$\DeclareMathOperator\SU{SU}$Informally speaking, the Casson invariant counts half the number of conjugacy classes of representations of the fundamental group of a homology $3$-sphere $M$ into the group $\SU(2)$. My question is for a knot $K$ inside $M$, what does the Casson's knot invariant count? Am I right to think that it counts half the the number of conjugacy classes of representations of the fundamental group of the knot complement $M\setminus K$ into the group $\SU(2)?$ Can anyone provide any heuristic and maybe a reference?
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- 1$\begingroup$ What is your definition of the "Casson's knot invariant"? Perhaps you mean the invariant mentioned here mathoverflow.net/questions/164357/casson-invariant and that is related to $\Delta_K''(1)$? Otherwise such a count of (traceless) representations was studied by Lin in his paper "a knot invariant by representation spaces". maths.ed.ac.uk/~v1ranick/papers/lin2.pdf and is related to the signature of the knot. $\endgroup$Anthony Conway– Anthony Conway2023-11-15 02:46:43 +00:00Commented Nov 15, 2023 at 2:46
- 2$\begingroup$ For knots in $S^3$ it's called the type-2 finite-type (Vassiliev) invariant. It's a count of round circles that intersect the knot in 5 points, such that if any two points are adjacent in the circle parametrization then they are not adjacent in the knot parametrization. I had a summer student create a demonstration here: sean564.github.io/top you'll need to click on a knot, or if you have a WebGPU enabled browser (and a GPU with a lot of cores) you can enter any knot you like. $\endgroup$Ryan Budney– Ryan Budney2023-11-15 04:06:26 +00:00Commented Nov 15, 2023 at 4:06
- $\begingroup$ @AnthonyConway, my definition is from Akbulut & McCarthy, I believe up to a sign it's the same one as the overflow link you mentioned. $\endgroup$Partha Ghosh– Partha Ghosh2023-11-15 08:40:35 +00:00Commented Nov 15, 2023 at 8:40
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I'm not sure that this directly answers your question, but section 3 of this paper by Morita might have something useful. He describes the Casson knot invariant in terms of the Alexander polynomial and then relates it to the regular Casson invariant.
