132 research outputs found
Extending Topological Surgery to Natural Processes and Dynamical Systems
Topological surgery is a mathematical technique used for creating new
manifolds out of known ones. We observe that it occurs in natural phenomena
where a sphere of dimension 0 or 1 is selected, forces are applied and the
manifold in which they occur changes type. For example, 1-dimensional surgery
happens during chromosomal crossover, DNA recombination and when cosmic
magnetic lines reconnect, while 2-dimensional surgery happens in the formation
of tornadoes, in the phenomenon of Falaco solitons, in drop coalescence and in
the cell mitosis. Inspired by such phenomena, we introduce new theoretical
concepts which enhance topological surgery with the observed forces and
dynamics. To do this, we first extend the formal definition to a continuous
process caused by local forces. Next, for modeling phenomena which do not
happen on arcs or surfaces but are 2-dimensional or 3-dimensional, we fill in
the interior space by defining the notion of solid topological surgery. We
further introduce the notion of embedded surgery in for modeling
phenomena which involve more intrinsically the ambient space, such as the
appearance of knotting in DNA and phenomena where the causes and effect of the
process lies beyond the initial manifold, such as the formation of black holes.
Finally, we connect these new theoretical concepts with a dynamical system and
we present it as a model for both 2-dimensional 0-surgery and natural phenomena
exhibiting a `hole drilling' behavior. We hope that through this study,
topology and dynamics of many natural phenomena, as well as topological surgery
itself, will be better understood.Comment: 54 pages, 34 figure
An important step for the computation of the HOMFLYPT skein module of the lens spaces via braids
We prove that, in order to derive the HOMFLYPT skein module of the lens
spaces from the HOMFLYPT skein module of the solid torus,
, it suffices to solve an infinite system of equations
obtained by imposing on the Lambropoulou invariant for knots and links in
the solid torus, braid band moves that are performed only on the first moving
strand of elements in a set , augmenting the basis of
.Comment: 20 pages, 5 figures. arXiv admin note: substantial text overlap with
arXiv:1702.06290, arXiv:1604.0616
p-adic framed braids
In this paper we define the -adic framed braid group , arising as the inverse limit of the modular framed braids and
we give topological generators for . We also give
geometric interpretations for the -adic framed braids. We then construct a
-adic Yokonuma-Hecke algebra as the inverse limit of
a family of classical Yokonuma-Hecke algebras. These are quotients of the
modular framed braid groups over a quadratic relation. We also give topological
generators for . Finally, we construct on this new
algebra a linear trace that supports the Markov property.Comment: 35 pages, 14 figures, LaTex documen
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