# Knot invariants that discern prime and composite knots.

Is there a list of knot invariants that can tell whether or not a knot is prime? Or at least partially so? i.e. invariants that have one or more of the following properties:

(a) The invariant has a certain value/form/etc. $\Rightarrow$ a knot is prime

(b) The invariant has a certain value/form/etc. $\Rightarrow$ a knot is composite

(c) A knot is prime $\Rightarrow$ the invariant has a certain value/form/etc.

(d) A knot is composite $\Rightarrow$ the invariant has a certain value/form/etc.

For example, genus falls into category (a) since $g(K) = 1 \Rightarrow K$ is prime

edit: I just realized that (a)$\Leftrightarrow$(d) and (b)$\Leftrightarrow$(c), but some things might be more natural to state in one form or the other.

• Please correct me if I am wrong... but since the composite knot $K_1 + K_2$ has an Alexander polynomial $A(K_1)A(K_2)$, if the Alexander polynomial of $K$ cannot be factorized then it is prime. There is a similar formula for satellite knots. And there are similar formulae for Jones polynomial, I guess? Commented Jun 18, 2015 at 9:28