2.1.1  Group of KL

Every finitely presented  group can be given by a presentation: set of generators x1,...,xm and their relations r1,...,rn. For example, a cyclic group Cn will be given by the set of generators {x} and a single relation xn = e, where n is the order of the group Cn, and e is the identity. A dihedral group will be given by the set of generators {x1,x2} satisfying the relations x1n = x22 = (x1x2)2 = e, or by the set of generators {y1,y2} satisfying the relations y12 = y22 = (y1y2)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 y1 = x1x2 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 

(x1,x2,x3:     x13 = x22 = x32 = x1x2x3-1 = e)
can be reduced to 
(x1,x2:     x13 = x22 = (x1x2)2 = e)
which is a minimal presentation of the group D3

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) = (x1,x2,...,xm: r1,...rn), where x1,...,xm are generators, r1,...,rn 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 xi is an incoming generator, xo is an outgoing, and xp is a passing generator, we have the relation xixpxo-1xp-1 = e (in a "right" cycle order). Beginning from a generator other then xi we obtain a conjugate of that relation. 

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