The Kauffman Polynomial

sony sdm-hx73b que viva film tomas milian chams motori e uomini verdi custodia chitarra canoa legno siti nazi dvpns32 tell him vonda shepard lavapiatti classe a all about incasso 30 cm piani cottura metallo urlante scarica programma gioco tombola toyota rav 4 usato lg lcd 19 flatron barlow, jane bicchiere birraia luomo in nero lg 8120 sblocco serie s dns western digital 160 jb cd per auto nikkor 10 5 riassunti decamerone cotonata udo samel cozumel cose fare scanner diapositive kandiramazsin beni jennilopez bruxelles expo alberghi e hotel provincia pesaro belkin tastiera lisa marie non partir uomo scarpe voli low cost australia porsche 997 lettori cd per dj scratch pole motomondiale www felitto it istruzione ricarica inchiostro cartuccia elettrodomestico per pop corn elenco presidenti usa i can t wait until today by max mass destruction faithless brother fax-t104 la pastora il mastino dei baskerville elicotteri da modellismo link torna in europa con il game boy advance sp televisori sony 21 just cavalli him jack the mp3 ripper fiat accessori lettori mp3 archos av profumo romeo gigli bluethooth samsung z500 wight flag sandra. the complete history ego sum panis verbi italiani barinight villaggio turistici puglia marito a sorpresa mp3 vasco rossi jugo de col roja crashbandicoot cd gemelli diversi honey soundtrack desperado antonio banderas ps2 rayman programma fax pogledni recupero dato informatico porto empedocle metalo slug inox 50 testo mad about you mousse t londra palermo biglietti aerei corso pompini fontina fotocamera digitali casio clito esercizi massa muscolare gambe western digital 200 gb basi midi free ricetta liquori posti di lavoro per i disabili palmari acer n35 con gps evahenger f 710 basket cafe olivetti jp 192 forno rustico cherokee limited i predatori della vena doro agent 2002- trainer messaggio visivo il bacio di gustav klimt pornod intel xeon 3 2ghz hit em up style pfalzen martini penna consulti cartomanzia amore cartomante bluetooth navigatore sigma 70 300 obiettivi dichiarazione dei rediti tunnel b1 www navy it mariam marina granata kamara femme like u video esecuzioni in irak www cassano it pb oro srl epson stylus cx3200 mp3 512mb creative muvo n200 figa and foto witch disney testo nelly furtado try poesia lettera d amore pictures e natale ma io non ci sto dentro nikon coolpix 7900 nital offerta vacanze settembre grotta del cane www murst it girls a verona konami supporta il nintendo ds aver media theme o c balestrini, nanni pc office chihuahua bianco a pelo lungo frank 1 fontana federica contratto preliminare www ari winans it elettronica flash reti doghe terzetti photo michel vieth aria ragazza sexi hp stampanti multifunzione foto e video dei vips volo aereo grecia frist date ilmercante rossella brescia strip porno gratis hentai passaggio di propieta intel 630 p4 reggae national tickets testimonial tim www ilcircolo it foto di cuckold risaia proiettore cinema scambisti night club napoli offline lagosta donne vecchie arrapate disney channel com summer estrada pelletterie in firenze jacknife corelli, marie rasoio elettrico braun intestatura say nothing at all testo della sigla o c devoto oli dizionario super campeones creare e stampare biglietti auguri topfield 4000 black eyed shut up criminali rischio di cambio e tecniche telefonini usim calientes mitili pamela cartoni monitor samsung 21 revista trip new yourk new yourk calcilutite posatore gps tom-tom navigator 5 www porca it playstation universale garlando 200 giovane ragazza loewe spheros 37 hd bo boss kurverwaltung

KnotTheory`/Navigation  The Mathematica Package KnotTheory` Acknowledgement Setup Naming and Enumeration Presentations Planar Diagrams Some further details Gauss Codes DT (Dowker-Thistlethwaite) Codes Braid Representatives MorseLink Presentations Arc Presentations Conway Notation Graphical Input Graphical Output Drawing Planar Diagrams How does it work? Drawing MorseLink Presentations Drawing Braids Structure and Operations Invariants Invariants from Braid Theory Three Dimensional Invariants The Alexander-Conway Polynomial The Multivariable Alexander Polynomial The Determinant and the Signature The Jones Polynomial How is the Jones polynomial computed? The Coloured Jones Polynomials The A2 Invariant Quantum knot invariants The HOMFLY-PT Polynomial The Kauffman Polynomial Finite Type (Vassiliev) Invariants Khovanov Homology Heegaard Floer Knot Homology R-Matrix Invariants Extras Included with KnotTheory` Drawing with TubePlot Standalone_TubePlot Using the LinKnot package WikiLink QuantumGroups` Lightly Documented Features A Sample KnotTheory` Session Further Usage Example Prime Links with a Non-Prime Component "Rubberband" Brunnian Links Identifying Knots within a List Cabling Burau's Theorem Further Knot Theory Software KnotPlot Knotscape Xknots Extending/Modifying KnotTheory` Bugs How to Edit this Manual...

The Kauffman polynomial $F(K)(a,z)$ (see [Kauffman]) of a knot or link $K$ is $a^{-w(K)}L(K)$ where $w(L)$ is the writhe of $K$ (see How is the Jones Polynomial Computed?) and where $L(K)$ is the regular isotopy invariant defined by the skein relations

$L(s_{)}=aL(s),\qquad L(s_{-})=a^{-1}L(s)$

(here $s$ is a strand and $s_{\pm }$ is the same strand with a $\pm$ kink added) and

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L(\backoverslash) L(\slashoverback) = z\left(L(\smoothing) L(\hsmoothing)\right)}

and by the initial condition $L(U)=1$ where $U$ is the unknot .

KnotTheory` knows about the Kauffman polynomial:

(For In[1] see Setup)

In[2]:= ?Kauffman Kauffman[K][a, z] computes the Kauffman polynomial of a knot or link K, in the variables a and z.

In[3]:= Kauffman::about The Kauffman program was written by Scott Morrison.

Thus, for example, here's the Kauffman polynomial of the knot 5_2:

In[4]:=	Kauffman[Knot[5, 2]][a, z]
Out[4]=	2 4 6 5 7 2 2 4 2 6 2 3 3 -a a a - 2 a z - 2 a z a z - a z - 2 a z a z 5 3 7 3 4 4 6 4 2 a z a z a z a z

5_2

T(8,3)

It is well known that the Jones polynomial is related to the Kauffman polynomial via

$J(L)(q)=(-1)^{c}L(K)(-q^{-3/4},\,q^{1/4}q^{-1/4})$,

where $K$ is some knot or link and where $c$ is the number of components of $K$. Let us verify this fact for the torus knot T(8,3):

In[5]:=

K = TorusKnot[8, 3];

In[6]:=	Simplify[{ (-1)^(Length[Skeleton[K]]-1)Kauffman[K][-q^(-3/4), q^(1/4) q^(-1/4)], Jones[K][q] }]
Out[6]=	7 9 16 7 9 16 {q q - q , q q - q }

[Kauffman] ^  L. H. Kauffman, An invariant of regular isotopy, Trans. Amer. Math. Soc. 312 (1990) 417-471.