Fundamental Concepts of Knot Theory
A knot, succinctly, is an entwined circle. However, throughout this book we shall think of a knot as an entwined polygon in 3-dimensional space, Figure 1.0.1(a). The reason for this is that it allows us, with recourse to combinatorial topology,1 to exclude wild knots. For an example of a wild knot, consider the knot in Figure 1.0.1(b). Close to the point P, in a sense we may take this to be a “limit” point, the knot starts to cluster together in a concertina fashion. Therefore, in the vicinity of such a point particular care needs to be taken with the nature of the knot. We shall not in this exposition apply or work within the constraints of such (wild) knots. In fact, since wild knots are not that common, this will be the only reference to these kind of knots.
KeywordsFundamental Concept Polygonal Line Original Orientation Combinatorial Topology Borromean Ring
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