affine_index_polynomial#

affine_index_polynomial(k)#

Compute the affine index polynomial of a knotoid.

The algorithm follows the standard labeling along the open arc starting from the outgoing terminal. Crossings receive weights that depend on the traversal label and crossing sign; the polynomial is then $\sum _{c\in \text{crossings}}\mathrm{sign}(c)(t^{w(c)}-1)$.

Parameters:

k (PlanarDiagram | OrientedPlanarDiagram) – A planar diagram of a knotoid. If not oriented, it will be oriented internally.

Returns:

A SymPy expression in the symbol t representing the affine index polynomial.

Raises:

ValueError – If a valid outgoing terminal endpoint of degree 1 cannot be found.

Return type:

Expr

Examples

>>> from knotpy.io import read_pd  # example-only; adjust to your API #TODO: !!!
>>> K = read_pd("knotoid_example")
>>> affine_index_polynomial(K)
t**2 - 2*t + 1