knot theory

knot theory

Knot theory

is the study of

knots

and

links

.

Roughly a

knot

is a

simple closed curve

in

ℝ

3

, and two knots are considered

equivalent

if and only if one can be smoothly deformed into another. This will often be used as a working definition as it is simple and appeals to intuition. Unfortunately this definition can not be taken too seriously because it includes many pathological cases, or

wild knots

, such as the

connected sum

of an

infinite

number of trefoils. Furthermore one must be careful about defining a “smooth deformation”, or all knots might turn out to be equivalent! (We shouldn’t be allowed to shrink part of a knot down to nothing.)

Links are defined in terms of knots, so once we have a definition for knots we have no trouble defining them.

Definition

.

A

link

is a set of

disjoint

knots.

Each knot is a

component

of the link. In particular a knot is a link of one component.

Luckily the knot theorist is not usually interested in the exact form of a knot or link, but rather the in its

equivalence class

. (Even so a possible formal definition for knots is given at the end of this entry.) All the “interesting” information about a knot or link can be described using a

knot diagram

. (It should be noted that the words “knot” and “link” are often used to mean an equivalence class of knots or links respectively. It is normally clear from

context

if this usage is intended.)

A knot diagram is a

projection

of a link onto a plane such that no more than two points of the link are projected to the same point on the plane and at each such point it is indicated which strand is closest to the plane (usually by erasing part of the lower strand). This can best be explained with some examples:

some knot diagrams

Two different knot diagrams may both represent the same knot — for example the last two diagrams both represent the unknot, although this is not obvious. Much of knot theory is devoted to telling when two knot diagrams represent the same link. In one sense this problem is easy — two knot diagrams represent equivalent links if and only if there exists a

sequence

of

Reidemeister moves

transforming one diagram to another.

Definition

.

A

Reidemeister move

consists of modifying a portion of knot diagram in one of the following ways:

1.

A single strand may be twisted, adding a crossing of the strand with itself.

2.

A single twist may be removed from a strand removing a crossing of the strand with itself.

3.

When two strands run

parallel

to each other one may be pushed under the other creating to new over-crossings.

4.

When one strand has two consecutive over-crossings with another strand the strands may be straightened so that the two run parallel.

5.

Given three strands

A

,

B

and

C

so that

A

passes below

B

and

C

,

B

passes between

A

and

C

, and

C

passes above

A

and

B

, the strand

A

may be moved to either side of the crossing of

B

and

C

.

Note that number 1. is the

inverse

of number 2. and number 3. is the inverse of number 4. Number 5 is its own inverse. In pictures:

⟷

⟷

⟷

Finding such a sequence of Reidemeister moves is generally not easy, and proving that no such sequence exists can be very difficult, so other approaches must be taken.

Knot theorists have accumulated a large number of

knot invariants

, values associated with a knot diagram which are unchanged when the diagram is modified by a Reidemeister move. Two diagrams with the same

invariant

may not represent the same knot, but two diagrams with different invariant never represent the same knot.

Knot theorists also study ways in which a complex knot may be described in terms of simple pieces — for example every knot is the connected sum of non

trivial

prime knots

and many knots can be described simply using Conway notation.

formal definitions of

knot

polygonal knots

This definition is used by Charles Livingston in his book

Knot Theory

. It avoids the problem of wild knots by restricting knots to piece-wise linear (polygonal) curves. Every knot that is intuitively “tame” can be approximated by such knot. We also define the

vertices

,

elementary deformation

, and

equivalence

of knots.

Definition

.

A

knot

is a simple closed polygonal curve in

𝕊

3

.

Definition

.

The vertices of a knot are the smallest ordered set of points such that the knot can be constructed by connecting them.

Definition

.

A knot

J

is an

elementary deformation

of a knot

K

if one is formed from the other by adding a single vertex

v

0

not on the knot such that the

triangle

formed by

v

0

together with its

adjacent vertices

v

1

and

v

2

intersects

the knot only along the segment

[

v

1

,

v

2

]

.

Definition

.

A knot

K

0

is

equivalent

to a knot

K

n

if there exists a sequence of knots

K

1

,

…

,

K

n

-

1

such that

K

i

is an elementary deformation of

K

i

-

1

for

1

<

i

≤

n

.

smooth submanifold

This definition is used by Raymond Lickorish in

An

Introduction

to Knot Theory

.

Definition

.

A

link

is a smooth one dimensional

submanifold

of the 3-sphere

S

3

. A

knot

is a link consisting of one component.

Definition

.

Links

L

1

and

L

2

are defined to be equivalent if there is an

orientation-preserving

homeomorphism

h

:

S

3

→

S

3

such that

h

⁢

(

L

1

)

=

L

2

.

Title

knot theory

Canonical name

KnotTheory

Date of creation

2013-03-22 13:17:27

Last modified on

2013-03-22 13:17:27

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

9

Author

CWoo (3771)

Entry type

Topic

Classification

msc 57M25

Related topic

Unknot

Related topic

ConnectedSum

Related topic

Tangle

Related topic

JonesPolynomialKnotTheory

Defines

knot

Defines

link

Defines

knot diagram

Defines

Reidemeister moves

Defines

vertices

Defines

elementary deformation

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