# Knots indistinguishable by HOMFLY

Is there any list (incomplete of course) of knots, that have similar HOMFLY polynomials? I am mainly interested in torus knots.

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Torus knots are distinguished by the Jones polynomial, which is contained in the HOMFLY(PT). Hence no torus knots have the same HOMFLY poly. – Jim Conant Nov 21 '13 at 10:20

## 1 Answer

There is some list on Thistlethwaite's knot page, I think, see "the knot atlas". Kanenobu has shown that there are infinitely many distinct knots with the same HOMFLY polynomial:

Kanenobu, T. "Infinitely Many Knots with the Same Polynomial." Proc. Amer. Math. Soc. 97, 158-161, 1986.

For knots with "similar" HOMFLY polynomial, in particular the Alexander polynomial, the Conway polynomial and the Jones polynomial coincide. For torus knots there are easy formulas, e.g., the Jones polynomial for a $(m,n)$-torus knot is $$V(t)=\frac{t^{{(m-1)(n-1)}/2}(1-t^{m+1}-t^{n+1}+t^{m+n})}{1-t^2},$$ and the Alexander polynomial is $$A(t)=\frac{(t^{nm}-1)(t-1)}{(t^m-1)(t^n-1)}.$$ This shows that the polynomials are different (except for obvious symmetries). In general, the HOMFLY polynomial for torus knots is computed in section 3 of the paper "The HOMFLY polynomial for torus links from Chern-Simons gauge theory", by J.M.F. Labastida and M. Marino (see formula $3.1$).

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