Presentation of a Knot Group
In this chapter we return to knot theory. The major objective here is the description and verification of a procedure for deriving from any polygonal knot K in regular position two presentations of the group of K, which are called respectively the over and under presentations. The classical Wirtinger presentation is obtained as a special case of the over presentation. In a later section we calculate over presentations of the groups of four separate knots explicitly, and the final section contains a proof of the existence of nontrivial knots, in that it is shown that the clover-leaf knot can not be untied. The fact that our basic description in this chapter is concerned with a pair of group presentations represents a concession to later theory. It is of no significance at this stage. One presentation is plenty, and, for this reason, Section 4 is limited to examples of over presentations. The existence of a pair of over and under presentations is the basis for a duality theory which will be exploited in Chapter IX to prove one of the important theorems.
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