Knot
Theory| Knot
Theory |
In General > s.a. Link
Theory.
* Idea: A knot is a continuous map from S1 to a 3-manifold S,
usually R3.
* Equivalence: Two knots k and k'
are equivalent iff there is a homeomorphism (diffeomorphism) h with h(k)
= k' (including
reflections!); Note:
In R3,
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, 
1(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: 1(R3 – unknot)
= Z; 1(R3 – trefoil)
= {a, b | aba = bab}, non-Abelian;
1(R3 – 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 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; knots in physics; types
of distances [between
knots/links].
* 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.
* Reidemeister moves:
There are three,
@ 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].
@ 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].
Generalizations > s.a. Ribbons.
* Virtual knot theory:
A generalization 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.
* 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].
@ 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 qp/04;
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.
@ I: Corrigan PW(93)jun; Menasco & Rudolph AS(95);
Fink & Mao 99 [and
tie knots]; in Casti 00.
@ 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];
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].
Main page – Abbreviations – Journals – Comments – Other
sites – Acknowledgements
Send feedback and suggestions to bombelli at olemiss.edu – Modified
12 may 2008