jones#
The bracket polynomial ⟨·⟩ (also called the Kauffman bracket) is a polynomial invariant of unoriented framed links.
See: L. H. Kauffman, State models and the Jones polynomial, Topology 26(3), 1987.
- jones(k)#
Compute the Jones polynomial via the normalized Kauffman bracket.
The Kauffman bracket polynomial ⟨·⟩ is characterized by:
\[\begin{split}\langle U \rangle &= 1, \quad \text{where $U$ is the unknot}, \\ \langle L_\times \rangle &= A \, \langle L_0 \rangle \;+\; A^{-1} \, \langle L_\infty \rangle, \\ \langle L \sqcup U \rangle &= \left(-A^2 - A^{-2}\right) \langle L \rangle.\end{split}\]The Jones polynomial is obtained from the normalized bracket polynomial by the specialization:
\[A = t^{-1/4}.\]- Parameters:
k (PlanarDiagram | OrientedPlanarDiagram) – Planar diagram of a knot or link (oriented or unoriented).
- Returns:
A SymPy expression in
trepresenting the Jones polynomial.- Return type:
Expr
Notes
Alternative (equivalent) substitution in the (l)–(m) variables: \(l = i \, t^{-1}\), \(m = i \, (t^{-1/2} - t^{1/2})\).