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.