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Difference between revisions of "Jones unknotting conjecture"

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Figure: j130050a
 
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]]].
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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 [[#References|[a5]]], [[#References|[a7]]]. Recently (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 [[#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]]].
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L.H. Kauffman showed that there are non-trivial virtual knots with Jones polynomial equal to $1$, [[#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.
 
It is still an open problem (as of 2001) whether a simple (non-satellite) link can have a Jones polynomial of an unlink.
  
 
====References====
 
====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>
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<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>
 +
 
 +
{{TEX|done}}

Revision as of 06:25, 3 October 2016

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]. Recently (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" preprint (2001)
[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
How to Cite This Entry:
Jones unknotting conjecture. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Jones_unknotting_conjecture&oldid=39362
This article was adapted from an original article by Jozef Przytycki (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article