Glossary of Knot Theory
 Level: 0 1 2 3 4 5 6 7
MSC Classification: 57 (Manifolds)

This glossary contains terms relevant to the study of knot theory.

 Rough Guides to Topology
 General Topology Algebraic Topology Manifold Theory Knot Theory

# Main Glossary

## 012

$\Delta$-unknotting operation
def
$\#$-unknotting operation
def
$\mu$-invariant
def

$2$-bridge knot (link)
def
$3$-colorability
def
$4$-plat
def

## Aa

Acheiral knot
def
def
def
def
def
Alexander ideal
def
\inv Alexander invariant
def
Alexander module
def
Alexander polynomial
def
Alexander's horned sphere
def
def
def
def
Ambient Isotopy
def
def
def
Antoine's horned sphere
def
Antoine's necklace
def
def
\inv Arf invariant
def

## Bb

Bicollar
def
def
def
\inv Bracket polynomial
def
Braid
def
Braid closure
def
Braid group
def
\inv Braid index
def
Braid notation
def
Braid representation
def
def
Branched cover
def
\inv Bridge index/number
def
\inv Bridge length
def
def

## Cc

Cable knot
def
\inv Catalan number
def
Category
def
Chebyshev polynomial
def
Cinquefoil knot
def
Clasp
def
Classification (of knots)
def
def
Classification (of rational knots)
def
Classification (of rational tangles)
def
Classification (of tangles)
def
Closed braid
def
def
\inv Coloring number
def
Companion knot
def
\inv Complement (of a knot)
def
Component
def
\inv Component number
def
Composite knot
def
Concordance (of knots)
def
Connected sum (of knots)
def
Continued fraction
def
Conway notation
def
\inv Conway polynomial
def
Conway sphere
def
Covering space
def
\inv Crookedness (of a knot)
def
Crossing/crossover
def
Crossing change
def
\inv Crossing number
def
Cyclic branched cover
def

## Dd

Dehn surgery
def
Dehn's homology sphere
def
Dehn's lemma
def
Delta-unknotting operation
def
Denominator (of a tangle)
def
\inv Determinant (of a knot)
def
def
Diagram (of a tangle)
def
DNA topology
def
Double branched cover
def
Doubled knot
def
Dowker notation
def

def
Essential (loop)
def
Exceptional
def

## Ff

\inv Fibered knot
def
Figure-eight knot
def
Flat (embedding)
def
Flype
def
Foliation
def
Four-plat
def
Fourier knot
def
Framing (of a knot)
def
Framing (of a torus)
def
Fundamental group (of a knot)
def
Fundamental theorem of surgery
def

## Gg

Generalized unknotting number
def
Generator (of a group)
def
\inv Genus (of a knot)
def
Genus (of $3$-manifolds
def
\inv Gordian distance/number
def
Granny knot
def
Group (of a knot)
def

## Hh

Handlebody
def
Handle surgery
def
Harmonic knot
def
Heegaard splitting/surface
def
Higher-dimensional knot
def
Homeomorphism
def
\inv HOMFLY polynomial
def
Homology/homotopy sphere
def
def
Horned sphere
def
def

## IiJjKk

Incompressible torus
def
def
Irreducible knot
def
Isotopy
def

Jones polynomial
def
Jordan curve theorem
def

\inv Kauffman bracket
def
\inv Kauffman polynomial
def
\inv Kauffman bracket skein module
def
Khovanov homology
def
Kinoshita-Terasaka knot
def
Kirby moves
def
Knot
A "loop" in space: a $1$-dimensional connected PL-submanifold of $S^3$.
Knot table
def

## Ll

\inv Laurent polynomial
def
Lens Space
The space obtained by gluing two solid tori together by a homeomorphism along their boundaries.
Several "loops" in space: an arbitrary $1$-dimensional PL-submanifold of $S^3$.
def
Locally flat
def
Locally [un]knotted tangle
def
Longitude (of a knot)
def
Longitude (of a torus)
def
Loop Theorem
A homotopy element nontrivial in $\del M$ but trivial in $M$ can be represented by a simple closed curve in $\del M$. This statement can be made slightly more general.
Lover's knot
def

## Mm

Mapping class group
def
Markov moves
def
Meridian (of a torus)
def
def
Monodromy
def
def
Multi-$\#$ unknotting operation
def
Mutation (of a knot)
def

## NnOo

\inv $n$-colorability (of a knot)
def
Non-cancellation theorem
def
Nugatory crossing
def
Numerator (of a tangle)
def

Obverse (of a knot)
def
Orientable double cover
def
Orientation
def
def

## PpQq

Partial sum (of tangles)
def
Pattern knot
def
Peripheral torus
def
Perko pair
def
Piecewise linear
def
PL
Short for piecewise linear.
Plumbing
def
Poincar/'e conjecture
def
Poincar/'e manifold
def
def
\inv Polynomial (of a knot)
def
def
def
Prime knot
def
Prime tangle
def
Projection (of a knot)
def
Property P
def

Quantum $SU_q(2)$ invariants
def

## Rr

$r$-parallel link
def
Rational decomposition (of knots)
def
def
Rational tangle
def
Reduced knot
def
\inv Reducible knot
def
Reef knot
def
Reflection (of a knot)
def
Regular isotopy
def
Regular projection
def
Reidemeister moves
def
def
Removable crossing
def
Reverse (of a knot)
def
Reversible (knot)
def
def
Ribbon-slice conjecture
def

## Ss

Satellite knot
def
SCC
Short for simple closed curve.
Sch\:onflies theorem
def
Seifert circuits
def
Seifert form/matrix
def
Seifert surface
A surface in $S^3$ whose boundary is a knot (this exists for every knot).
Sign (of a crossing)
def
\inv Signature (of a knot)
def
Singular
def
Singular knot
def
Skein formula/relation
def
\inv Slice knot
def
Slicing disk
def
Slope (of a surgery)
def
Smith conjecture
def
Smooth
def
Sphere Theorem
If $\pi_2(M) \neq 0$, then there is a noncontractible sphere embedded in $M$.
Spinning
def
def
Split diagram
def
Spun knot
def
\inv Square bracket polynomial
def
Square knot
def
Standard position
def
State (of a knot diagram)
def
Stevedore's knot
def
\inv Stick number
def
\inv Strand passage distance
def
Strongly prime (knot diagram)
def
Sum (of knots)
def
Sum (of tangles)
def
Surgery
def
Suspended knot
def
Suspension
def

## Tt

def
Tangle
def
\inv Tangle distance
def
Tangle surgery
def
\inv Three-colorability
def
Torsion number
def
\inv Torsion invariant
def
def
def
Trefoil knot
def
\inv Tri-colorability
def
Trivial knot
def
def
Trivial tangle
def
\inv Tunnel number
def
def
Twist-equivalent
def
Twist-spun knot
def
Twisted double
def
def

## UuVvWw

Universal covering
def
Unknot
def
\inv Unknotting number
def
def
Unknotting operation
def
def
def

\inv Vassiliev invariants
def
Viergeflechte
def
\inv Volume (of a knot)
def

\inv $w$-signature
def
def
def
def
def
def
Wirtinger presentation
def
\inv Writhe (of a knot)
def

# Theorems

In this section, we give some important theorems used in knot theory. Where possible, the statements are given in language rather than notation. In some cases we give two versions of the theorem, one “intuitive” and the other “semantic,” the second usually including extra information. An indication of the proof is also given or, if not, a reference.

## Embedding Theorems

The following theorems give ways to translate algebraic information into geometric information. Basically, if the fundamental group is nontrivial (in dimensions one or two), they tell how to find a representative of the group which is an embedding, namely one-to-one.

Dehn's Lemma
If a disk $D$ is mapped into a \thrmm, such that its boundary is mapped injectively into $M$, then the map can be adjusted on the interior of the disk to make it an embedding.
Proof: corollary of the Loop Theorem.
Loop Theorem
Intuitive: A homotopy element nontrivial in $\del M$ but trivial in $M$ can be represented by a simple closed curve in $\del M$.
Semantic: If $N$ is a normal subgroup of $\pi_1(S)$ for a surface $S \subset \del M$ and $\ker(\pi_1(S)\to\pi_1(M)) \not \subset N$, then there is a proper embedding of the disk in $M$ whose boundary is not an element of $N$.
Sphere Theorem
Intuitive: If $\pi_2(M) \neq 0$, then there is a noncontractible sphere embedded in $M$.
Semantic: If $N$ is a $\pi_1(M)$ invariant subgroup of $\pi_2(M)$ with $\pi_2(M) \not \subset N$, then there is an embedding of the sphere in $M$ which is not an element of $N$.

## Classification Theorems

Theorem (lens space classification)
The lens spaces $L(p,q)$ are classified by (i) $L(1,q)=S^3$ and (ii) $L(p,q)=L(p,q')$ if either $q \equiv \pm q' \mod p$ or $qq' \equiv \pm 1 \mod p.$
Proof: ask when a homeomorphism on the boundary of a torus can be extended to a homeomorphism on the whole torus.

# Knot Theorists

Author of The Knot Book. Has made significant contributions to the connections between knot theory and hyperbolic geometry.
Alexander, J
def
Artin, Emil
Credited with inventing braids.
Conway, John
def
Dehn, ??
def
Heegaard, ??
def
Jones, Vaughan
def
Kauffman, Louis
Author of Knots and Physics and several other books on knot theory. Namesake of the Kauffman bracket and the Kauffman bracket skein module.
Khovanov
def
Lickorish, ??
def
Lord Kelvin
def
Perko, Kenneth
def
Poincare/'e, ??
def
Przytycki, Jozef
def
Reidemeister, ??
def
Rolfsen, Dale
Seifert, ??
def
Tait, P
def
Thurston, William
def
Wirtinger, K
def

# References

1. D. Rolfsen, Knots and Links.
2. Lickorish.
3. Colin Adams, The Knot Book.
4. Burde and Zieschang.
page revision: 14, last edited: 19 Sep 2009 23:24