Non-orientable genus of knots in punctured Spin 4-manifolds

Abstract

For a closed 4-manifold $X$ and a knot $K$ in the boundary of punctured $X$, we define ${\gamma }_X^0(K)$ to be the smallest first Betti number of non-orientable and null-homologous surfaces in punctured $X$ with boundary $K$. Note that ${\gamma }_{S^4}^0$ is equal to the non-orientable 4-ball genus and hence ${\gamma }_X^0$ is a generalization of the non-orientable 4-ball genus. While it is very likely that for given $X$, ${\gamma }_X^0$ has no upper bound, it is difficult to show it. In fact, even in the case of ${\gamma }_{S^4}^0$, its non-boundedness was shown for the first time by Batson in 2012. In this paper, we prove that for any Spin 4-manifold $X$, ${\gamma }_X^0$ has no upper bound.

4 pages, 1 figure

References in corpus (3)

• The nonorientable four-genus of knots
• Nonorientable four-ball genus can be arbitrarily large
• Non-orientable genus of a knot in punctured $\mathbb{C}{P}^2$