Knot Genus

The least genus of any Seifert surface for a given knot. The unknot is the only knot with genus 0.

Usually, one denotes by  the genus of the knot . The knot genus has the pleasing additivity property that if  and  are oriented knots, then

where the sum on the left hand side denotes knot sum. This additivity implies immediately, by induction, that any oriented knot can be factored into a sum of prime knots. Indeed, by the additivity of knot genus, any knot of genus 1 is prime. Furthermore, given any knot  of genus , either  itself is prime, or  can be written as a sum of knots of lesser genus, each of which can be decomposed into a sum of prime knots, by induction.

A nonobvious fact is that the prime decomposition is also unique.

REFERENCES:

Lickorish, W. B. R. Ch. 2 in An Introduction to Knot Theory. New York: Springer-Verlag, 1997.