Formal Knot Theory
This exploration of combinatorics and knot theory is geared toward advanced undergraduates and graduate students. The author, Louis H. Kauffman, is a professor in the Department of Mathematics, Statistics, and Computer Science at the University of Illinois at Chicago. Kauffman draws upon his work as a topologist to illustrate the relationships between knot theory and statistical mechanics, quantum theory, and algebra, as well as the role of knot theory in combinatorics.
Featured topics include state, trails, and the clock theorem; state polynomials and the duality conjecture; knots and links; axiomatic link calculations; spanning surfaces; the genus of alternative links; and ribbon knots and the Arf invariant. Key concepts are related in easy-to-remember terms, and numerous helpful diagrams appear throughout the text. The author has provided a new supplement, entitled "Remarks on Formal Knot Theory," as well as his article, "New Invariants in the Theory of Knots," first published in The American Mathematical Monthly, March 1988.
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Alexander matrix Alexander polynomial algorithm alternating knot alternating link ambient isotopy Arf invariant ARF(K axioms black holes boundary bounded region calculation Clock Theorem clockwise moves column of index combinatorial completes the proof connected sum Conway polynomial corresponding counterclocked counterclockwise moves crossing cusps defined Definition denote the number disk edge example exchange Figgre Figure fixed stars formula genus geometry given graph Hence identity immersion induction interaction rules introduction Jordan curves Jordan trail knot diagram knot or link knot theory knots and links labelling Lemma link diagram linking number markers Math mathematics midline mirror image moves of type obtained oriented universe pass-equivalent permutation plane Proposition reassemblies Remark ribbon knot riders Seifert circles Seifert pairing Seifert surface sequence shell composition slice knot space spanning surface split strands string with sites topology transpositions tree trefoil knot unknot vertex vertices vK(z white holes