3,210 research outputs found
Embeddings and immersions of tropical curves
We construct immersions of trivalent abstract tropical curves in the
Euclidean plane and embeddings of all abstract tropical curves in higher
dimensional Euclidean space. Since not all curves have an embedding in the
plane, we define the tropical crossing number of an abstract tropical curve to
be the minimum number of self-intersections, counted with multiplicity, over
all its immersions in the plane. We show that the tropical crossing number is
at most quadratic in the number of edges and this bound is sharp. For curves of
genus up to two, we systematically compute the crossing number. Finally, we use
our immersed tropical curves to construct totally faithful nodal algebraic
curves via lifting results of Mikhalkin and Shustin.Comment: 23 pages, 14 figures, final submitted versio
Deformation of string topology into homotopy skein modules
Relations between the string topology of Chas and Sullivan and the homotopy
skein modules of Hoste and Przytycki are studied. This provides new insight
into the structure of homotopy skein modules and their meaning in the framework
of quantum topology. Our results can be considered as weak extensions to all
orientable 3-manifolds of classical results by Turaev and Goldman concerning
intersection and skein theory on oriented surfaces.Comment: Published by Algebraic and Geometric Topology at
http://www.maths.warwick.ac.uk/agt/AGTVol3/agt-3-34.abs.htm
On invariants and homology of spaces of knots in arbitrary manifolds
Finite-order invariants of knots in arbitrary 3-manifolds (including
non-orientable ones) are constructed and studied by methods of the topology of
discriminant sets. Obstructions to the integrability of admissible weight
systems to well-defined knot invariants are identified as 1-dimensional
cohomology classes of generalized loop spaces of the manifold. Unlike the case
of the 3-sphere, these obstructions can be non-trivial and provide invariants
of the manifold itself.
The corresponding algebraic machinery allows us to obtain on the level of the
``abstract nonsense'' some of results and problems of the theory, and to
extract from other the essential topological part
HOMFLYPT Skein Theory, String Topology and 2-Categories
We show that relations in Homflypt type skein theory of an oriented
-manifold are induced from a -groupoid defined from the fundamental
-groupoid of a space of singular links in . The module relations are
defined by homomorphisms related to string topology. They appear from a
representation of the groupoid into free modules on a set of model objects. The
construction on the fundamental -groupoid is defined by the singularity
stratification and relates Vassiliev and skein theory. Several explicit
properties are discussed, and some implications for skein modules are derived.Comment: 55 pages, 1 figur
Chord Diagrams and Gauss Codes for Graphs
Chord diagrams on circles and their intersection graphs (also known as circle
graphs) have been intensively studied, and have many applications to the study
of knots and knot invariants, among others. However, chord diagrams on more
general graphs have not been studied, and are potentially equally valuable in
the study of spatial graphs. We will define chord diagrams for planar
embeddings of planar graphs and their intersection graphs, and prove some basic
results. Then, as an application, we will introduce Gauss codes for immersions
of graphs in the plane and give algorithms to determine whether a particular
crossing sequence is realizable as the Gauss code of an immersed graph.Comment: 20 pages, many figures. This version has been substantially
rewritten, and the results are stronge
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