5,186 research outputs found
Self-Assembly of DNA Graphs and Postman Tours
DNA graph structures can self-assemble from branched junction molecules to yield solutions to computational problems. Self-assembly of graphs have previously been shown to give polynomial time solutions to hard computational problems such as 3-SAT and k-colorability problems. Jonoska et al. have proposed studying self-assembly of graphs topologically, considering the boundary components of their thickened graphs, which allows for reading the solutions to computational problems through reporter strands. We discuss weighting algorithms and consider applications of self-assembly of graphs and the boundary components of their thickened graphs to problems involving minimal weight Eulerian walks such as the Chinese Postman Problem and the Windy Postman Problem
Genus Ranges of Chord Diagrams
A chord diagram consists of a circle, called the backbone, with line
segments, called chords, whose endpoints are attached to distinct points on the
circle. The genus of a chord diagram is the genus of the orientable surface
obtained by thickening the backbone to an annulus and attaching bands to the
inner boundary circle at the ends of each chord. Variations of this
construction are considered here, where bands are possibly attached to the
outer boundary circle of the annulus. The genus range of a chord diagram is the
genus values over all such variations of surfaces thus obtained from a given
chord diagram. Genus ranges of chord diagrams for a fixed number of chords are
studied. Integer intervals that can, and cannot, be realized as genus ranges
are investigated. Computer calculations are presented, and play a key role in
discovering and proving the properties of genus ranges.Comment: 12 pages, 8 figure
Unsigned state models for the Jones polynomial
It is well a known and fundamental result that the Jones polynomial can be
expressed as Potts and vertex partition functions of signed plane graphs. Here
we consider constructions of the Jones polynomial as state models of unsigned
graphs and show that the Jones polynomial of any link can be expressed as a
vertex model of an unsigned embedded graph.
In the process of deriving this result, we show that for every diagram of a
link in the 3-sphere there exists a diagram of an alternating link in a
thickened surface (and an alternating virtual link) with the same Kauffman
bracket. We also recover two recent results in the literature relating the
Jones and Bollobas-Riordan polynomials and show they arise from two different
interpretations of the same embedded graph.Comment: Minor corrections. To appear in Annals of Combinatoric
Twisty itsy bitsy topological field theory
We extend the topological field theory (``itsy bitsy topological field
theory"') of our previous work from mod-2 to twisted coefficients. This
topological field theory is derived from sutured Floer homology but described
purely in terms of surfaces with signed points on their boundary (occupied
surfaces) and curves on those surfaces respecting signs (sutures). It has
information-theoretic (``itsy'') and quantum-field-theoretic (``bitsy'')
aspects. In the process we extend some results of sutured Floer homology,
consider associated ribbon graph structures, and construct explicit admissible
Heegaard decompositions.Comment: 52 pages, 26 figure
What is a virtual link?
Several authors have recently studied virtual knots and links because they
admit invariants arising from R-matrices. We prove that every virtual link is
uniquely represented by a link L in S X I, a thickened, compact, oriented
surface S, such that the link complement (S X I) - L has no essential vertical
cylinder.Comment: Published by Algebraic and Geometric Topology at
http://www.maths.warwick.ac.uk/agt/AGTVol3/agt-3-20.abs.htm
- …