For any knot $K$ does there exist some diagram for $K$ such that Seifert's algorithm produces a Seifert surface for $K$ where the genus of said surface is $g(K)$ - the $3$-genus of $K$?

I have seen some knot blogs that claim it doesn't, but have yet to see an example where it will not produce one with minimal genus.


The minimum genus of any Seifert surface coming from Seifert's algorithm is known as the canonical genus $g_c(K)$ of the knot. An example where the canonical genus is strictly greater than the $3$-genus is a Whitehead double of the trefoil (pictured below).

enter image description here

This post gives an explanation of why the genus of a Whitehead double is one. Hugh Morton proved that the highest degree of one of the two variables of the HOMFLY polynomial gives a lower bound on twice the canonical genus $2g_c(K)$ (see the paper). This bound gives that the canonical genus of the Whitehead double of the trefoil is $3$.

See this paper of Brittenham and Jensen for more examples.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.