Kouki Sato — Page 2

arXiv2014

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

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

math.GT↗

arXiv2014

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

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 $k$.

math.GT↗