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On Knots is a journey through the theory of knots, starting from the simplest combinatorial ideas--ideas arising from the representation of weaving patterns. From this beginning, topological invariants are constructed directly: first linking numbers, then the Conway polynomial and skein theory. This paves the way for later discussion of the recently discovered Jones and generalized polynomials. The central chapter, Chapter Six, is a miscellany of topics and recreations. Here the reader will find the quaternions and the belt trick, a devilish rope trick, Alhambra mosaics, Fibonacci trees, the topology of DNA, and the author's geometric interpretation of the generalized Jones Polynomial.Then come branched covering spaces, the Alexander polynomial, signature theorems, the work of Casson and Gordon on slice knots, and a chapter on knots and algebraic singularities.The book concludes with an appendix about generalized polynomials.

Knot theory --- Knots (Topology) --- Low-dimensional topology --- Knot theory. --- Algebraic topology --- 3-sphere. --- Addition theorem. --- Addition. --- Alexander polynomial. --- Algebraic variety. --- Algorithm. --- Ambient isotopy. --- Arf invariant. --- Basepoint. --- Bijection. --- Bilinear form. --- Borromean rings. --- Bracket polynomial. --- Braid group. --- Branched covering. --- Chiral knot. --- Chromatic polynomial. --- Cobordism. --- Codimension. --- Combination. --- Combinatorics. --- Complex analysis. --- Concentric. --- Conjecture. --- Connected sum. --- Conway polynomial (finite fields). --- Counting. --- Covering space. --- Cyclic group. --- Dense set. --- Determinant. --- Diagram (category theory). --- Diffeomorphism. --- Dimension. --- Disjoint union. --- Disk (mathematics). --- Dual graph. --- Elementary algebra. --- Embedding. --- Enumeration. --- Existential quantification. --- Exotic sphere. --- Fibration. --- Formal power series. --- Fundamental group. --- Geometric topology. --- Geometry and topology. --- Geometry. --- Group action. --- Homotopy. --- Integer. --- Intersection form (4-manifold). --- Isolated singularity. --- Jones polynomial. --- Knot complement. --- Knot group. --- Laws of Form. --- Lens space. --- Linking number. --- Manifold. --- Module (mathematics). --- Morwen Thistlethwaite. --- Normal bundle. --- Notation. --- Obstruction theory. --- Operator algebra. --- Pairing. --- Parity (mathematics). --- Partition function (mathematics). --- Planar graph. --- Point at infinity. --- Polynomial ring. --- Polynomial. --- Quantity. --- Rectangle. --- Reidemeister move. --- Remainder. --- Root of unity. --- Saddle point. --- Seifert surface. --- Singularity theory. --- Slice knot. --- Special case. --- Statistical mechanics. --- Substructure. --- Summation. --- Symmetry. --- Theorem. --- Three-dimensional space (mathematics). --- Topological space. --- Torus knot. --- Trefoil knot. --- Tubular neighborhood. --- Underpinning. --- Unknot. --- Variable (mathematics). --- Whitehead link. --- Wild knot. --- Writhe. --- Variétés topologiques --- Topologie combinatoire --- Theorie des noeuds

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This book develops some of the extraordinary richness, beauty, and power of geometry in two and three dimensions, and the strong connection of geometry with topology. Hyperbolic geometry is the star. A strong effort has been made to convey not just denatured formal reasoning (definitions, theorems, and proofs), but a living feeling for the subject. There are many figures, examples, and exercises of varying difficulty.

Topology --- Differential geometry. Global analysis --- Geometry, Hyperbolic --- Three-manifolds (Topology) --- Géométrie hyperbolique --- Variétés topologiques à 3 dimensions --- Geometry, Hyperbolic. --- 514.1 --- 3-manifolds (Topology) --- Manifolds, Three dimensional (Topology) --- Three-dimensional manifolds (Topology) --- Low-dimensional topology --- Topological manifolds --- Hyperbolic geometry --- Lobachevski geometry --- Lobatschevski geometry --- Geometry, Non-Euclidean --- General geometry --- Three-manifolds (Topology). --- 514.1 General geometry --- Géométrie hyperbolique --- Variétés topologiques à 3 dimensions --- 3-sphere. --- Abelian group. --- Affine space. --- Affine transformation. --- Atlas (topology). --- Automorphism. --- Basis (linear algebra). --- Bounded set (topological vector space). --- Brouwer fixed-point theorem. --- Cartesian coordinate system. --- Characterization (mathematics). --- Compactification (mathematics). --- Conformal map. --- Contact geometry. --- Curvature. --- Cut locus (Riemannian manifold). --- Diagram (category theory). --- Diffeomorphism. --- Differentiable manifold. --- Dimension (vector space). --- Dimension. --- Disk (mathematics). --- Divisor (algebraic geometry). --- Dodecahedron. --- Eigenvalues and eigenvectors. --- Embedding. --- Euclidean space. --- Euler number. --- Exterior (topology). --- Facet (geometry). --- Fiber bundle. --- Foliation. --- Fundamental group. --- Gaussian curvature. --- Geometry. --- Group homomorphism. --- Half-space (geometry). --- Holonomy. --- Homeomorphism. --- Homotopy. --- Horocycle. --- Hyperbolic geometry. --- Hyperbolic manifold. --- Hyperbolic space. --- Hyperboloid model. --- Interior (topology). --- Intersection (set theory). --- Isometry group. --- Isometry. --- Jordan curve theorem. --- Lefschetz fixed-point theorem. --- Lie algebra. --- Lie group. --- Line (geometry). --- Linear map. --- Linearization. --- Manifold. --- Mathematical induction. --- Metric space. --- Moduli space. --- Möbius transformation. --- Norm (mathematics). --- Pair of pants (mathematics). --- Piecewise linear manifold. --- Piecewise linear. --- Poincaré disk model. --- Polyhedron. --- Projection (linear algebra). --- Projection (mathematics). --- Pseudogroup. --- Pullback (category theory). --- Quasi-isometry. --- Quotient space (topology). --- Riemann mapping theorem. --- Riemann surface. --- Riemannian manifold. --- Sheaf (mathematics). --- Sign (mathematics). --- Simplicial complex. --- Simply connected space. --- Special linear group. --- Stokes' theorem. --- Subgroup. --- Subset. --- Tangent space. --- Tangent vector. --- Tetrahedron. --- Theorem. --- Three-dimensional space (mathematics). --- Topological group. --- Topological manifold. --- Topological space. --- Topology. --- Transversal (geometry). --- Two-dimensional space. --- Uniformization theorem. --- Unit sphere. --- Variable (mathematics). --- Vector bundle. --- Vector field. --- Topologie algébrique --- Topologie combinatoire --- Algebraic topology. --- Combinatorial topology. --- Variétés topologiques --- Geometrie --- Theorie des noeuds