How to prove a knot with genus larger than 1 is prime, such as Miller Institute Knot?

Question

It is easy to show that a knot with genus 1 is a prime knot because the genus is additive under direct sum. However, I found that some prime knot, for example, $6_2$ the Miller Institute Knot have genus larger than 1(Specifically, $6_2$ has genus 2). How can we prove that MIK is a prime knot? I so far do not have any ideas. Also, will the method used to prove certain knots are prime vary for each prime knot? Or do mathematicians already have a general way to identify a prime knot?

Answer 1

As you guessed, there are a lot of ways to show a knot is prime and depending on the knot, certain ways are much easier than others. In the case for this knot, the easiest way that I know of is to realize it has a bridge index of 2. We see that from this diagram, it must be $\leq 2$ since there are 2 maxima, and the unknot is the only knot with bridge number less than 2. And it is known that all 2-bridge knots are prime.

Off-hand, I know of an algorithmic way to test if a knot is composite or prime, but it is from a paper not published and therefore, does not lend well to being explained here. Maybe someone else here knows of a more classical method.

Actually, I believe that the crossed out statement is false. I don't think there is an algorithmic way, or at least not one that is computationally viable for any given knot. Again, maybe someone else knows better than I.

Answer 2

There is a (not computationally viable) algorithm due to Lackenby from https://arxiv.org/pdf/0805.4706.pdf. He gives an upper bound on the crossing number of components of a composite knot. So one needs to check that for all pairs of knots with less than this crossing number bound are not isotopic to your original knot. An algorithm for the knot isotopy problem (due to Coward and Lackenby) is given here (https://arxiv.org/pdf/1104.1882.pdf) .