Editing Jones polynomial (section)

==Properties==
The Jones polynomial is characterized by taking the value 1 on any diagram of the unknot and satisfies the following [[skein relation]]:

::<math> (t^{1/2} - t^{-1/2})V(L_0)  = t^{-1}V(L_{+}) - tV(L_{-}) \,</math>

where <math>L_{+}</math>, <math>L_{-}</math>, and <math>L_{0}</math> are three oriented link diagrams that are identical except in one small region where they differ by the crossing changes or smoothing shown in the figure below:

[[Image:Skein (HOMFLY).svg|center|200px]]

The definition of the Jones polynomial by the bracket makes it simple to show that for a knot <math>K</math>, the Jones polynomial of its mirror image is given by substitution of <math>t^{-1}</math> for <math>t</math> in <math>V(K)</math>. Thus, an '''[[Chiral knot|amphicheiral knot]]''', a knot equivalent to its mirror image, has [[palindromic]] entries in its Jones polynomial. See the article on [[skein relation]] for an example of a computation using these relations.

Another remarkable property of this invariant states that the Jones polynomial of an alternating link is an [[alternating polynomial]]. This property was proved by [[Morwen Thistlethwaite]]<ref>{{cite journal|last=Thistlethwaite|first=Morwen B.|author-link=Morwen Thistlethwaite| title=A spanning tree expansion of the Jones polynomial|journal=[[Topology (journal)|Topology]]| date=1987|volume=26|issue=3|pages=297–309|doi=10.1016/0040-9383(87)90003-6|doi-access=free}}</ref> in 1987. Another proof of this last property is due to [[Hernando Burgos-Soto]], who also gave an extension of the property to tangles.<ref>{{cite journal|last=Burgos-Soto|first=Hernando|author-link=Hernando Burgos-Soto | title=The Jones polynomial and the planar algebra of alternating links|journal=Journal of Knot Theory and Its Ramifications|date=2010|volume=19|issue=11|pages=1487–1505|doi=10.1142/s0218216510008510|arxiv=0807.2600|s2cid=13993750}}</ref>

The Jones polynomial is not a complete invariant. There exist an infinite number of non-equivalent knots that have the same Jones polynomial. An example of two distinct knots having the same Jones polynomial can be found in the book by Murasugi.<ref>{{Cite book |last=Murasugi |first=Kunio |title=Knot theory and its applications |publisher=Birkhäuser Boston, MA |year=1996 |isbn=978-0-8176-4718-6 |pages=227}}</ref>