Knot Theory| 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 the identity.
* Continuity: It is often convenient
to work with polygonal or tame knots, equivalent to polygonal ones.
* Knot group: The fundamental group
of its complement, π1(S−k);
A presentation is generated by x1, ...,
xn (loops based at some fixed
p ∈ S 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 G&T(17)-a1511,
Greene DMJ(17)-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];
Gukov et al a2010 [natural language processing].
@ 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.
main page
– abbreviations
– journals – comments
– other sites – acknowledgements
send feedback and suggestions to bombelli at olemiss.edu – modified 7 nov 2020