23 research outputs found
Knot Graphs
We consider the equivalence classes of graphs induced by the unsigned
versions of the Reidemeister moves on knot diagrams.
Any graph which is
reducible by some finite sequence of these moves, to a graph with no
edges is called a knot graph. We show that the class of knot graphs
strictly contains the set of delta-wye graphs. We prove that the
dimension of the intersection of the cycle and cocycle spaces is an
effective numerical invariant of these classes
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Time-dependent shortest paths in treelike graphs
We present a proof that the number of breakpoints in the arrival function between two terminals in graphs of treewidth ω is n^(O(log²ω) when the edge arrival functions are piecewise linear. This is an improvement on the bound of n^(Θ(log n))by Foschini, Hershberger, and Suri for graphs without any bound on treewidth. We provide an algorithm for calculating this arrival function using star-mesh transformations, a generalization of the wye-delta-wye transformations.
Key Words: treewidth, time-dependent shortest paths, star-mesh transformation
Bond-Propagation Algorithm for Thermodynamic Functions in General 2D Ising Models
Recently, we developed and implemented the bond propagation algorithm for
calculating the partition function and correlation functions of random bond
Ising models in two dimensions. The algorithm is the fastest available for
calculating these quantities near the percolation threshold. In this paper, we
show how to extend the bond propagation algorithm to directly calculate
thermodynamic functions by applying the algorithm to derivatives of the
partition function, and we derive explicit expressions for this transformation.
We also discuss variations of the original bond propagation procedure within
the larger context of Y-Delta-Y-reducibility and discuss the relation of this
class of algorithm to other algorithms developed for Ising systems. We conclude
with a discussion on the outlook for applying similar algorithms to other
models.Comment: 12 pages, 10 figures; submitte
Tightening curves and graphs on surfaces
Any continuous deformation of closed curves on a surface can be decomposed into a finite sequence of local changes on the structure of the curves; we refer to such local operations as homotopy moves. Tightening is the process of deforming given curves into their minimum position; that is, those with minimum number of self-intersections. While such operations and the tightening process has been studied extensively, surprisingly little is known about the quantitative bounds on the number of homotopy moves required to tighten an arbitrary curve.
An unexpected connection exists between homotopy moves and a set of local operations on graphs called electrical transformations. Electrical transformations have been used to simplify electrical networks since the 19th century; later they have been used for solving various combinatorial problems on graphs, as well as applications in statistical mechanics, robotics, and quantum mechanics. Steinitz, in his study of 3-dimensional polytopes, looked at the electrical transformations through the lens of medial construction, and implicitly established the connection to homotopy moves; later the same observation has been discovered independently in the context of knots.
In this thesis, we study the process of tightening curves on surfaces using homotopy moves and their consequences on electrical transformations from a quantitative perspective. To derive upper and lower bounds we utilize tools like curve invariants, surface theory, combinatorial topology, and hyperbolic geometry. We develop several new tools to construct efficient algorithms on tightening curves and graphs, as well as to present examples where no efficient algorithm exists. We then argue that in order to study electrical transformations, intuitively it is most beneficial to work with monotonic homotopy moves instead, where no new crossings are created throughout the process; ideas and proof techniques that work for monotonic homotopy moves should transfer to those for electrical transformations. We present conjectures and partial evidence supporting the argument