An m-generator presentation obtained in this way can always be reduced to a minimal presentation. One group can have several isomorphic minimal presentations, as we noticed in the case of dihedral group. For example, for the cinquefoil knot 5 in Conway notation (or 5₁ in the classical notation) we have the presentation 

For every KL, the group G(L) is not dependent on the projection used for its calculation. G(L) is an invariant of L, so if L₁ and L₂ are equal (ambient isotopic) KLs, their corresponding groups G(L₁) and G(L₂) will be isomorphic. The inverse theorem does not hold: two different KLs can have isomorphic KL-groups (Crowell and Fox, 1963). For example, a granny knot 3#3 and a square knot 3#(-3) have isomorphic groups, with the same presentation (x,y,z:  z^-1xz = xzx^-1,  z^-1yz = yzy^-1). Again, it is not a surprise that KL-groups agree with KL-families. For example, for every knot of the family (2k+1) the knot group is G((2k+1)) = (a,b: a²b^2k+1); for every link of the family (2k) it is G((2k)) = (a,b: ab^ka^-1b^-k); for every (2k+1,2) torus knot it is G(K) = (a,b: a^2k+1 = b²), (k ³ 1), etc.. 

To work with fundamental groups and many other topological properties of KLs (like a hyperbolic volume, symmetry group, Dirichlet domain, etc.) one can use the program SnapPea by J. Weeks. A conversion from P-data to data in SnapPea format or vice versa is provided by the Knot 2000 (K2K) functions SnapPeaDataFromPLData and PLDataFromSnapPeaData.