A characterization of alternating links in thickened surfaces – CORRIGENDUM

Recall that a spanning surface for a link $L$ is by assumption a connected unoriented surface with boundary equal to $L$.

Theorem 1.1 and Corollary 4.9 from the paper are incorrect as stated. For example, one can construct counterexamples to Theorem 1.1 using links $L$ contained in 3-balls, so-called local links. Let $L \subset B^{3}$ be a link with an alternating projection on $S^{2} = \partial B^{3}.$ Under inclusion $B^{3} \subset \Sigma \times I$, we obtain a local link $L$ in $\Sigma \times I$ which bounds definite spanning surfaces of opposite sign. However, if the genus $g(\Sigma )>0$, then $L$ does not have minimal genus.

To correct for this issue, we need to add the assumption that $L$ is not a local link in the case $g(\Sigma )>0$. The corrected statement of the theorem is as follows.

Theorem 1.1. Let $L$ be a link in $\Sigma \times I$, and assume that $L$ bounds a positive definite spanning surface and a negative definite spanning surface. Then $L \subset \Sigma \times I$ is a non-split alternating link which either has minimal genus or is contained in a 3-ball.

A few remarks on the proof are in order. For $g(\Sigma ) = 0$, the proof is the same as before. For $g(\Sigma ) \geq 1$, then arguing as before, we see that $P$ and $N$ are not $S^{*}$-equivalent, unless the core surface $S$ of $\nu (P \cup N)$ is a 2-sphere. In the latter case, $L$ is contained in a 3-ball, since $\Sigma \times I$ is irreducible, and $L$ has a connected alternating diagram on $S,$ implying that $L$ is non-split and has an alternating diagram on $\Sigma$.

Otherwise, assuming that $P$ and $N$ are not $S^{*}$-equivalent, then the argument goes through as before.

Below is a corrected statement of the corollary.

Corollary 4.9. A link $L \subset \Sigma \times I$ in a thickened surface of positive genus is alternating and has minimal genus if and only if $L$ bounds definite spanning surfaces of opposite sign and is not contained in a 3-ball.