Knot Theory |

**In General** > s.a. Braids; knots in physics;
Link Theory.

* __Idea__: A knot is a continuous map
from S^{1} 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, π_{1}(*S*–*k*);
A presentation is generated by *x*_{1},
..., *x*_{n} (loops based
at some fixed *p* ∈ *S* and
winding – like a right-handed screw – around
each overpass), subject to relations *r*_{1},
..., *r*_{n–1},
where *r*_{i} is
*x*_{i} *x*_{k}
= *x*_{k} *x*_{i+1},
where the *k*-th overpass goes between the *i*-th and the (*i*+1)-th,
or *x*_{i} = *x*_{i+1} if
no such *x*_{k} exists.
__Examples__: π_{1}(\(\mathbb R\)^{3} – unknot)
= \(\mathbb Z\); π_{1}(\(\mathbb R\)^{3} – trefoil)
= {*a*, *b* | *aba* = *bab*}, non-Abelian;
π_{1}(\(\mathbb R\)^{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) [S^{3}, 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.

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**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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send feedback and suggestions to bombelli at olemiss.edu – modified 28 jul 2017