The Kauffman Polynomial

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 Threading a link by a polynomial Further Knot Theory Software KnotPlot Knotscape Xknots Extending/Modifying KnotTheory` Bugs How to Edit this Manual...

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

[math]\displaystyle{ L(s_+)=aL(s), \qquad L(s_-)=a^{-1}L(s) }[/math]

(here [math]\displaystyle{ s }[/math] is a strand and [math]\displaystyle{ s_\pm }[/math] is the same strand with a [math]\displaystyle{ \pm }[/math] kink added) and

[math]\displaystyle{ L(T_1)+L(T_2) = z\left(L(T_3)+L(T_4)\right) }[/math]

(here [math]\displaystyle{ T_1 }[/math], [math]\displaystyle{ T_2 }[/math], [math]\displaystyle{ T_3 }[/math] and [math]\displaystyle{ T_4 }[/math] are , , and , respectively), and by the initial condition [math]\displaystyle{ L(U)=1 }[/math] where [math]\displaystyle{ U }[/math] is the unknot .

KnotTheory` knows about the Kauffman polynomial:

(For In[1] see Setup)

In[1]:= ?Kauffman Kauffman[K][a, z] computes the Kauffman polynomial of a knot or link K, in the variables a and z.

In[2]:= Kauffman::about The Kauffman program was written by Scott Morrison.

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

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

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

[math]\displaystyle{ J(L)(q) = (-1)^cL(K)(-q^{-3/4},\,q^{1/4}+q^{-1/4}) }[/math],

where [math]\displaystyle{ K }[/math] is some knot or link and where [math]\displaystyle{ c }[/math] is the number of components of [math]\displaystyle{ K }[/math]. Let us verify this fact for the torus knot T(8,3):

In[4]:=

K = TorusKnot[8, 3];

In[5]:=	Simplify[{ (-1)^(Length[Skeleton[K]]-1)Kauffman[K][-q^(-3/4), q^(1/4)+q^(-1/4)], Jones[K][q] }]
Out[5]=	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.