View source for Jones unknotting conjecture

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

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/j130050a.gif" />

Figure: j130050a

The conjecture has been confirmed for several families of knots, including alternating and adequate knots, knots up to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130050/j1300501.png" /> crossings and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130050/j1300502.png" />-algebraic knots (cf. [[Knot theory|Knot theory]]) up to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130050/j1300503.png" /> crossings [[#References|[a5]]], [[#References|[a7]]]. Recently (2001), S. Yamada announced that the conjecture holds for knots with up to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130050/j1300504.png" /> crossings. The analogous conjecture for links does not hold, as M.B. Thistlethwaite found a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130050/j1300505.png" />-crossing link whose Jones polynomial coincides with a trivial link of two components, cf. Fig.a1. This and similar examples constructed since are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130050/j1300506.png" />-satellites on a Hopf link [[#References|[a6]]], [[#References|[a1]]].

L.H. Kauffman showed that there are non-trivial virtual knots with Jones polynomial equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130050/j1300507.png" />, [[#References|[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====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S. Eliahou,   L.H. Kauffman,   M. Thistlethwaite,   "Infinite families of links with trivial Jones polynomial"  ''preprint''  (2001)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  V.F.R. Jones,   "Hecke algebra representations of braid groups and link polynomials"  ''Ann. of Math.'' , '''126''' :  2  (1987)  pp. 335–388</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  V.F.R. Jones,   "Ten problems" , ''Mathematics: Frontiers and Perspectives'' , Amer. Math. Soc.  (2000)  pp. 79–91</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  L.H. Kauffman,   "A survey of virtual knot theory" , ''Knots in Hellas '98'' , ''Ser. on Knots and Everything'' , '''24'''  (2000)  pp. 143–202</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  M.B. Thistlethwaite,   "Links with trivial Jones polynomial"  ''J. Knot Th. Ramifications'' , '''10''' :  4  (2001)  pp. 641–643</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  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</TD></TR></table>