A Knot-vice’s Guide to Untangling Knot Theory

Rebecca Hardenbrook

Department of Mathematics University of Utah

Rebecca Hardenbrook A Knot-vice’s Guide to Untangling Knot Theory 1 / 26 What is Not a Knot?

Rebecca Hardenbrook A Knot-vice’s Guide to Untangling Knot Theory 2 / 26 What is a Knot?

2 A knot is an embedding of the circle in the Euclidean plane (R ). 3 Also defined as a closed, non-self-intersecting curve in R . 2 Represented by knot projections in R .

Rebecca Hardenbrook A Knot-vice’s Guide to Untangling Knot Theory 3 / 26 Why Knots?

Late nineteenth century chemists and physicists believed that a substance known as aether existed throughout all of space. Could knots represent the elements?

Rebecca Hardenbrook A Knot-vice’s Guide to Untangling Knot Theory 4 / 26 Why Knots?

Rebecca Hardenbrook A Knot-vice’s Guide to Untangling Knot Theory 5 / 26 Why Knots?

Unfortunately, no.

Nevertheless, mathematicians continued to study knots!

Rebecca Hardenbrook A Knot-vice’s Guide to Untangling Knot Theory 6 / 26 Current Applications

Natural knotting in DNA molecules (1980s).

Credit: K. Kimura et al. (1999)

Rebecca Hardenbrook A Knot-vice’s Guide to Untangling Knot Theory 7 / 26 Current Applications

Chemical synthesis of knotted molecules – Dietrich-Buchecker and Sauvage (1988).

Credit: J. Guo et al. (2010)

Rebecca Hardenbrook A Knot-vice’s Guide to Untangling Knot Theory 8 / 26 Current Applications

Use of lattice models, e.g. the Ising model (1925), and planar projection of knots to find a knot invariant via statistical mechanics.

Credit: D. Chicherin, V.P. Spiridonov (2015)

Rebecca Hardenbrook A Knot-vice’s Guide to Untangling Knot Theory 9 / 26 The Unknot

2 The unknot is the standard circle in R ; has no crossings. Could every knot secretly be the unknot?

The Thistlethwaite Unknot

Rebecca Hardenbrook A Knot-vice’s Guide to Untangling Knot Theory 10 / 26 Reidemeister Moves

Kurt Reidemeister (1927) and James Alexander & G.B. Briggs (1926) independently showed that if two knot diagrams are the same, they can be related by three moves:

Rebecca Hardenbrook A Knot-vice’s Guide to Untangling Knot Theory 11 / 26 Tricolorability

A knot is tricolorable if each strand can be colored one of three colors At least two colors are used. At each crossing, all three strands are either the same color or are three different colors.

Rebecca Hardenbrook A Knot-vice’s Guide to Untangling Knot Theory 12 / 26 Tricolorability

Reidemeister moves preserve the tricolorability of a knot. Either every projection of a knot is tricolorable or no projection of that knot is tricolorable. The unknot is not the only knot!

Tricolorability is known as a knot invariant: a certain trait obtained by knots that is the same for equivalent knots.

Rebecca Hardenbrook A Knot-vice’s Guide to Untangling Knot Theory 13 / 26 The Unknotting Number

A knot K is assigned the unknotting number n if there exists some projection of K such that exactly n crossings can be changed to give the unknot.

The trefoil knot has unknotting number n = 1:

Rebecca Hardenbrook A Knot-vice’s Guide to Untangling Knot Theory 14 / 26 The Bridge Number

An overpass is a subarc of a knot that goes over one or more crossings but does not go under a crossing.

A maximal overpass is an overpass that cannot be extended any further.

The bridge number of a given knot K, denoted b(K), is the least number of maximal overpasses over all projections of K.

Rebecca Hardenbrook A Knot-vice’s Guide to Untangling Knot Theory 15 / 26 Lucky Knot

The Lucky Knot Bridge in Changsha, China

Rebecca Hardenbrook A Knot-vice’s Guide to Untangling Knot Theory 16 / 26 Other Knot Invariants

Knot genus.

Knot polynomials.

Rebecca Hardenbrook A Knot-vice’s Guide to Untangling Knot Theory 17 / 26 An Unsolved Problem!

Create a computer program that will tell you whether a given knot is actually the unknot. Wolfgang Haken (1961) came up with an algorithm that could do this.

However, it is so complicated that no program has been written to efficiently apply it!

Rebecca Hardenbrook A Knot-vice’s Guide to Untangling Knot Theory 18 / 26 Knot Composition

The composition of two knots J and K is the knot resulting from cutting one strand of each knot and combining the loose ends, denoted J#K. J and K are called factor knots of J#K.

The crossing number of a knot K, c(K), is the least number of crossings in any projection of K.

Rebecca Hardenbrook A Knot-vice’s Guide to Untangling Knot Theory 19 / 26 Keeping Track of Knots

If a knot can only be factored into itself and the unknot, then it is called a prime knot.

There was a need to find a way to list all possible knots, particularly prime knots, of varying crossing numbers; called tabulation.

John Conway (1969) invented a new notation, which he used to find all prime knots K where c(K) ≤ 11 by hand. Corrected by Alain Caudron (1978).

Rebecca Hardenbrook A Knot-vice’s Guide to Untangling Knot Theory 20 / 26 Keeping Track of Knots

Hugh Dowker (1978) created another notation based on Tait’s ideas.

Morwen Thistlethwaite used Dowker’s notation to create a computer program that generated all prime knots K such that c(K) ≤ 13 (1981-82).

Jim Hoste, Thistlethwaite, and Jeff Weekes were able to then generate all prime knots K where c(K) ≤ 16 (1998). The First 1,701,936 Knots, The Mathematical Intelligencer.

Rebecca Hardenbrook A Knot-vice’s Guide to Untangling Knot Theory 21 / 26 Dowker Notation

The below knot has the Dowker notation: 8 10 2 12 4 6.

Rebecca Hardenbrook A Knot-vice’s Guide to Untangling Knot Theory 22 / 26 Unsolved Problems

Is it true that c(K1#K2) = c(K1) + c(K2) for all knots K1,K2?

Determine the following sequence of integers:

1, 1, 2, 3, 7, 21, 49, 165, 552, 2176, 9988, ···

Can this sequence be represented by a function?

+ Show that, for all n ∈ Z , the number of unique prime (n + 1)-crossing knots is greater than the number of unique prime n-crossing knots.

Rebecca Hardenbrook A Knot-vice’s Guide to Untangling Knot Theory 23 / 26 Sources

J. C. Cha and C. Livingston, KnotInfo: Table of Knot Invariants, http://www.indiana.edu/ knotinfo, February 10, 2018. C.C. Adams, The Knot Book, W.H. Freeman and Company, 1994. Adams’ cousin-in-law, Sir Randolph Bacon III, gives a great lecture: Math Encounters - Blown Away: What Knot to Do When Sailing.

Rebecca Hardenbrook A Knot-vice’s Guide to Untangling Knot Theory 24 / 26 Learn More About Knots

Take an introductory topology course! (Math 5510/20)

Take a reading course on knot theory!

Do a knot theory REU project!

Read a ton of books: The Knot Book by Colin Adams Why Knot? by Colin Adams Knots and Links by Dale Rolfsen On Knots by Louis Kauffman ...and more!

Rebecca Hardenbrook A Knot-vice’s Guide to Untangling Knot Theory 25 / 26 Thank You!

Rebecca Hardenbrook A Knot-vice’s Guide to Untangling Knot Theory 26 / 26