Knot Theory  

In General > s.a. Braids; knots in physics; Link Theory.
* Idea: A knot is a continuous map from S1 to a 3-manifold S, usually \(\mathbb R\)3.
* Equivalence: Two knots k and k' are equivalent iff there is a homeomorphism (diffeomorphism) h with h(k) = k' (including reflections!); Note: In \(\mathbb R\)3, any orientation-preserving homeomorphism is isotopic to id.
* Continuity: Often convenient to work with polygonal or tame knots, equivalent to polygonal ones.
* Knot group: The fundamental group of its complement, π1(Sk); A presentation is generated by x1, ..., xn (loops based at some fixed pS and winding – like a right-handed screw – around each overpass), subject to relations r1, ..., rn–1, where ri is xi xk = xk xi+1, where the k-th overpass goes between the i-th and the (i+1)-th, or xi = xi+1 if no such xk exists. Examples: π1(\(\mathbb R\)3 – unknot) = \(\mathbb Z\); π1(\(\mathbb R\)3 – trefoil) = {a, b | aba = bab}, non-Abelian; π1(\(\mathbb R\)3 – square knot) = {a, b, c | aba = bab, aca = cac}. Notes: Inequivalent knots can have the same knot group (e.g. square knot and granny knot), and of course we have the word problem; The abelianization of the knot group is always \(\mathbb Z\).
* Nice projection: Gives a finite number of intersections of two lines at a time (at non-zero angles and without kinks at the intersections).
* Seifert surface: A compact, connected, orientable 2-surface with the knot k as its boundary; It always exists (non-unique in fact); A knot is the unknot if it can be spanned by a (tame) disk.
* Genus: The smallest genus of a Seifert surface for a knot; The genus g(k) is additive.

Special Types of Knots and Related Topics > s.a. knot invariants; Reidemeister Moves; types of distances [between knots/links]; types of orders.
* Unknot: The usual, unknotted loop.
* Other knots: Trefoil knot; Square knot; Granny knot.
* Braided knot: One in which all lines move around a point (axis) in the same direction; All knots can be braided.
* Torus knot: A knot that can live on the surface of a torus.
* Alternating knot: One that has an alternating diagram, a diagram in which crossings alternate over and under along the knot.
@ Torus knots: Etnyre G&T(99) [transversal]; Labastida & Mariño CMP(01) [invariants]; > s.a. knot invariants.
@ On other manifolds: Greene & Wiest G&T(98) [S3, framing]; Kalfagianni Top(98) [irreducible 3-manifolds]; Christensen Top(98) [lens spaces]; Matsuda T&A(04) [small knots on Haken 3-manifolds]; Cattabriga et al T&IA(13) [lens spaces].
@ Reidemeister moves: Carter et al T&A(06) [number of type III].
@ Related topics: Churchard & Spring T&A(90) [classifying]; Jones 93 [subfactors]; Birman & Hirsch G&T(98) [recognizing the unknot]; Askitas & Kalfagianni T&A(02) [knot adjacency]; Ozsváth & Szabó Top(05) [unknotting no 1, and Heegaard Floer homology]; Morishita 09 [and number theory, arithmetic topology]; Howie a1511, Greene a1511 + Moskovich blog(15)nov [alternating knots, characterization].

Generalizations > s.a. Ribbons.
* Virtual knot theory: A generalization, discovered by Louis Kauffman in 1996, of knot theory to the study of all oriented Gauss codes (classical knot theory is a study of planar Gauss codes); It studies non-planar Gauss codes via knot diagrams with virtual crossings; It helps better understand some aspects of classical knot theory; > s.a. knot invariants.
* s-knot: A diffeomorphism equivalence class of embedded spin networks.
@ Virtual knots: Kauffman EJC(99)m.GT/98; Goussarov et al Top(00) [invariants]; Fenn & Turaev JGP(07) [and Weyl algebras]; Manturov & Ilyutko 12.
@ Intersecting loops: Armand-Ugón et al PLB(93)ht/92, Grot & Rovelli JMP(96)gq [invariants].
@ Higher-dimensional: Hillman 89 [2-knots]; Ng Top(98) [groups of ribbon knots]; Cattaneo & Rossi CMP(05)mp/02 [invariants from BF theory].
@ Quantum knots: Kauffman & Lomonaco SPIE(04)qp; Lomonaco & Kauffman a0805 [and mosaics].
@ Other generalizations: Griego NPB(96)gq [extended knots]; Gambini et al PLB(98) [spin networks].

References > s.a. algebraic topology; Theta Functions; types of homology.
@ I: Corrigan PW(93)jun; Menasco & Rudolph AS(95); Fink & Mao 99 [and tie knots]; in Casti 00; Kawauchi & Yanagimoto ed-12 [from elementary to high school].
@ Books, II: in Armstrong 83; Adams 94; Gilbert & Porter 95.
@ Books and reviews: Reidemeister 32; Birman 74; Neuwirth 75; Crowell & Fox 77; in Fenn 83; Kauffman 83 [formal]; Eisenbud & Neumann 85; Kauffman 87; Birman BAMS(88) ["book review"]; Birman BAMS(93); Kawauchi 96 [survey]; Lickorish 97; in Kauffman 01; Burde & Zieschang 03; Cromwell 04 [and links].
@ Books, IV: Rolfsen 76; Yetter 01 [categories of tangles, etc].
@ Computational aspects: Millett & Sumners ed-94 [random knots/links]; Aneziris 99; Arsuaga et al JPA(07) [generating large knots].
@ History: Turner & van de Griend ed-96; Silver AS(06).
@ Representations of knot group: Frohman Top(93) [unitary].
> Online resources: see Ho Hon Leung's lessons; Wikipedia page.


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