Given a knot diagram, it is possible to construct a collection of variables and equations, and given such a collection, a group
naturally arises that is known as the group of the knot. While the group itself depends
on the choices made in the construction, any two groups that arise in this way are
isomorphic (Livingston 1993, p. 103).

For example, the knot group of the trefoil knot is

The group of a knot is not a complete knot invariant (Rolfsen 1976, p. 62). Furthermore, it is often quite difficult to prove that two knot group presentations represent nonisomorphic groups (Rolfsen 1976, p. 63).

Livingston, C. Knot Theory. Washington, DC: Math. Assoc. Amer., 1993.Rolfsen, D.
"The Knot Group." §3B in Knots
and Links. Wilmington, DE: Publish or Perish Press, pp. 51-52, 1976.