Fun with knots: Invariants and applications

Mark Ioppolo

School of Mathematics and Statistics University of Western Australia

August 18, 2011

Mark Ioppolo Fun with knots: Invariants and applications 1867: Mathematician PG Tait demonstrated that vortex lines in a fluid (lines along ∇ × ~v) can form loops. Physicist Lord Kelvin proposed atoms were vortices in the ether, ie: knotted electric field lines.

Figure: Vortices in the air around aeroplane wings

Origins of knot theory

Early 19th century: Maxwell’s equations suggested light was an electromagnetic wave travelling in the 0luminiferous ether0. Physicist Lord Kelvin proposed atoms were vortices in the ether, ie: knotted electric field lines.

Figure: Vortices in the air around aeroplane wings

Origins of knot theory

Early 19th century: Maxwell’s equations suggested light was an electromagnetic wave travelling in the 0luminiferous ether0. 1867: Mathematician PG Tait demonstrated that vortex lines in a fluid (lines along ∇ × ~v) can form loops. Figure: Vortices in the air around aeroplane wings

Origins of knot theory

Early 19th century: Maxwell’s equations suggested light was an electromagnetic wave travelling in the 0luminiferous ether0. 1867: Mathematician PG Tait demonstrated that vortex lines in a fluid (lines along ∇ × ~v) can form loops. Physicist Lord Kelvin proposed atoms were vortices in the ether, ie: knotted electric field lines. Origins of knot theory

Early 19th century: Maxwell’s equations suggested light was an electromagnetic wave travelling in the 0luminiferous ether0. 1867: Mathematician PG Tait demonstrated that vortex lines in a fluid (lines along ∇ × ~v) can form loops. Physicist Lord Kelvin proposed atoms were vortices in the ether, ie: knotted electric field lines.

Figure: Vortices in the air around aeroplane wings A semi-formal definition: a knot is a simple closed curve in either 3 3 4 R or S = {x ∈ R : ||x|| = 1}

A link is a disjoint union of knots

Knots are covered more formally in Baez and Munian’s Gauge Fields, Knots and Gravity. Note: much of the material from this talk has been stolen borrowed from this book.

What is a knot?

Intuitively: a knot is a tangle of string sitting in 3-space with its ends glued together. A link is a disjoint union of knots

Knots are covered more formally in Baez and Munian’s Gauge Fields, Knots and Gravity. Note: much of the material from this talk has been stolen borrowed from this book.

What is a knot?

Intuitively: a knot is a tangle of string sitting in 3-space with its ends glued together.

A semi-formal definition: a knot is a simple closed curve in either 3 3 4 R or S = {x ∈ R : ||x|| = 1} Knots are covered more formally in Baez and Munian’s Gauge Fields, Knots and Gravity. Note: much of the material from this talk has been stolen borrowed from this book.

What is a knot?

Intuitively: a knot is a tangle of string sitting in 3-space with its ends glued together.

A semi-formal definition: a knot is a simple closed curve in either 3 3 4 R or S = {x ∈ R : ||x|| = 1}

A link is a disjoint union of knots What is a knot?

Intuitively: a knot is a tangle of string sitting in 3-space with its ends glued together.

A semi-formal definition: a knot is a simple closed curve in either 3 3 4 R or S = {x ∈ R : ||x|| = 1}

A link is a disjoint union of knots

Knots are covered more formally in Baez and Munian’s Gauge Fields, Knots and Gravity. Note: much of the material from this talk has been stolen borrowed from this book. Example 1: The figure 8

Figure: A figure 8 knot in 3-space Example 2: The Borromean rings

Figure: The Borromean rings Topological equivalence is completely useless in classifying knots. This is because all knots are homeomorphic to S1!

Two knots K1 and K2 are called isotopic if is possible to deform K1 into K2 without tearing or gluing.

Isotopy: When are two knots the same?

Figure: Topology in a nutshell - artist unknown Two knots K1 and K2 are called isotopic if is possible to deform K1 into K2 without tearing or gluing.

Isotopy: When are two knots the same?

Figure: Topology in a nutshell - artist unknown

Topological equivalence is completely useless in classifying knots. This is because all knots are homeomorphic to S1! Isotopy: When are two knots the same?

Figure: Topology in a nutshell - artist unknown

Topological equivalence is completely useless in classifying knots. This is because all knots are homeomorphic to S1!

Two knots K1 and K2 are called isotopic if is possible to deform K1 into K2 without tearing or gluing. Knot diagrams

A knot diagram is a projection onto the plane where the neighbourhood of every point is one of the following:

Figure: Allowed neighbourhoods in a knot diagram Theorem Every knot is isotopic to a knot whose projection contains no catastrophes.

Knot diagrams cont.

Sometimes a projection will generate a knot diagram with a neighbourhood of the form:

Figure: Uh oh, try again

This is called a catastrophe. Knot diagrams cont.

Sometimes a projection will generate a knot diagram with a neighbourhood of the form:

Figure: Uh oh, try again

This is called a catastrophe. Theorem Every knot is isotopic to a knot whose projection contains no catastrophes. Example 3: Some knot diagrams

Figure: The unknot (i), Haken’s knot (ii) and Goeritz’s knot (iii) are isotopic to each other Note that for any knot K we have

K ? U = U ? K = K

where U denotes the unknot.

Prime knots: If K = K1 ? K2 for knots K1, K2 then K1 = K and K2 = U.

Connected sums

Notation: K1 ? K2

1 Remove a point from K1 and K2 (away from crossings!) 2 Glue resulting free strands Prime knots: If K = K1 ? K2 for knots K1, K2 then K1 = K and K2 = U.

Connected sums

Notation: K1 ? K2

1 Remove a point from K1 and K2 (away from crossings!) 2 Glue resulting free strands

Note that for any knot K we have

K ? U = U ? K = K

where U denotes the unknot. Connected sums

Notation: K1 ? K2

1 Remove a point from K1 and K2 (away from crossings!) 2 Glue resulting free strands

Note that for any knot K we have

K ? U = U ? K = K

where U denotes the unknot.

Prime knots: If K = K1 ? K2 for knots K1, K2 then K1 = K and K2 = U. Example 4: Prime knots

Figure: Prime knot diagrams with 7 crossings or less Theorem Two knots are isotopic if and only if the corresponding knot diagrams can be connected by a finite sequence of Reidemeister moves

Reidemeister moves

Figure: The three kinds of Reidemeister moves Reidemeister moves

Figure: The three kinds of Reidemeister moves

Theorem Two knots are isotopic if and only if the corresponding knot diagrams can be connected by a finite sequence of Reidemeister moves There is an algorithm that will test whether or not a knot with n crossings is isotopic to an unknot

I would not recommend trying this.

Unknot recognition

Theorem (Hass,Lagarias) There is a positive constant c, such that for each n ≥ 1, any unknotted knot diagram n crossings can be transformed to the trivial knot diagram using at most k = 2cn Reidemeister moves Unknot recognition

Theorem (Hass,Lagarias) There is a positive constant c, such that for each n ≥ 1, any unknotted knot diagram n crossings can be transformed to the trivial knot diagram using at most k = 2cn Reidemeister moves

There is an algorithm that will test whether or not a knot with n crossings is isotopic to an unknot

I would not recommend trying this. Side note: Hass and Lagarias built their 3-sphere by gluing tetrahedra.

A knot is represented by a sequence of edges.

Unknot recognition cont.

Hass and Lagarias’ least upper bound was

 1 2 2dt 6 · 840n + 2dt + 1 2

where d = 107 and t is the number of tetrahedra. Taking c = 1011 is sufficient. Unknot recognition cont.

Hass and Lagarias’ least upper bound was

 1 2 2dt 6 · 840n + 2dt + 1 2

where d = 107 and t is the number of tetrahedra. Taking c = 1011 is sufficient.

Side note: Hass and Lagarias built their 3-sphere by gluing tetrahedra.

A knot is represented by a sequence of edges. Knot tabulation is difficult; we need an easy way of checking if two knots are isotopic.

Figure: The Perko pair: Perko discovered an error in Little’s tables in 1974

Knot tabulation

Tait, Little and other mathematicians tried to create a so called periodic table of elements by tabulating knots. Wikipedia: As of May 2008 all prime knots up to 16 crossings have been tabulated. Figure: The Perko pair: Perko discovered an error in Little’s tables in 1974

Knot tabulation

Tait, Little and other mathematicians tried to create a so called periodic table of elements by tabulating knots. Wikipedia: As of May 2008 all prime knots up to 16 crossings have been tabulated. Knot tabulation is difficult; we need an easy way of checking if two knots are isotopic. Knot tabulation

Tait, Little and other mathematicians tried to create a so called periodic table of elements by tabulating knots. Wikipedia: As of May 2008 all prime knots up to 16 crossings have been tabulated. Knot tabulation is difficult; we need an easy way of checking if two knots are isotopic.

Figure: The Perko pair: Perko discovered an error in Little’s tables in 1974 Knot tabulation cont.

Figure: Number of prime alternating links per crossing number and number of components. From The Enumeration and Classification of Knots and Links by Jim Hoste. Right handed crossings are assigned the value +1 and left handed crossings are assigned -1.

Orientation

Knot crossings are assigned a 0handedness0

Figure: Crossing orientations Orientation

Knot crossings are assigned a 0handedness0

Figure: Crossing orientations

Right handed crossings are assigned the value +1 and left handed crossings are assigned -1. The link number is defined to be 1 X L(L) = sign(c). 2 c∈C

The writhe number w(L) of a link L is obtained by summing up the signs of all crossings.

Writhe and link numbers

Consider a link L with two components L1, L2.

Let C denote the set of crossings involving both L1 and L2. The writhe number w(L) of a link L is obtained by summing up the signs of all crossings.

Writhe and link numbers

Consider a link L with two components L1, L2.

Let C denote the set of crossings involving both L1 and L2.

The link number is defined to be 1 X L(L) = sign(c). 2 c∈C Writhe and link numbers

Consider a link L with two components L1, L2.

Let C denote the set of crossings involving both L1 and L2.

The link number is defined to be 1 X L(L) = sign(c). 2 c∈C

The writhe number w(L) of a link L is obtained by summing up the signs of all crossings. This is not an invariant of Reidemeister move 1! But all is not lost....

Knot polynomials 1: The bracket polynomial

The bracket polynomial is the unique Laurent polynomial defined by the following relations: Knot polynomials 1: The bracket polynomial

The bracket polynomial is the unique Laurent polynomial defined by the following relations:

This is not an invariant of Reidemeister move 1! But all is not lost.... The writhe number w(L) and the bracket polynomial hLi are both preserved by Reidemeister moves 2 and 3.

Noting that Reidemeister move 1 changes w(L) by +/ − 1 and

it can be shown that XL(A) is preserved by move 1.

XL(A) is an invariant of ALL knots and links.

Knot polynomials 2: The X polynomial

Define a polynomial XL(A) by

3 −w(L) XL(A) = (−A ) hLi . Noting that Reidemeister move 1 changes w(L) by +/ − 1 and

it can be shown that XL(A) is preserved by move 1.

XL(A) is an invariant of ALL knots and links.

Knot polynomials 2: The X polynomial

Define a polynomial XL(A) by

3 −w(L) XL(A) = (−A ) hLi .

The writhe number w(L) and the bracket polynomial hLi are both preserved by Reidemeister moves 2 and 3. Knot polynomials 2: The X polynomial

Define a polynomial XL(A) by

3 −w(L) XL(A) = (−A ) hLi .

The writhe number w(L) and the bracket polynomial hLi are both preserved by Reidemeister moves 2 and 3.

Noting that Reidemeister move 1 changes w(L) by +/ − 1 and

it can be shown that XL(A) is preserved by move 1.

XL(A) is an invariant of ALL knots and links. Figure: Vaughan Jones

Very powerful: Every knot with 10 or fewer crossings has a distinct Jones polynomial!

Knot polynomials 3: The Jones polynomial

The Jones polynomial JL(t) of a link L is obtained from XL(A) by − 1 setting A = t 4 . Very powerful: Every knot with 10 or fewer crossings has a distinct Jones polynomial!

Knot polynomials 3: The Jones polynomial

The Jones polynomial JL(t) of a link L is obtained from XL(A) by − 1 setting A = t 4 .

Figure: Vaughan Jones Knot polynomials 3: The Jones polynomial

The Jones polynomial JL(t) of a link L is obtained from XL(A) by − 1 setting A = t 4 .

Figure: Vaughan Jones

Very powerful: Every knot with 10 or fewer crossings has a distinct Jones polynomial! Physics: Topological quantum field theory, loop quantum gravity, statistical mechanics

Chemistry: Chirality of molecules

Applications to biology: Knotting in DNA

Why study knots?

It is good clean fun and keeps knot theorists of the street Chemistry: Chirality of molecules

Applications to biology: Knotting in DNA

Why study knots?

It is good clean fun and keeps knot theorists of the street

Physics: Topological quantum field theory, loop quantum gravity, statistical mechanics Applications to biology: Knotting in DNA

Why study knots?

It is good clean fun and keeps knot theorists of the street

Physics: Topological quantum field theory, loop quantum gravity, statistical mechanics

Chemistry: Chirality of molecules Why study knots?

It is good clean fun and keeps knot theorists of the street

Physics: Topological quantum field theory, loop quantum gravity, statistical mechanics

Chemistry: Chirality of molecules

Applications to biology: Knotting in DNA Biology

See http://www.maths.ed.ac.uk/ s0681349/DNAtalk.pdf

Figure: Knotted DNA I will induce a magnetic field B.

I B · ds = L(L). K2

A cute result of Gauss

Figure: A knotty circuit A cute result of Gauss

Figure: A knotty circuit

I will induce a magnetic field B.

I B · ds = L(L). K2 For a classical theory you will need: 1 A 3 dimensional manifold (the universe) 2 A gauge group G (encodes symmetry) 3 A matrix representation of G

Chern-Simons theory

Chern-Simons theory: A 3D topological quantum field theory constructed by Ed Witten based on the Chern-Simons action: k Z 2 S = tr(A ∧ dA + A ∧ A ∧ A) 4π M 3 Chern-Simons theory

Chern-Simons theory: A 3D topological quantum field theory constructed by Ed Witten based on the Chern-Simons action: k Z 2 S = tr(A ∧ dA + A ∧ A ∧ A) 4π M 3

For a classical theory you will need: 1 A 3 dimensional manifold (the universe) 2 A gauge group G (encodes symmetry) 3 A matrix representation of G For the case M = S3 G = U(1) = S1 produces the linking number

G = U(2) = S3 produces the Jones polynomial

† † G = U(n) = {U ∈ Mn×n(C): UU = U U = In produces the HOMFLY polynomial

T T G = SO(n) = {A ∈ Mn×n(R): AA = A A = In produces the Kauffman polynomial

Chern-Simons theory cont.

Now quantize the classical theory. Topological invariants are obtained as physical observables. G = U(2) = S3 produces the Jones polynomial

† † G = U(n) = {U ∈ Mn×n(C): UU = U U = In produces the HOMFLY polynomial

T T G = SO(n) = {A ∈ Mn×n(R): AA = A A = In produces the Kauffman polynomial

Chern-Simons theory cont.

Now quantize the classical theory. Topological invariants are obtained as physical observables.

For the case M = S3 G = U(1) = S1 produces the linking number † † G = U(n) = {U ∈ Mn×n(C): UU = U U = In produces the HOMFLY polynomial

T T G = SO(n) = {A ∈ Mn×n(R): AA = A A = In produces the Kauffman polynomial

Chern-Simons theory cont.

Now quantize the classical theory. Topological invariants are obtained as physical observables.

For the case M = S3 G = U(1) = S1 produces the linking number

G = U(2) = S3 produces the Jones polynomial T T G = SO(n) = {A ∈ Mn×n(R): AA = A A = In produces the Kauffman polynomial

Chern-Simons theory cont.

Now quantize the classical theory. Topological invariants are obtained as physical observables.

For the case M = S3 G = U(1) = S1 produces the linking number

G = U(2) = S3 produces the Jones polynomial

† † G = U(n) = {U ∈ Mn×n(C): UU = U U = In produces the HOMFLY polynomial Chern-Simons theory cont.

Now quantize the classical theory. Topological invariants are obtained as physical observables.

For the case M = S3 G = U(1) = S1 produces the linking number

G = U(2) = S3 produces the Jones polynomial

† † G = U(n) = {U ∈ Mn×n(C): UU = U U = In produces the HOMFLY polynomial

T T G = SO(n) = {A ∈ Mn×n(R): AA = A A = In produces the Kauffman polynomial Baez claims there is a relationship between knots and loop quantum gravity. Look into this if you dare...

There appears to be a deep relationship between the mathematical theory of knots and theoretical physics but I am getting too far into material which I do not understand. Now is a good place to stop.

Loop representation of quantum gravity

An attempt at quantum gravity - does not require the extra dimensions of string theory There appears to be a deep relationship between the mathematical theory of knots and theoretical physics but I am getting too far into material which I do not understand. Now is a good place to stop.

Loop representation of quantum gravity

An attempt at quantum gravity - does not require the extra dimensions of string theory

Baez claims there is a relationship between knots and loop quantum gravity. Look into this if you dare... Loop representation of quantum gravity

An attempt at quantum gravity - does not require the extra dimensions of string theory

Baez claims there is a relationship between knots and loop quantum gravity. Look into this if you dare...

There appears to be a deep relationship between the mathematical theory of knots and theoretical physics but I am getting too far into material which I do not understand. Now is a good place to stop. References

[1] J Baez, J Munian. Gauge Fields, Knots and Gravity (1994)

[2] C Adams. The Knot Book (1994)

[3] J Hass, J Lagarias. The number of Reidemeister Moves Needed for Unknotting (2008)