How can I compute knots?
The most natural ways to represent knots are either to embed them piecewise linearly in $\mathbb{R}^3$ (just store the coordinates of the vertices and where you want to put segments) (any tame knot can be embedded piecewise linearly) or with a knot diagram, i.e. storing a projection on $\mathbb{R}^2$ as a graph where at every crossing you specify which strand is above.

As Suresh pointed out, checking knot equivalence is highly non-trivial (not known to be in P) but experimental results for unknot recognition are polynomial-like -- knot equivalence looks much harder though. If you are looking for software, peek at Regina .