Page 252 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
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APPLIED COMBINATORIAL AND GEOMETRIC TOPOLOGY (MS-34)
manifolds up to complexity 12 (see http://matlas.math.csu.ru/?page=search). In this talk, after
recalling some general result about complexity, I will focus on an important class of manifolds:
graph manifolds. These manifolds have been introduced and classified by Waldhausen in the
sixties and are defined as compact 3-manifolds obtained by gluing Seifert fibre spaces along
toric boundary components. I will present an upper bound for their complexity obtained in a
joint work with Michele Mulazzani (University of Bologna) that is sharp for all the 14502 graph
manifolds included in the Recognizer catalogue
Periodic projections of alternating knots
Antonio F. Costa, antoniofelixcostagonzalez@gmail.com
UNED, Spain
Coauthor: Cam Van Quach-Hongler
Let K be an oriented prime alternating knot that is q-periodic with q ≥ 3, i.e. K admits a
symmetry that is a rotation of order q. We present a proof that K has an alternating q-periodic
projection.
The main tool is the Menasco-Thistlethwaite’s Flyping theorem.
We present also some results about the visibility on projections of q-freely periodicity of
alternating knots.
Classifying compact PL 4-manifolds according to generalized regular
genus and G-degree
Paola Cristofori, paola.cristofori@unimore.it
University of Modena and Reggio Emilia, Italy
Coauthor: Maria Rita Casali
(d+1)-colored graphs, that is (d+1)-regular graphs endowed with a proper edge-coloration, are
the objects of a long-studied representation theory for closed PL d-manifolds, which has been
recently extended to the whole class of compact PL d-manifolds.
In this context, combinatorially defined PL invariants play a relevant role; in this talk we
will focus on two of them: the generalized regular genus and the G-degree. The former ex-
tends to higher dimension the classical notion of Heegaard genus for 3-manifolds; the latter has
arisen in connection with Colored Tensor Models (CTM), a particular kind of tensor models,
that have been intensively studied in the last years, mainly as an approach to quantum gravity in
dimension greater than two. CTMs established a link between colored graphs and tensor mod-
els, since the Feynman graphs of a d-dimensional CTM are precisely (d + 1)-colored graphs.
Furthermore, the G-degree of a colored graph is a crucial quantity driving the 1/N expansion of
the free energy of a CTM.
This talk will mainly concern recent results achieved in dimension 4: in particular, the clas-
sification of all compact PL 4-manifolds with generalized regular genus at most one or with
G-degree at most 18. Furthermore, we will discuss interesting classes of 5-colored graphs
(semi-simple and weak semi-simple crystallizations), representing compact PL 4-manifolds
with empty or connected boundary and minimizing the invariants. In the simply-connected
case they also belong to a wider class of 5-colored graphs which are proved to induce handle-
decompositions of the represented 4-manifold lacking in 1- and/or 3-handles; therefore their
250
manifolds up to complexity 12 (see http://matlas.math.csu.ru/?page=search). In this talk, after
recalling some general result about complexity, I will focus on an important class of manifolds:
graph manifolds. These manifolds have been introduced and classified by Waldhausen in the
sixties and are defined as compact 3-manifolds obtained by gluing Seifert fibre spaces along
toric boundary components. I will present an upper bound for their complexity obtained in a
joint work with Michele Mulazzani (University of Bologna) that is sharp for all the 14502 graph
manifolds included in the Recognizer catalogue
Periodic projections of alternating knots
Antonio F. Costa, antoniofelixcostagonzalez@gmail.com
UNED, Spain
Coauthor: Cam Van Quach-Hongler
Let K be an oriented prime alternating knot that is q-periodic with q ≥ 3, i.e. K admits a
symmetry that is a rotation of order q. We present a proof that K has an alternating q-periodic
projection.
The main tool is the Menasco-Thistlethwaite’s Flyping theorem.
We present also some results about the visibility on projections of q-freely periodicity of
alternating knots.
Classifying compact PL 4-manifolds according to generalized regular
genus and G-degree
Paola Cristofori, paola.cristofori@unimore.it
University of Modena and Reggio Emilia, Italy
Coauthor: Maria Rita Casali
(d+1)-colored graphs, that is (d+1)-regular graphs endowed with a proper edge-coloration, are
the objects of a long-studied representation theory for closed PL d-manifolds, which has been
recently extended to the whole class of compact PL d-manifolds.
In this context, combinatorially defined PL invariants play a relevant role; in this talk we
will focus on two of them: the generalized regular genus and the G-degree. The former ex-
tends to higher dimension the classical notion of Heegaard genus for 3-manifolds; the latter has
arisen in connection with Colored Tensor Models (CTM), a particular kind of tensor models,
that have been intensively studied in the last years, mainly as an approach to quantum gravity in
dimension greater than two. CTMs established a link between colored graphs and tensor mod-
els, since the Feynman graphs of a d-dimensional CTM are precisely (d + 1)-colored graphs.
Furthermore, the G-degree of a colored graph is a crucial quantity driving the 1/N expansion of
the free energy of a CTM.
This talk will mainly concern recent results achieved in dimension 4: in particular, the clas-
sification of all compact PL 4-manifolds with generalized regular genus at most one or with
G-degree at most 18. Furthermore, we will discuss interesting classes of 5-colored graphs
(semi-simple and weak semi-simple crystallizations), representing compact PL 4-manifolds
with empty or connected boundary and minimizing the invariants. In the simply-connected
case they also belong to a wider class of 5-colored graphs which are proved to induce handle-
decompositions of the represented 4-manifold lacking in 1- and/or 3-handles; therefore their
250
