Related papers: Adequate links in thickened surfaces and the gener…
We define smooth notions of concordance and sliceness for spatial graphs. We prove that sliceness of a spatial graph is equivalent to a condition on a set of linking numbers together with sliceness of a link associated to the graph. This…
Let $K$ be a link of Conway's normal form $C(m)$, $m \geq 0$, or $C(m,n)$ with $mn\textgreater{}0$, and let $D$ be a trigonal diagram of $K.$ We show that it is possible to transform $D$ into an alternating trigonal diagram, so that all…
Two skew diagrams are defined to be equivalent if their corresponding skew Schur functions are equal. The equivalence classes for ribbons have been classified by Billera, Thomas and van Willigenburg in 2006. In this paper, we provide a…
Edge connectivity and vertex connectivity are two fundamental concepts in graph theory. Although by now there is a good understanding of the structure of graphs based on their edge connectivity, our knowledge in the case of vertex…
It is well known that the braid index of a link equals the minimum number of Seifert circles among all link diagrams representing it. For a link with a reduced alternating diagram $D$, $s(D)$, the number of Seifert circles in $D$, equals…
We consider triangulations of closed surfaces S with a given set of vertices V; every triangulation can be branched that is enhanced to a Delta-complex. Branched triangulations are considered up to the b-transit equivalence generated by…
We give a combinatorial description of the ``$D_{2n}$ planar algebra,'' by generators and relations. We explain how the generator interacts with the Temperley-Lieb braiding. This shows the previously known braiding on the even part extends…
We establish a criterion that ensures a bounded almost complex curve in a bounded almost complex 4-manifold minimizes genus amongst all smooth surfaces that share its homology class and the transverse link on its boundary. An immediate…
The statistical mechanics of polymers grafted on surfaces has been the subject of intense research activity because of many potential applications. In this paper, we analytically investigate the conformational changes caused by a single…
This paper defines several algebras associated to an oriented surface $S$ with a finite set of marked points on the boundary. The first is the skein algebra $Sk_q(S)$, which is spanned by links in the surface which are allowed to have…
The class of +adequate links contains both alternating and positive links. Generalizing results of Tanaka (for the positive case) and Ng (for the alternating case), we construct fronts of an arbitrary +adequate link A so that the diagram…
Tutte's spring embedding theorem states that, for a three-connected planar graph, if the outer face of the graph is fixed as the complement of some convex region in the plane, and all other vertices are placed at the mass center of their…
Let $G$ be a graph and $T$ be a vertex subset of $G$ with even cardinality. A $T$-join of $G$ is a subset $J$ of edges such that a vertex of $G$ is incident with an odd number of edges in $J$ if and only if the vertex belongs to $T$.…
We formulate a very general conjecture relating the analytical invariants of a normal surface singularity to the Seiberg-Witten invariants of its link provided that the link is a rational homology sphere. As supporting evidence, we…
There is a well-known way to describe a link diagram as a (signed) plane graph, called its Tait graph. This concept was recently extended, providing a way to associate a set of embedded graphs (or ribbon graphs) to a link diagram. While…
In this paper, we introduce a bisected vertex leveling of a plane graph. Using this planar embedding, we present elementary proofs of the well-known upper bounds in terms of the minimal crossing number on braid index $b(L)$ and arc index…
We provide two constructions for $t$ edge-disjoint maximal outerplanar graphs on every number of $n \geq 4t$ vertices. The bound on the minimum number of vertices is tight. These constructions yield the existence of optimal…
We consider network design problems in which we are given a graph and seek a min-size $2$-connected subgraph that satisfies a prescribed property. $\bullet$ In the 1-Connectivity Augmentation problem the goal is to augment a connected graph…
We show how the theory of the critical behaviour of $d$-dimensional polymer networks of arbitrary topology can be generalized to the case of networks confined by hyperplanes. This in particular encompasses the case of a single polymer chain…
We derive attainable upper bounds on the algebraic connectivity (spectral gap) of a regular graph in terms of its diameter and girth. This bound agrees with the well-known Alon-Boppana-Friedman bound for graphs of even diameter, but is an…