Please consider the following link
its corresponding Jones polynomial is
Could you please identify such link as a member of the Thistlethwaite Link Table?
Many thanks.
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asked Apr 25, 2017 at 21:34
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2 Answers
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for a list of all links up to 11 crossings with their corresponding Jones' polynomial and here http://www.indiana.edu/~linkinfo/descriptions/jones_homfly_kauffman_description/polynomial_defn.html to see the conventions used in determining the actual polynomial (which I am pretty sure are different then your conventions). Linkinfo and knotinfo make pretty short work of these types of questions.
answered Apr 25, 2017 at 22:27
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Using SnapPy, recommended by N. Owad (Are isomorphic the following two links?) and Jim Belk (What knot is this?); I am obtaining the following result.
The link is drawing inside the SnapPy environment as
Now using the commands
N= Manifold()
N.identify()
we obtain the output
[7^2_6(0,0)(0,0), L7a1(0,0)(0,0)]
It is to say the link L7a1 in the Thistlethwaite Link Table.
The Jones polynomial for L7a1 computed using the Mathematica package KnotTheory is
The ratio between the Jones polynomial of the original link and the Jones polynomial of L7a1 is the monomial $q^{-3}$.
Such ratio confirms that the original link is L7a1. Do you agree?
answered Apr 25, 2017 at 23:54
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$\begingroup$ That appears to have a different Jones polynomial then you suggested, so you might want to recheck your computation. $\endgroup$ Commented Apr 26, 2017 at 5:06
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$\begingroup$ @PVAL-inactive, thanks for your comment. As yo can see in the answer the two Jones polynomials only are different by a multiplicative monomial- $\endgroup$ Commented Apr 26, 2017 at 10:57
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1$\begingroup$ @PVAL-inactive For links, the Jones polynomial depends on the orientation. Changing the orientation of one component of the link, but not the other, can multiple the Jones polynomial by a power of $q$. $\endgroup$ Commented Nov 10, 2017 at 19:49