Does the Jones Polynomial Detect Unknottedness?
Experimental Mathematics 6 (1997), 51-56
Abstract:
There were many attempts to settle the question whether there exist
non-trivial knots with trivial
Jones polynomial. In this paper we show that such a knot must have
crossing number at least 18.
Furthermore we give the number of prime alternating knots and an upper
bound for the number of prime knots
up to 17 crossings. We also compute the number of different Homfly,
Jones and Alexander polynomials for
knots up to 15 crossings.
Keywords:
Knots, links, Jones polynomial, unknottedness, alternating knots,
HOMFLY polynomial
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