KNOTS AND MATHEMATICS LINH TRUONG COLUMBIA UNIVERSITY MARCH

KNOTS AND MATHEMATICS LINH TRUONG COLUMBIA UNIVERSITY MARCH 27, 2018 S. T. E. M. outreach colloquium

Knot Plot Knots: closed loops Two knots are equivalent if one can be transformed to the other without cutting the loop.

How to distinguish two knots? ❖ It’s not always easy to tell when a given diagram is the simplest possible picture for that knot. = ? Knot Plot ❖ For example, these two knots are the same.

Knot Plot How to distinguish two knots? Use invariants! ❖ An invariant is a quantity, such as a number or a polynomial, that is defined for every knot.

An Invariant: The crossing number ❖ The crossing number of a knot is the minimal number of crossings among all diagrams for the knot. unknot diagram with 0 crossings ❖ unknot diagram with 1 crossing The crossing number of the unknot is 0. Knot Plot ❖ Knots with crossing number up to seven.

An invariant: The unknotting number ❖ The unknotting number of a knot is the minimum number of crossing changes to change the knot to the unknot, among all diagrams for the knot. = trefoil ❖ The unknotting number of the trefoil is 1. unknot

An invariant: The Jones Polynomial ❖ The Jones polynomial of any knot or link, denoted VK(t), is characterized by the rules: ❖ Vunknot(t) = 1 ❖ t VL (t) + t VL (t) = (t - t ) VL (t) -1 1/2 + ❖ -1/2 - 0 The diagrams for L+ , L- , and L 0 are the same except in one spot where they differ as follows: L + L - L 0

Jones Polynomial: an example ❖ The Jones polynomial of the left-handed trefoil, shown to the right, is V (t) = −t-4+t-3+t-1 Figure by Jim. belk, Wikipedia

Mathematical Questions ❖ How can you tell two knots apart? ❖ What kinds of properties do knots have? ❖ Can you classify all knots? ❖ ❖ Knot Plot What kind of algebraic structure and operations exist for knots? Adding, subtracting? How are knots related to surfaces or 3 -dimensional spaces?

Chirality ❖ An achiral (or amphichiral) knot is the same as its mirror image. = Knot Plot ❖ The figure eight knot is achiral.

Symmetry ❖ What kind of symmetry does a particular knot have? ❖ A knot is n-periodic if it is preserved by a rotation of degree 2*pi/n Knot Atlas A 7 -periodic knot.

How can you “add” two knots? figure from Wikipedia Connected sum of two knots K 1#K 2 ❖ Question: What is the knot corresponding to “zero” with the connected sum operation?

What is a “prime” or a “composite” knot? ❖ A prime knot is one that cannot be decomposed as the “connected sum” of two other knots. ❖ Knots have a “prime factorization” theorem, much like numbers, e. g. 20 = 2*2*5. Knot Atlas A composite knot that is a sum of three prime knots.

My research ❖ I apply powerful invariants to study properties of knots and their relationship to three- and four-dimensional spaces. Figures by Jack van Wijk, Seifert. View

Connections to biology ❖ DNA can be tangled/knotted. ❖ Tightly packed DNA in the genes must quickly unknot itself in order for replication or transcription to occur. ❖ Knot theory is used to estimate how hard DNA is to unknot.

Many more knot invariants exist! Many applications to other branches of mathematics! Many open questions about knots! i. e. lots more to figure out… Knot. Plot Thanks for listening!