Louis H. Kauffman- Knot theory and the heuristics of functional integration
Louis H. Kauffman- Topics in combinatorial knot theory
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CHAPTER 1 Topics in combinatorial knot theory The purpose of this part of these notes is to record concisely a number of diff ere nt parts of basic kno t the ory . Muc h of thi s is relate d to ph ysi cal models and to functional integration, as some side comments along the way will show. Nevertheless, we develop the knot theory here as a chapter in pure combinatorial topology. 1. A. Coloring knots and graphs A knot is an embedding in three-dimensional space of a single circle. A link is an embeddi ng of a collection of circles. Two knot s (links) are ambient isotopic if there is a continuously varying family of embeddings connecting one to the other. Unless otherwise specified I shall deal only with tame knots and links. In a tame knot every point on the knot has a neighborhood in 3-space that is equiv alent to the standard (ball, arc-diameter) pair. Tame knots (links)can be represented up to ambient isotopy by piecewise linear knots and links. A link is piecewise linear if the embedding consist s in straigh t line segme nts. Thu s a piece wise linear link is an embedding of a collection of boundaries of n-gons (different n for different components and the n’s are not fixed). A piecewise linear knot (link) is made from “straight sticks.” Exercise. Show that any “knot” made with ≤ 5 straight sticks is necessarily unknot ted. (I. e. ambi ent isoto pic to a circle in a plane .) What is misleading about this picture? Reidemeister, in the 1920–1930’s, studied knots and links from the PL view- point. He defined PL-equivalence of knots and links via the following 3-dimensional move: 1
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8/3/2019 Louis H. Kauffman- Topics in combinatorial knot theory
http://slidepdf.com/reader/full/louis-h-kauffman-topics-in-combinatorial-knot-theory 130/159
8/3/2019 Louis H. Kauffman- Topics in combinatorial knot theory
http://slidepdf.com/reader/full/louis-h-kauffman-topics-in-combinatorial-knot-theory 131/159
8/3/2019 Louis H. Kauffman- Topics in combinatorial knot theory
http://slidepdf.com/reader/full/louis-h-kauffman-topics-in-combinatorial-knot-theory 132/159
8/3/2019 Louis H. Kauffman- Topics in combinatorial knot theory
http://slidepdf.com/reader/full/louis-h-kauffman-topics-in-combinatorial-knot-theory 133/159
8/3/2019 Louis H. Kauffman- Topics in combinatorial knot theory
http://slidepdf.com/reader/full/louis-h-kauffman-topics-in-combinatorial-knot-theory 134/159
8/3/2019 Louis H. Kauffman- Topics in combinatorial knot theory
http://slidepdf.com/reader/full/louis-h-kauffman-topics-in-combinatorial-knot-theory 135/159
8/3/2019 Louis H. Kauffman- Topics in combinatorial knot theory
http://slidepdf.com/reader/full/louis-h-kauffman-topics-in-combinatorial-knot-theory 136/159
8/3/2019 Louis H. Kauffman- Topics in combinatorial knot theory
http://slidepdf.com/reader/full/louis-h-kauffman-topics-in-combinatorial-knot-theory 137/159
8/3/2019 Louis H. Kauffman- Topics in combinatorial knot theory
http://slidepdf.com/reader/full/louis-h-kauffman-topics-in-combinatorial-knot-theory 138/159
8/3/2019 Louis H. Kauffman- Topics in combinatorial knot theory
http://slidepdf.com/reader/full/louis-h-kauffman-topics-in-combinatorial-knot-theory 139/159
8/3/2019 Louis H. Kauffman- Topics in combinatorial knot theory
http://slidepdf.com/reader/full/louis-h-kauffman-topics-in-combinatorial-knot-theory 140/159
8/3/2019 Louis H. Kauffman- Topics in combinatorial knot theory
http://slidepdf.com/reader/full/louis-h-kauffman-topics-in-combinatorial-knot-theory 141/159
8/3/2019 Louis H. Kauffman- Topics in combinatorial knot theory
http://slidepdf.com/reader/full/louis-h-kauffman-topics-in-combinatorial-knot-theory 142/159
8/3/2019 Louis H. Kauffman- Topics in combinatorial knot theory
http://slidepdf.com/reader/full/louis-h-kauffman-topics-in-combinatorial-knot-theory 143/159
8/3/2019 Louis H. Kauffman- Topics in combinatorial knot theory
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8/3/2019 Louis H. Kauffman- Topics in combinatorial knot theory
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8/3/2019 Louis H. Kauffman- Topics in combinatorial knot theory
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8/3/2019 Louis H. Kauffman- Topics in combinatorial knot theory
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8/3/2019 Louis H. Kauffman- Topics in combinatorial knot theory
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8/3/2019 Louis H. Kauffman- Topics in combinatorial knot theory
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8/3/2019 Louis H. Kauffman- Topics in combinatorial knot theory
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8/3/2019 Louis H. Kauffman- Topics in combinatorial knot theory
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8/3/2019 Louis H. Kauffman- Topics in combinatorial knot theory
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8/3/2019 Louis H. Kauffman- Topics in combinatorial knot theory
http://slidepdf.com/reader/full/louis-h-kauffman-topics-in-combinatorial-knot-theory 153/159
8/3/2019 Louis H. Kauffman- Topics in combinatorial knot theory
http://slidepdf.com/reader/full/louis-h-kauffman-topics-in-combinatorial-knot-theory 154/159
8/3/2019 Louis H. Kauffman- Topics in combinatorial knot theory
http://slidepdf.com/reader/full/louis-h-kauffman-topics-in-combinatorial-knot-theory 155/159
8/3/2019 Louis H. Kauffman- Topics in combinatorial knot theory
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8/3/2019 Louis H. Kauffman- Topics in combinatorial knot theory
http://slidepdf.com/reader/full/louis-h-kauffman-topics-in-combinatorial-knot-theory 157/159
8/3/2019 Louis H. Kauffman- Topics in combinatorial knot theory
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8/3/2019 Louis H. Kauffman- Topics in combinatorial knot theory
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