Knot Polynomial Calculator
Knot Polynomial Calculator computes the Alexander polynomial for a torus knot \\( T_{p,q} \\) with presentation \\( \langle x, y \mid x^p = y^q \rangle \\), where \\( p \\) and \\( q \\) are coprime positive integers.
Alexander Polynomial for Torus Knots
The Alexander polynomial for a torus knot \\( T_{p,q} \\) with coprime integers \\( p \\) and \\( q \\) is given by:
\\[
\Delta(T_{p,q}) = \frac{(t^{pq} – 1)(t – 1)}{(t^p – 1)(t^q – 1)}
\\]
Where:
- \\( p, q \\): Coprime positive integers defining the torus knot.
- \\( t \\): Polynomial variable.
- \\( \Delta(T_{p,q}) \\): Alexander polynomial of the knot.
This formula is derived using Fox free calculus on the knot group presentation \\( \langle x, y \mid x^p = y^q \rangle \\).
[](https://mathoverflow.net/questions/129717/how-to-compute-the-alexander-polynomial-of-general-torus-knot)