A [Seifert surface](https://en.wikipedia.org/wiki/Seifert_surface) of a knot is a surface whose boundary is the knot. The genus of a knot is the minimal genus among all the Seifert surfaces of the knot. My question is, is any algorithm known to find the genus of a knot?


Note that it’s been known since the 1980’s that Seifert’s algorithm for finding a Seifert surface does not suffice to find the genus of a knot. To wit, there even exist knots for which Seifert’s algorithm never produces a minimal-genus Seifert surface no matter what diagram of the knot you take.


In any case, is there some diagrammatic knot invariant that allows you to calculate the genus?



Is there an algorithm for the genus of a knot?
Jaco and Oertel's paper *[An algorithm to decide if a three-manifold is a Haken manifold](https://www.sciencedirect.com/science/article/pii/0040938384900399)* [1984], plus a bit of work, gives a doubly exponential time algorithm to compute the Seifert genus. (In practice their algorithm lies in exp-poly.)


Agol, Hass, and Thurston's paper *[The computational complexity of knot genus and spanning area](https://www.ams.org/journals/tran/2006-358-09/S0002-9947-05-03919-X/S0002-9947-05-03919-X.pdf)* reduces the time required to exp-poly (and proves that "genus less than $g$" lies in NP).


Recently announced work of Lackenby *[Unknot recognition in quasi-polynomial time](http://people.maths.ox.ac.uk/lackenby/quasipolynomial-talk.pdf)* [2021] gives a Haken hierarchy of the initial knot complement. If this can be improved to be a taut sutured manifold hierarchy, then that will reduce the time required (for the genus problem of a knot in the three-sphere given as a diagram) to quasi-polynomial.




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The above (and the other answers) answer the question as asked in its first paragraph. However, its final paragraph asks if the genus can be computed from a "diagrammatic knot invariant". The answer there seems to be "not in general", but this is not a theorem. In particular there are families of knots (fibered, alternating) where the spans of certain polynomial invariants (Alexander, Jones) record the genus.


