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.
