2.1.1 Group of KLEvery finitely presented group can be given by a presentation: set of generators x_{1},...,x_{m} and their relations r_{1},...,r_{n}. For example, a cyclic group C_{n} will be given by the set of generators {x} and a single relation x^{n} = e, where n is the order of the group C_{n}, and e is the identity. A dihedral group will be given by the set of generators {x_{1},x_{2}} satisfying the relations x_{1}^{n} = x_{2}^{2} = (x_{1}x_{2})^{2} = e, or by the set of generators {y_{1},y_{2}} satisfying the relations y_{1}^{2} = y_{2}^{2} = (y_{1}y_{2})^{n} = e. Two presentations are isomorphic iff one is algebraically equivalent to the other. From the first presentation of the dihedral group, by the replacement y_{1} = x_{1}x_{2} we obtain the second. Working with presentations, we can reduce them in order to obtain minimal presentations, i.e., presentations with a minimum number of generators. For example, the presentation
A Wirtinger presentation of a group G(L) of the link L can be defined for every knot or link diagram L. The group G(L) does not depend on the choice of link diagram. It has a presentation G(L) = (x_{1},x_{2},...,x_{m}: r_{1},...r_{n}), where x_{1},...,x_{m} are generators, r_{1},...,r_{n} are relations satisfied by generators in crossing points, and n is a number of crossings. The rule for making relations is: for each crossing write down the corresponding generator with exponent 1 if the arc is entering the crossing and 1 if it is leaving it. For all crossings do that in the same cyclic order ("left" or "right"). If x_{i} is an incoming generator, x_{o} is an outgoing, and x_{p} is a passing generator, we have the relation x_{i}x_{p}x_{o}^{1}x_{p}^{1} = e (in a "right" cycle order). Beginning from a generator other then x_{i} we obtain a conjugate of that relation.
