kauffman#
The Kauffman 2-variable polynomial.
Definition:
\[\begin{split}F(K)(a, z) &= a^{-\mathrm{wr}(K)} \, L(K), \\
F(\text{unknot}) &= 1, \\
F(s^-) &= a \, F(s), \\
F(s^+) &= a^{-1} \, F(s), \\
L(\times) + L(\times) &= z \, L(+) + z \, L(-).\end{split}\]
Personal notes:
F(k)(a,z) = a^-wr(K) * L(k)
L(O) = 1 L(s-) = a L(s) L(s+) = a^-1 L(s)
L(X) + L(X) = z (X+) + z (X-)
If K is an oriented knot/link, and K* the mirror image, then L(K*) = L(a^-1, z) F(K*) = L(a^-1, z) (this is wr(K*) = -wr(K) F(K*) != L(K) when they are not isotopic
L(K # K’) = L(K) * L(K’) L(K U K’) = (z^-1 (a + a^-1) - 1) L(K) * L(K’)
- kauffman(k)#
- Parameters:
- Return type:
Expr