arrow

Arrow polynomial for knotoids (per Kauffman et al.).

Functions

arrow_polynomial(k[, normalize])	Compute the arrow polynomial of a knotoid (arXiv:1602.03579).
disoriented_smoothing(k, crossing)	Apply the disoriented smoothing at a crossing.

arrow_polynomial(k, normalize=True)

Compute the arrow polynomial of a knotoid (arXiv:1602.03579).

Parameters:

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

• normalize (bool) – If True, multiply by (-A^3)^(-writhe) to ignore framing.

Returns:

A SymPy expression in A and formal variables K_i, L_j.

Return type:

Expr

Notes

The state sum expands over oriented/disoriented smoothings. After smoothing, consecutive opposite-acuteness cusps are reduced; each component contributes a factor as described in _generator_to_variables().