Related papers: Recognizing trace graphs of closed braids
We consider the class of graphs for which the edge connectivity is equal to the maximum number of edge-disjoint spanning trees, and the natural generalization to matroids, where the cogirth is equal to the number of disjoint bases. We…
A connected graph can be associated with two distinct evolution algebras. In the first case, the structural matrix is the adjacency matrix of the graph itself. In the second case, the structural matrix is the transition probabilities matrix…
We report on the geometry and mechanics of knotted stiff strings. We discuss both closed and open knots. Our two main results are: (i) Their equilibrium energy as well as the equilibrium tension for open knots depend on the type of knot as…
We provide linear lower bounds for the signature of positive braids in terms of the three genus of their braid closure. This yields linear bounds for the topological slice genus of knots that arise as closures of positive braids.
This is an expository article on diagrammatic representations of knots and links in various settings via braids.
A map graph is a graph admitting a representation in which vertices are nations on a spherical map and edges are shared curve segments or points between nations. We present an explicit fixed-parameter tractable algorithm for recognizing map…
The Rickard complex of a braid with strands colored by positive integers is a chain complex of singular Soergel bimodules. The complex determines the colored triply-graded homology and colored sl(N) homology of the braid closure, when…
Given a graph, does there exist an orientation of the edges such that the resulting directed graph is strongly connected? Robbins' theorem [Robbins, Am. Math. Monthly, 1939] states that such an orientation exists if and only if the graph is…
Many well studied knots can be realized as positive braid knots where the braid word contains a positive full twist; we say that such knots are twist positive. Some important families of knots are twist positive, including torus knots,…
Triadic closure, the formation of a connection between two nodes in a network sharing a common neighbor, is considered a fundamental mechanism determining the clustered nature of many real-world topologies. In this work we define a static…
For every knot K with stick number k there is a knotted polyhedral torus of knot type K with 3k vertices. We prove that at least 3k-2 vertices are necessary.
Traditionally, knot theorists have considered projections of knots where there are two strands meeting at every crossing. A triple crossing is a crossing where three strands meet at a single point, such that each strand bisects the…
We solve the isomorphism problem for braid groups on trees with $n = 4$ or 5 strands. We do so in three main steps, each of which is interesting in its own right. First, we establish some tools and terminology for dealing with computations…
The Markov Theorem Without Stabilization (MTWS) established the existence of a calculus of braid isotopies that can be used to move between closed braid representatives of a given oriented link type without having to increase the braid…
A scramble on a connected multigraph is a collection of connected subgraphs that generalizes the notion of a bramble. The maximum order of a scramble, called the scramble number of a graph, was recently developed as a tool for lower…
A simple graph is $3$-rigid if its generic bar-joint frameworks in $R^3$ are infinitesimally rigid. Necessary and sufficient conditions are obtained for the minimal $3$-rigidity of a simple graph which is obtained from the $1$-skeleton of a…
A tree is called k-ended tree if it has at most k leaves, where a leaf is a vertex of degree one. In this paper we prove that every 3-regular connected graph with n vertices such that n is greater than 8 has spanning sub tree with at most…
We give a simple characterization of braids that can be unplaited keeping separately their upper ends and their lower ends tied together
In the following text we show if $X$ is an Alexandroff space, then $f:X\to Y$ has closed graph if and only if it has constant closed value on each connected component of $X$. Moreover, if $X$ an Alexandroff space and $f:X\to Y$ has closed…
Two signed graphs are called switching isomorphic if one of them is isomorphic to a switching equivalent of the other. To determine the number of switching non-isomorphic signed graphs on a specific graph, we will establish a method based…