Workshop on "Computational Knot Theory"
I am organizing an online workshop on
"Computational Knot Theory" hosted by
Ziegler's lab at KAIST,
School of Computing,
and would be very happy to invite you to join.
The emphasis should still be on "knot theory",
while we like to consider "computational" in the
loosest possible sense. Our goal is also to promote the
topic with talks accessible to (and whet interest of)
computer science students (who have not heard much
about knot theory). To students: for a very basic introduction,
you may consult notes and links to my
previous talks at the dept,
to Computational Knot Theory",
and knot tables" (see the bottom of
my talk page).
Our agreement is that we
use zoom, that I would like you to install
ID 875 1903 7390, password "knotty".
I am planning a week in late May-early June,
May 26-June 2, 2021.
Talk time should be 25-50 minutes (+ possible questions).
Schedule and abstracts will be completed below. Times are KST
(= GMT +0900). There are reasons like time(/time zone) difference
why I cant attain a uniform talk schedule every day.
Looking forward to seeing you -at least virtually.
Please forward this page to whoever you think may be interested.
You're welcome to write me (address at the bottom of the page)
if you have questions.
Orientation and introduction (by myself and Martin; please hear before joining a talk!)
report on the workshop on the dept webpage (also in Korean)
follow-up event (planned)
Colin Adams (Williams College) 5/26 10am*
Multi-crossing Number and Petal Number for Knots
Instead of considering pictures of knots with two strands
crossing at every crossing, we can ask for n strands to cross at every
crossing. We will show that every knot and link has such an n-crossing
projection for all integers n greater than 1 and therefore an n-crossing
number. We also show that every knot has a projection with a single such
multicrossing and no nested loops, resembling a daisy. This a petal
projection which generates a petal number. We will discuss work done by
undergraduates on determining these numbers, and mention a variety of open
Stepan Orevkov (Universite P. Sabatier Toulouse) 5/26 8pm
Computation of multivariable signatures of colored links
A colored link is a link with a fixed partition of
its connected components into several sublinks
(which we call monochrome components).
Multivariable signatures of were introduced and studied by
Oleg Viro and Vincent Florens. They serve as an upper bound for
the Euler characteristic of a slice surface whose connected
components have monochrome boundaries.
David Cimasoni and Vincent Florens computed the multivariable
signatures via the signatures of a generalized Seifert form
on the 1-st homologies of a C-complex of a colored link.
In my talk I discuss the computational problems appearing
in this context.
Gyo Taek Jin (KAIST) 5/27 10am*
Arc index and minimal grid diagrams of prime knots
Every knot in space can be embedded in a finitely many half planes attached
to the z-axis in such a way that each half plane contains a single arc
whose ends are on the z-axis. Such embeddings are called arc presentations.
The minimal number of such arcs for all possible arc presentations of a
knot is called its arc index. A grid diagram is a knot diagram with
finitely many horizontal line segments and the same number of
vertical line segments such that at each crossing the vertical segment goes
over the horizontal segment. Grid diagrams are easily converted to arc
presentations and vice versa. We describe how the arc index of prime knots
up to certain ranges are obtained. We also describe how minimal grid
diagrams of the 11 crossing prime alternating knots and the 12 crossing
prime alternating knots are obtained.
Roland van der Veen (University of Groningen) 5/27 8pm
OU tangles and effective universal knot invariants
Studying knots through planar projections (knot diagrams) is
many diagrams represent the same knot. Is there some way of selecting a
preferred diagram for a knot?
We attempt to bring knot diagrams in a unique standard form by forcing the
knot to first run through all
overpasses, postponing any underpasses. While this algorithm may not
terminate, it does provide
an intuitive basis for an important class of knot invariants known as
universal (quantum) knot invariants.
Universal invariants include the Jones polynomial and many of its
generalizations and behave well under many
topological operations on knots. We will illustrate these ideas with a
computer implementation of a particular
universal invariant that is distinguishes all knots in the Rolfsen table
yet is computationally effective.
Unlike all other known powerful knot invariants it runs in polynomial time
in the complexity of the knot diagrams.
This is joint work with Dror Bar-Natan and parts of
our work appeared in
Alexander Mednykh (Novosibirsk State University) 5/28 10am
Volumes of knots and links in spaces of constant curvature
We investigate the existence of hyperbolic, spherical or Euclidean structure on cone-manifolds whose underlying space is the three-dimensional sphere and singular set is a given knot or link. For two-bridge knots with not more than seven crossings we present trigonometrical identities involving the lengths of singular geodesics and cone angles of such cone-manifolds. Then these identities are used to produce exact integral formulae for the volume of the corresponding cone-manifold modeled in the hyperbolic, spherical and Euclidean geometries.
Thomas Fiedler (Universite P. Sabatier Toulouse) 5/28 8pm
1-cocycles and knot invariants
We introduce an appropriate moduli space for diagrams of classical knots and we lift the Chmutov-Khoury-Rossi-Brandenbursky Gauss diagram formulas for the coefficients of the Conway polynomial to combinatorial 1-cocycles on this moduli space. In contrast to the Conway polynomial they can distinguish mirror images of knots when they are evaluated on canonical loops in the moduli space, and at least as 1-cocycles they are sensitive to the orientation of the knots.
Sergei Chmutov (The Ohio State University, Mansfield) 5/31 8pm
Construction of links from Thompson's group
Vaughan Jones introduces an algorithmic construction of links from elements of Thompson's group and demonstrate the Alexander type theorem. His algorithm is highly inefficient. One of the main problems in this direction is to find an efficient algorithm. I will discuss a definition of the Thompson group and combinatorial description of its elements. Then I will explain the Jones algorithm and discuss the actual computational problems.
Sang Youl Lee (Pusan National University) 6/1 10am*
Biquandle cocycle invariants for surface-links in
ℝ4 via marked
A surface-link is a closed 2-manifold smoothly embedded in 4-space
A marked graph diagram is a link diagram possibly with
4-valent vertices with markers. It is known that a surface-link can be
described by a marked graph diagram modulo Yoshikawa moves.
On the other hand, it is well known that the (bi)quandle cocycle invariants
and shadow (bi)quandle cocycle invariants are defined for oriented classical
links in 3-space and surface-links in 4-space by using broken surface
diagrams and cohomology theory of (bi)quandles.
In this talk, I would like to introduce representation of surface-links via
marked graph diagrams and then discuss a method of computing (bi)quandle
cocycle invariants from marked graph diagrams.
Hwa Jeong Lee (Dongguk University - Gyeongju) 6/1 8pm
Quantum knots and the number of knot mosaics
Lomonaco and Kauffman developed a knot mosaic system to
introduce a precise and workable definition of a quantum knot system. This
definition is intended to represent an actual physical quantum system. A
knot m×n-mosaic is an m×n matrix of mosaic tiles
(T0 through T10 depicted in the introduction) representing a knot or a link by adjoining properly that is called suitably connected. D(m,n) is the total number of all knot (m, n)-mosaics. In this talk, we construct an algorithm producing the precise value of D(m,n) for m, n≥ 2 that uses recurrence relations of state matrices that turn out to be remarkably efficient to count knot mosaics.
Toshitake Kohno (Meiji University, The University of Tokyo) 6/2 10am
Quantum computation and homological representations of braid groups
In the theory of topological quantum computation
unitary representations of the braid groups play an
important role. In this talk I will explain a method to
obtain unitary representations from homological
representations of braid groups by investigating
a relation to conformal field theory.
Hugh Morton (University of Liverpool) 6/2 8pm
Using knot invariants
Knot invariants come in many different forms. They may reflect geometric or combinatorial features of the knot. Calculating and comparing them is not always easy.
Strategies for constructing them and test cases for comparing them will be described. I will look particularly at the case of satellites and symmetric mutants.
The general guidance when looking at an invariant is to suggest and subsequently identify geometric features which it can show up. Equally, for more geometrically defined invariants the aim is to target their computation.
* = please allow for a possible 3-4 min delay
for greeting/because of overlapping lecture time at the dept
Co-organizer: Hyunwoo Lee