Non-orientable genus of a knot in punctured $\mathbb{C}P ^2$

TL;DR

This paper establishes a lower bound on the first Betti number of non-orientable, null-homologous surfaces bounded by knots in punctured complex projective planes, linking knot invariants with surface topology.

Contribution

It introduces a new lower bound involving the knot signature and Heegaard Floer invariants, providing obstructions to certain surface bounds in punctured complex projective planes.

Findings

Derived a lower bound for the first Betti number of surfaces bounded by knots.

Proved that certain knots cannot bound surfaces with small first Betti number under specified conditions.

Connected knot invariants with topological constraints in complex projective spaces.

Abstract

For any knot $K$ which bounds non-orientable and null-homologous surfaces $F$ in punctured $n\mathbb{C}P^2$, we construct a lower bound of the first Betti number of $F$ which consists of the signature of $K$ and the Heegaard Floer $d$-invariant of the integer homology sphere obtained by $1$-surgery along $K$. By using this lower bound, we prove that for any integer $k$, a certain knot cannot bound any surface which satisfies the above conditions and whose first Betti number is less than $k$.