This page is devoted to the following odd relationship between the Jones polynomial of a knot at -1 and the hyperbolic volume its complement. The point is to show the graphs from certain computer experiments.

First, look at all (approx 5000) alternating knots with exactly 13 crossing. Bellow is a plot of the volume of the complement vs. Pi * Log(JonesPoly(-1)).

Notice the near linear relationship between these two quantities. It suggests that for alternating knots Log(JonesPoly(-1)) is almost a linear function of volume. A little thought shows that this can't be true, but that there might be a relationship of the form: log(J(-1)) is almost a linear function of Vol * log(deg(J)). In the next plot we examine the relationship for a) all alternating knots w/ <= 12 crossings (labled 12_alt), b) all alternating knots w/ 13 crossings, c) samples of alternating knots with 14, 15 and 16 crossings.

What about non-alternating knots? Well, these seem to lie "below" the alternating knots. The following two plots of the same data (on all knots with 13 or fewer crossings), but layered in different ways.

All of this suggests that there should be an inequality:

log(J(-1))/log(deg(J)) < a * Volume + b

Why?

Notes:

• log(J(-1)) is one of the first terms in Kashaev's conjecture about the relationship between the colored Jones polynomials and hyperbolic volume. However, the above doesn't appear to simply be saying that you have fast convergence in Kashaev's conjecture as the slope of the line is not what you would expect.
• J(-1) is just the Alexander polynomial at -1, which is the order of the torision in the 2-fold cyclic cover branched over the knot.
• It is possible to prove such an inequality for two-bridge knots, but the proof is not very informative. (Surprisingly, the constants one gets are off by less than an order of magnitude.)
• (Added 2000/7/17) There was an error in my data for Jones polys of knots with < 10 crossings. This has been corrected, and only improves the correlation.

-Nathan