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.
Acknowledgements
We would like to thank the referee for bringing these issues to our attention.
