What is the Jones Polynomial for the Borromean Link?

Question

I was looking up the Jones Polynomial for a project I’m working on and came up with this equation from the knot atlas:

$$ -q^3-q^{-3}+3q^2+3q^{-2}-2q-2q^{-1}+4 $$

However, I know that when entering VL(1), you should get the number of components required for the knot/link, and this equation churns out a 4 and not a 3 for a simple borromean.

Is this the actual Jones Polynomial? And if so, why does it produce this result? I’m simply an enthusiast and not a mathematician, so any explanation is beneficial.

Answer 1

when entering VL(1), you should get the number of components required for the knot/link

Actually, $V_L(1)=(-2)^{\text{number of components}-1}$, which agrees with the $4$ for the Borromean link. See e.g. the answer to https://mathoverflow.net/questions/176862/the-jones-polynomial-at-specific-values-of-t ; for a disjoint union of unknots, this is easy to see from relations for the Kauffman bracket.

Answer 2

For an independent verification, I drew the Borromean link in KnotFolio (https://knotfol.io) and found that the Jones polynomial was $-t^3+3t^2-2t+4-2t^{-1}+3t^{-2}-t^{-3}$, which is exactly the same (with $q=t$).