non-orientable 4-genus of a knot $K$ in $S^{3}$, denoted $\gamma_4(K)$, measures the minimum genus of a non-orientable surface in $B^{4}$ bounded by $K$. We compute bounds for the non-orientable 4-genus of knots $T_{5, q}$ and $T_{6, q}$, extending previous research. Additionally, we provide a generalized, non-recursive formula for $d(S^{3}_{-1}(T_{p,q}))$, the $d$-invariant of -1-surgery on torus knots.