Knots and LinksKnots and links are studied by mathematicians, and are also finding increasing application in chemistry and biology. Many naturally occurring questions are often simple to state, yet finding the answers may require ideas from the forefront of research. This readable and richly illustrated 2004 book explores selected topics in depth in a way that makes contemporary mathematics accessible to an undergraduate audience. It can be used for upper-division courses, and assumes only knowledge of basic algebra and elementary topology. Together with standard topics, the book explains: polygonal and smooth presentations; the surgery equivalence of surfaces; the behaviour of invariants under factorisation and the satellite construction; the arithmetic of Conway's rational tangles; arc presentations. Alongside the systematic development of the main theory, there are discussion sections that cover historical aspects, motivation, possible extensions, and applications. Many examples and exercises are included to show both the power and limitations of the techniques developed. |
Contents
Introduction | 1 |
1 | 11 |
A Topologists Toolkit | 32 |
3 | 40 |
13 | 50 |
Link Diagrams | 51 |
16 | 64 |
20 | 70 |
Rational Tangles | 189 |
90 | 194 |
HR 2222 | 201 |
31 | 208 |
More Polynomials | 215 |
Closed Braids and Arc Presentations | 241 |
Appendix A Knot Diagrams | 286 |
93 | 289 |
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Common terms and phrases
1-cycle 2-tangles adequate diagram Alexander polynomial algebraic algorithm alternating diagram amphicheiral arc presentation axis boundary braid index braid presentation cheiral closed braid coefficients coloured components construction continued fraction Conway polynomial Corollary crossing number defined denote disc edge embedded endpoints equivalent example Exercise factorisation finite genus genus-1 H₁(F half-twists hence homeomorphism Homfly polynomial homology integers intersection isotopic Jones polynomial knot theory labelled Lemma linear link diagram link invariant linking number loops Math minimal number of crossings orientable surface oriented link plane points polygon positive produce projection surface PROOF rational knots rational link rational tangles Reidemeister moves satellite Seifert circles Seifert graph Seifert matrix sequence Show shown in Figure signature skein relation spanning surface sphere split link strand strings surface F surface spanning Theorem topology torus knot trefoil trivial knot trivial link twist twist-boxes unknotting unknotting number vertex vertices zero