If K is a knot diagram, and K' is the knot diagram obtained by changing one of its crossings. Why does applying Seifert's algorithm to both diagrams produce surfaces of the same genus?
2026-03-30 08:21:17.1774858877
Why does Seifert algorithm give surfaces with the same genus?
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The genus of a surface produced by Seifert's algorithm only depends on the number of crossing in your diagram (which is the number of twisted bands you use to build the surface) and the number of closed circles you get after resolving all the crossings in the diagram.
Changing a crossing doesn't alter the number of crossings in your diagram. You can check locally that applying the algorithm before and after a crossing change in a diagram will give you the same number of closed circles because the resolution you do is the same in both cases. Because both of these numbers haven't changed, the genus of the surface won't change.