Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

I need to compute a series of Tristram-Levine signatures for a family of torus knots. I was wondering if this has already been done or whether there is a good way to streamline the computation.

I am aware that Tristram's original paper on the signature invariants provides a formula for the $e^{\frac{2\pi i}{3}}$-signature of $(2, odd)$ knots, and I am looking for generalizations. The most relevant reference I found is the following paper by Maciej Borodzik and Krzysztof Oleszkiewicz: http://arxiv.org/abs/1002.4500 . It seems to me that a less complex (if less general) solution should exist.

To make the question a little more specific, the simplest example I'm working with is the $e^{\frac{2\pi i}{3}}$-signature of the $(5,6)$-torus knot, which has a $20\times20$ Seifert Matrix. In general, I'd like to be able to find the $e^{\frac{2\pi i}{3}}$- signature of various $(m, mn +1)$-torus knots where $m$ and $n$ are any integers.

share|improve this question
    
I suppose with a little work one could make the software Regina compute Tristram-Levine signatures. All the core code is implemented but I haven't set up a single procedure to do those computations in precisely that form. But the intersection forms and abelian covering spaces are all set-up. The webpage is here: regina.sourceforge.net –  Ryan Budney May 7 at 1:43

1 Answer 1

up vote 5 down vote accepted

In general, closed formulas for such things tend to be rather slow, computationally. I would suggest that you look at the paper of Litherland: Signatures of iterated torus knots. Topology of low-dimensional manifolds, pp. 71–84, Lecture Notes in Math., 722, Springer, Berlin, 1979. Litherland starts with a formula, due to Brieskorn, for the signature function of a torus knot on the circle, that involves counting lattice points; this is easily implemented on a computer.

On the unit circle (where you are computing Levine-Tristram signatures) the function is piecewise constant with jumps at certain roots of unity. Litherland computes the jump function, which gives the signature function. I believe that this is computationally much faster. It's quite possible that for special classes of knots, such as the $(m,mn+1)$ torus knots you mention, that you can get a simple explicit answer.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.