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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.

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  • $\begingroup$ 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? $\endgroup$
    – wilsonw
    Commented Jun 18, 2015 at 9:28

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There are other invariants, like your example genus, where the first value that the unknot does not take is guarnteed prime. Bridge number is also like this. The more useful answer however, is in the form of being able to determine hyperbolic knots.

Knots come in three types: Torus, Satellite (which includes composite knots), and Hyperbolic.

Any hyperbolic knot is prime. There is a program, originally developed by Jeff Weeks, called SnapPea, and the new incarnation is called SnapPy (download here) , that determines can determine the hyperbolic volume, (which is always >2) of a knot - IF that knot is a hyperbolic knot. Otherwise, it returns zero, in which case it is either torus or satellite.

Now you have answered your question, or gave yourself a new one. To determine if you have a torus knot or satellite knot, you can use the fact that a knot is a torus knot if and only if it has non-trivial center of its fundamental group. Unfortunately, this isn't that easy. But it will mostly answer your question, you will then have to determine the index of your satellite knot. If it is index 1, you have a composite, if >1, it is (I will use the word) proper satellite. This whole process shouldn't be too bad for a knot with small crossing number.

Hope this helps. Let me know if something was not clear here.

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