The Non-Orientable Four-Ball Genus of a New Infinite Family of Torus Knots

We extend previous work by using a combination of band surgeries and known bounds to compute ${\gamma }_4(T_{4n,(2n\pm 1{)}^2+4n-2})=2n-1$ for all $n\ge 1$. We further generalize this result by showing that ${\gamma }_4(T_{4n+2k,n(4n+2k)-1})={\gamma }_4(T_{4n+2k,(n+2)(4n+2k)-1})=2n-1+k$ for all $n\ge 1$ and $k\ge 0$. All knots in this family are counterexamples to Batson’s conjecture.