There are many results for the knots winded around a torus, but what about knots winded around surfaces of a higher genus? Is there any classification of such knots? I would be glad to see any review on this subject. To be more specific, I'm precisely interested in any results regarding Alexander polynomials and Seifert surfaces for such knots.
2026-03-25 14:33:19.1774449199
Knots winded around surfaces of a higher genus
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I believe what you are looking for is something called the Heegaard Genus of a knot $K$, which we denote $h(K)$. This is a relatively new invariant that Morimoto defined in 92 in his paper titled On the Additivity of $h$-Genus of Knots. He has some follow up papers about the topic too that you can find with minimal effort now that you know the name.
As to your specific interests in Alexander polynomials and Seifert surfaces, I am not sure what is known, but would not be surprised if there is little to no published work relating the them to $h$-genus. If you do find something though, I would be intersted in seeing them. Good luck.