Mini-Workshop "Knots + More"
This online mini workshop is hosted by
Ziegler's lab at KAIST,
School of Computing.
It is a follow-up to the previous
workshop on "Computational Knot Theory".
But the emphasis should now be less on basics, or either "knot theory",
or "computational". Other modalities, though, are similar.
(Please see "orientation" on the previous workshop website.)
Again we use zoom
ID 350 042 7768, no password.
no-sign-up, no-ap link:
Click "Join from Your Browser".
You must get after typing your name
"Please wait, the meeting host will let you in soon".
I am planning a week for late August, August 18th-25th, 2021.
Talk time should be 25-50 minutes (+ possible questions).
Schedule and abstracts will be completed below. Times are KST
(= GMT +0900).
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.
Due to a serious health issue, S.Baader's talk has
to be (at least for the time frame of the workshop) cancelled.
I apologise very much for the inconvenience.
(please hear before joining a talk!)
Eleni Panagiotou (University of Tennessee, Chattanooga)
Knot polynomials and Vassiliev measures of open and closed curves in 3-space and their applications
Many physical systems are composed by entangled filamentous structures whose entanglement greatly affects their mechanical properties. Measuring entanglement complexity of physical systems has been a challenge because these can be seen as open curves in 3-space for which the conventional measures of topological complexity do not apply. In this talk we introduce the Jones polynomial for open curves in 3-space. We show it is a continuous function of the curve coordinates that, as the ends tend to coincide, it tends to the Jones polynomial of the resulting knot. We show how this Jones polynomial can be applied to diblock polymers to detect phase transitions. The calculation of the Jones polynomial for real physical filaments can be intractable. In this talk we also introduce Vassiliev measures of open curves in 3-space. These are also continuous functions of the curve coordinates and tend to the Vassiliev invariant of the resulting knot as the endpoints coincide. We show that the second Vassiliev measure can provide a measure of entanglement complexity of open curves in 3-space that may be preferable for applications.
Homayun Karimi (McMaster University) 8/18 8pm
The Gordon-Litherland pairing for links in surfaces, and applications
to the alternating links and the concordance
We introduce the Gordon-Litherland (GL) pairing for knots and
links in thickened surfaces that bound unoriented spanning surfaces. Using
the GL pairing, we define several invariants (signature, determinant, and
nullity). We will discuss several applications to problems such as detecting
the minimal supporting genus, determining sliceness, and slice genus of
virtual knots. The GL pairing can also be regarded as the relative
intersection pairing on a 4-manifold obtained as the 2-fold cover along the
surface. We will also give an application to the problem of characterization
of alternating links.
This talk represents three papers, one is a joint work in progress with Hans
Boden, and Micah Chrisman and the other two are joint work with Hans Boden.
Jieon Kim (Pusan National University) 8/19 10am
On symmetric biquandles
A biquandle is a non-empty set with
two binary operations. When an invariant is constructed by using biquandles,
orientations are needed. Biquandles are generalizations of quandles. For
constructing invariants by using quandles, we need orientations. Kamada and
Oshiro defined symmetric quandles, which will be used for constructing
invariants for unoriented (surface-)links. In this talk, we define symmetric
biquandles, which are generalizations of symmetric quandles. Also we
construct invariants for unoriented (surface-)links by using symmetric
Nafaa Chbili (United Arab Emirates University) 8/19 8pm
Quasi-alternating links, new examples and obstructions
Quasi-alternating links represent an important class of links that has been introduced by Ozsváth and Szabó while studying the Heegaard Floer homology of the branched double-covers of alternating links. This new class of links which share many homological properties with alternating links, is defined in a recursive way which is not easy to use in order to determine whether a given link is quasi-alternating. In this talk, we shall review some of the main obstruction criteria that have been established to characterize quasi-alternating links. Then we shall explain how to extend the twisting technique of Champanerkar and Kofman to introduce new examples of quasi-alternating links.
Serge Tabachnikov (Pennsylvania State University) 8/20 10am
Cusps of caustics by reflection and Legendrian knots
The boundary of a planar oval is an ideal mirror, and one has a
point source of light inside the oval. Consider the rays of light that have
undergone N reflection in the mirror, where N=1,2,... The envelope of this
system of rays is the Nth caustic by reflection. I shall explain why, for
every N, this caustic has at least four cusps. This result is a consequence
of a far reaching generalization of the 4-vertex theorem, conjectured by
Arnold and proved by Chekanov and Pushkar' using Legendrian knot theory.
Similar problems for convex surfaces were considered before: Caratheodory
proved that the locus of points conjugated to a given point has at least
four cusps, and Jacobi stated, in his "Lectures on dynamics", that this
number is exactly four in the case of the ellipsoid (this is known as the
"Last Geometric Statement of Jacobi"). Our problem is a billiard version of
these problems of differential geometry of surfaces. Conjecturally, ellipses
are characterized by the property that all caustics by reflection have
exactly four cusps.
Ken Perko (Juris Doctor, New York) 8/20 8pm
Looking at Linking Numbers
Linking numbers between branch curves of non-cyclic covering
spaces were first discovered by Reidemeister and used to complete the
classification of prime knots through 9 crossings. Algorithms for
calculating them were originally limited to 2-bridge examples (then known
as "Viergeflechte") and set forth rather confusingly in a 1934 paper by
Bankwitz and Schumann that never got properly cited in Reidemeister's 1932
book on "Knotentheorie." At Fox's suggestion, our 1964 Princetion senior
thesis set forth a general algorithm for their calculation (when they
exist). This talk will discuss some short cuts for intuitively "seeing"
such linking numbers, when they are integers, along with various
relationships between them when crossings of diagrams are reversed.
Ken Millett (University of California, Santa Barbara) 8/24 8pm
Some observations concerning the Conway and Kinoshita-Terasaka Knots
We discuss the 11 edge equilateral polygonal representations of the Conway and Kinoshita-Terasaka knots. By removing edges of the configurations, we study the HOMFLY-PT spectrum of the resulting open arc conformations using DMS closures whose superposition defines an average HOMFLY-PT polynomial of the open arc. Defining the spread of this polynomial gives a new measure of the complexity of the knotting entanglement of the open arc. The spectrum, its superposition polynomial and the associated spread provide new methods to compare the Conway and Kinoshita-Terasaka knots and provide new information supporting the view that despite their similarities, the Conway knot is more entangled and more complex than the Kinoshita-Tereasaka knot. This work is in collaboration with Eleni Panagiotou.
Teruhisa Kadokami (Kanazawa University) 8/25 10am
On amphicheirality of links
An n-component link L in S3 is amphicheiral
if there is an orientation-reversing homeomorphism
f of S3 such that f(L) is equivalent to L.
We show some results on amphicheirality of links.
Mikami Hirasawa (Nagoya Institute of Technology) 8/25 8pm
On Alexander polynomials of degree over 764 of alternating knots with
over 778 crossings
Let K be an alternating knot with Alexander polynomial f(t).
Since J. Hoste raised the question, it had been believed over 17 years that the
of the zeros of f(t) are greater than -1.
Note that there are 491327 alternating knots up to 16 crossings available to check.
The conjecture has evaded many attacks and yielded only partial positive answers
but no concrete counter example.
To find a counter examples, we narrowed the candidate to a special subclass of
410141 alternating Montesinos knots up to 1000 crossings, and found first 45 counter examples.
The minimal one has 778 crossings and the degree of f equals 764.
We made explicit Seifert surfaces for the candidates, made formula for the polynomial,
and after numerical calculations, gave a rigorous proof that they have a zero whose real part is less than -1.
This is a joint work with Katsumi Ishikawa (Kyoto University) and Masaaki Suzuki (Meiji University).
* = please allow for a possible 3-4 min delay
Co-organizer: Hyunwoo Lee