Jones unknotting conjecture

Every non-trivial knot has a non-trivial Jones polynomial.

Figure: j130050a

The conjecture has been confirmed for several families of knots, including alternating and adequate knots, knots up to $18$ crossings and $2$-algebraic knots (cf. Knot theory) up to $21$ crossings [a5], [a7]. In 2001, S. Yamada announced that the conjecture holds for knots with up to $20$ crossings. The analogous conjecture for links does not hold, as M.B. Thistlethwaite found a $15$-crossing link whose Jones polynomial coincides with a trivial link of two components, cf. Fig.a1. This and similar examples constructed since are $2$-satellites on a Hopf link [a6], [a1].

L.H. Kauffman showed that there are non-trivial virtual knots with Jones polynomial equal to $1$, [a4].

It is still an open problem (as of 2001) whether a simple (non-satellite) link can have a Jones polynomial of an unlink.

References

[a1]	S. Eliahou, L.H. Kauffman, M. Thistlethwaite, "Infinite families of links with trivial Jones polynomial" Topology 42, No. 1, 155-169 (2003) Zbl 1013.57005
[a2]	V.F.R. Jones, "Hecke algebra representations of braid groups and link polynomials" Ann. of Math. , 126 : 2 (1987) pp. 335–388
[a3]	V.F.R. Jones, "Ten problems" , Mathematics: Frontiers and Perspectives , Amer. Math. Soc. (2000) pp. 79–91
[a4]	L.H. Kauffman, "A survey of virtual knot theory" , Knots in Hellas '98 , Ser. on Knots and Everything , 24 (2000) pp. 143–202
[a5]	W.B.R. Lickorish, M.B. Thistlethwaite, "Some links with non-trivial polynomials and their crossing-numbers" Comment. Math. Helv. , 63 (1988) pp. 527–539
[a6]	M.B. Thistlethwaite, "Links with trivial Jones polynomial" J. Knot Th. Ramifications , 10 : 4 (2001) pp. 641–643
[a7]	S. Yamada, "How to find knots with unit Jones polynomials" , Knot Theory, Proc. Conf. Dedicated to Professor Kunio Murasugi for his 70th Birthday (Toronto, July 13th-17th 1999) (2000) pp. 355–361

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