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

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 $k$.

11 pages, 6 figures

References in corpus (2)

• A concordance invariant from the Floer homology of +/- 1 surgeries
• The nonorientable four-genus of knots

Cited by in corpus (2)

• On eigenvalues of double branched covers
• Non-orientable genus of knots in punctured Spin 4-manifolds