Page 314 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 314
GRAPHS, POLYNOMIALS, SURFACES, AND KNOTS (MS-49)
Framed- and Biframed Knotoids
Wout Moltmaker, woutmoltmaker@gmail.com
University of Oxford, United Kingdom
In this talk I will recall the definition of a spherical knotoid and modify this definition to include
a framing, in analogy to framed knots. I also define a further modification that includes a
secondary ’coframing’ to obtain ’biframed’ knotoids. Afterwards I will give topological spaces
whose ambient isotopy classes are in one-to-one correspondence with framed- and biframed
knotoids respectively. Finally I will mention how biframed knotoids allow for the construction
of quantum invariants.
Coloring quadrangulations of the projective space
Atsuhiro Nakamoto, nakamoto@ynu.ac.jp
Yokohama National University, Japan
Coauthor: Kenta Ozeki
A quadrangulation on a surface F 2 is a fixed embedding of a simple graph such that each
face is quadrilateral. It is known that every quadrangulation on the sphere is bipartite, but ev-
ery non-spherical surface admits non-bipartite quadrangulations. For the projective plane P 2,
Young pointed out an interesting fact that every non-bipartite quadrangulation is 4-chromatic.
Kaiser and Stehlík considered a higher dimensional quadrangulations in the projective space,
and proved that every non-bipartite d-dimensional quadrangulation in the d-dimensional pro-
jective space P d has chromatic number exactly d + 2. In our talk, we will give another proof
to Young’s result, focusing the dual map of quadrangulations. Moreover, giving a new defini-
tion of a higher dimensional quadrangulations different from those by Kaiser and Stehlík, we
prove that 3-dimensional quadrangulations of P 3 in a certain class have chromatic number 4,
and conjecture that this can be extended to all of our quadrangulations in P 3.
The two-variable Bollobás–Riordan polynomial of a connected even
delta-matroid is irreducible
Steven Noble, s.noble@bbk.ac.uk
Birkbeck, University of London, United Kingdom
Coauthors: Joanna Ellis-Monaghan, Andrew Goodall, Iain Moffatt, Lluís Vena
One of the most striking results concerning the Tutte polynomial is that the Tutte polynomial of
a matroid is irreducible if and only if the matroid is connected.
The most natural analogue of the Tutte polynomial for an even delta-matroid is perhaps a
normalized two variable specialization
(x − 1)w(D)/2RD(x, y − 1, 1/ (x − 1)(y − 1), 1)
of the Bollobás–Riordan polynomial. We show that for even delta-matroids this two-variable
Bollobás–Riordan polynomial is irreducible if and only if the delta-matroid is connected.
312
Framed- and Biframed Knotoids
Wout Moltmaker, woutmoltmaker@gmail.com
University of Oxford, United Kingdom
In this talk I will recall the definition of a spherical knotoid and modify this definition to include
a framing, in analogy to framed knots. I also define a further modification that includes a
secondary ’coframing’ to obtain ’biframed’ knotoids. Afterwards I will give topological spaces
whose ambient isotopy classes are in one-to-one correspondence with framed- and biframed
knotoids respectively. Finally I will mention how biframed knotoids allow for the construction
of quantum invariants.
Coloring quadrangulations of the projective space
Atsuhiro Nakamoto, nakamoto@ynu.ac.jp
Yokohama National University, Japan
Coauthor: Kenta Ozeki
A quadrangulation on a surface F 2 is a fixed embedding of a simple graph such that each
face is quadrilateral. It is known that every quadrangulation on the sphere is bipartite, but ev-
ery non-spherical surface admits non-bipartite quadrangulations. For the projective plane P 2,
Young pointed out an interesting fact that every non-bipartite quadrangulation is 4-chromatic.
Kaiser and Stehlík considered a higher dimensional quadrangulations in the projective space,
and proved that every non-bipartite d-dimensional quadrangulation in the d-dimensional pro-
jective space P d has chromatic number exactly d + 2. In our talk, we will give another proof
to Young’s result, focusing the dual map of quadrangulations. Moreover, giving a new defini-
tion of a higher dimensional quadrangulations different from those by Kaiser and Stehlík, we
prove that 3-dimensional quadrangulations of P 3 in a certain class have chromatic number 4,
and conjecture that this can be extended to all of our quadrangulations in P 3.
The two-variable Bollobás–Riordan polynomial of a connected even
delta-matroid is irreducible
Steven Noble, s.noble@bbk.ac.uk
Birkbeck, University of London, United Kingdom
Coauthors: Joanna Ellis-Monaghan, Andrew Goodall, Iain Moffatt, Lluís Vena
One of the most striking results concerning the Tutte polynomial is that the Tutte polynomial of
a matroid is irreducible if and only if the matroid is connected.
The most natural analogue of the Tutte polynomial for an even delta-matroid is perhaps a
normalized two variable specialization
(x − 1)w(D)/2RD(x, y − 1, 1/ (x − 1)(y − 1), 1)
of the Bollobás–Riordan polynomial. We show that for even delta-matroids this two-variable
Bollobás–Riordan polynomial is irreducible if and only if the delta-matroid is connected.
312
