# Connected Sum Decomposition of a Knot

Given a composite knot, is it possible to decompose it in prime knots by an algorithm that runs in polynomial time?

• What is your motivation for this? And what is your understanding of the situation up until now? Commented Dec 13, 2011 at 23:56
• My understanding of present knowledge is that the answer is it's not known, but highly doubtful that such an algorithm exists. Commented Dec 14, 2011 at 0:40
• I think current run-time estimates for known algorithms to construct the prime decomposition of a knot tend to be at least exponential. I'm thinking of normal surface theory arguments. Commented Dec 14, 2011 at 0:41
• A straightforward comment is that this problem is at least as hard as recognizing the unknot. (Given a diagram D for knot, form a larger diagram as the "connected sum" of D with a trefoil diagram. The larger diagram represents a composite knot iff D represents the unknot.) Commented Dec 14, 2011 at 13:45
• I know that in computational topology, there are several softwares like SnapPea and Regina, designed to help topologists in the resolutions of hard problems. I am studying knot theory in this period and then i would like to know if it is possbile (thanks to these programs or other softwares) to decompose a (non prime) knot by an algorithm that runs in polynomial time!
– Gian
Commented Dec 14, 2011 at 17:53