Genus of knot is defined to be the least genus among all Seifert surfaces of knot. Crossing number is the minimal number of crossings over all possible diagrams. Both genus of knot and crossing number are known to be invariants of knots. I ask whether there is a known relationship between these two invariants. I could not find in the literature review and at the same time I have feeling that there is kind of relationship between them. Any idea about this?
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2 Answers
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Given any knot $K$, we can form $K'$, the Whitehead Double of $K$, which increases crossing number and $g(K')=1$. You can find a more of that here and other here. So there are knots with arbitrarily large crossing number and genus 1.
In the other direction, I am not sure what is out there. There is probably some silly upper bound, like $g(K)<2c(K)$ but I have no proof of that... So don't quote it. Maybe someone else can weigh in.
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1$\begingroup$ If you do Seifert's algorithm on a minimal crossing number diagram, you get a surface whose decomposition is some number of disks attached by $c(K)$ handles. A tree's worth of handles are used just to join the disks into a single disk (and there are at least two disks if the knot is not the unknot) so $g(K) < c(K)/2$ from the worst case of every pair of handles after the first going into the genus. $\endgroup$ Apr 20, 2019 at 19:34
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In "The Additivity of Crossing Numbers" by Yuanan Diao it is shown among others that
$$g \le 1 + \frac {c-b-\mu}{2},$$
where $g$ is 3-genus, $c$ is crossing number, $b$ is braid index and $\mu$ denotes number of components. This generalised the classical inequality that you expect: $2g \le c$ ($\mu = 1$, $b \ge 1$ for knots).
