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Difference between revisions of "Brandt-Lickorish-Millett-Ho polynomial"

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An invariant of non-oriented links in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130240/b1302401.png" />, invented at the beginning of 1985 [[#References|[a1]]], [[#References|[a2]]] and generalized by L.H. Kauffman (the [[Kauffman polynomial|Kauffman polynomial]]; cf. also [[Link|Link]]).
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{{TEX|done}}{{MSC|57M27}}
  
It satisfies the four term skein relation (cf. also [[Conway skein triple|Conway skein triple]])
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An invariant of non-oriented links in $\mathbf{R}^3$, invented at the beginning of 1985 [[#References|[a1]]], [[#References|[a2]]] and generalized by L.H. Kauffman (the [[Kauffman polynomial]]; cf. also [[Link]]).
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130240/b1302402.png" /></td> </tr></table>
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It satisfies the four term skein relation for a Kauffman skein quadruple (cf. also [[Conway skein triple]])
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$$
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Q_{L_{+}}(z)  + Q_{L_{-}}(z) = z\left({ Q_{L_{0}}(z)  + Q_{L_{\infty}}(z) }\right)
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$$
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and is normalized to be $1$ for the trivial knot.
  
and is normalized to be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130240/b1302403.png" /> for the trivial knot.
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<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/c130240b.gif" />
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R.D. Brandt,  W.B.R. Lickorish,  K.C. Millett,  "A polynomial invariant for unoriented knots and links"  ''Invent. Math.'' , '''84'''  (1986)  pp. 563–573</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  C.F. Ho,  "A new polynomial for knots and links; preliminary report"  ''Abstracts Amer. Math. Soc.'' , '''6''' :  4  (1985)  pp. 300</TD></TR></table>
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<table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  R.D. Brandt,  W.B.R. Lickorish,  K.C. Millett,  "A polynomial invariant for unoriented knots and links"  ''Invent. Math.'' , '''84'''  (1986)  pp. 563–573</TD></TR>
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<TR><TD valign="top">[a2]</TD> <TD valign="top">  C.F. Ho,  "A new polynomial for knots and links; preliminary report"  ''Abstracts Amer. Math. Soc.'' , '''6''' :  4  (1985)  pp. 300</TD></TR>
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</table>

Latest revision as of 19:49, 15 December 2016

2010 Mathematics Subject Classification: Primary: 57M27 [MSN][ZBL]

An invariant of non-oriented links in $\mathbf{R}^3$, invented at the beginning of 1985 [a1], [a2] and generalized by L.H. Kauffman (the Kauffman polynomial; cf. also Link).

It satisfies the four term skein relation for a Kauffman skein quadruple (cf. also Conway skein triple) $$ Q_{L_{+}}(z) + Q_{L_{-}}(z) = z\left({ Q_{L_{0}}(z) + Q_{L_{\infty}}(z) }\right) $$ and is normalized to be $1$ for the trivial knot.

References

[a1] R.D. Brandt, W.B.R. Lickorish, K.C. Millett, "A polynomial invariant for unoriented knots and links" Invent. Math. , 84 (1986) pp. 563–573
[a2] C.F. Ho, "A new polynomial for knots and links; preliminary report" Abstracts Amer. Math. Soc. , 6 : 4 (1985) pp. 300
How to Cite This Entry:
Brandt-Lickorish-Millett-Ho polynomial. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Brandt-Lickorish-Millett-Ho_polynomial&oldid=18535
This article was adapted from an original article by Jozef Przytycki (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article