Related papers: Intersection Graphs for String Links
Random intersection graphs have received much interest and been used in diverse applications. They are naturally induced in modeling secure sensor networks under random key predistribution schemes, as well as in modeling the topologies of…
We publish a table of primitive finite-type invariants of order less than or equal to six, for knots of ten or fewer crossings. We note certain mod-2 congruences, one of which leads to a chirality criterion in the Alexander polynomial. We…
This paper leverages linear systems theory to propose a principled measure of complexity for network systems. We focus on a network of first-order scalar linear systems interconnected through a directed graph. By locally filtering out the…
We give a detailed explicit computation of weights of Kontsevich graphs which arise from connection and curvature terms within the globalization picture for the special case of symplectic manifolds. We will show how the weights for the…
In real-world networks, predicting the weight (strength) of links is as crucial as predicting the existence of the links themselves. Previous studies have primarily used shallow graph features for link weight prediction, limiting the…
This paper contains linear systems of equations which can distinguish knots without knot invariants. Let $M_n$ be the topological moduli space of all n-component string links and such that a fixed projection into the plane is an immersion.…
We introduce a method of computing biquandle brackets of oriented knots and links using a type of decorated trivalent spatial graphs we call trace diagrams. We identify algebraic conditions on the biquandle bracket coefficients for moving…
Each knot invariant can be extended to singular knots according to the skein rule. A Vassiliev invariant of order at most $n$ is defined as a knot invariant that vanishes identically on knots with more than $n$ double points. A chord…
Interconnected networks of rigid struts are critical for application in lightweight, load-bearing structures. However, accurately modeling stress distribution in these strut lattices poses significant computational challenges due to its…
We present recent advances in harmonic analysis on infinite graphs. Our approach combines combinatorial tools with new results from the theory of unbounded Hermitian operators in Hilbert space, geometry, boundary constructions, and spectral…
We present two models for the space of knots which have endpoints at fixed boundary points in a manifold with boundary, one model defined as an inverse limit of spaces of maps between configuration spaces and another which is cosimplicial.…
Holonomy invariants in strict higher gauge theory have been studied in depth, aiming to applications to higher Chern-Simons theory. For a flat 2-connection, the holonomy of surface knots of arbitrary genus has been defined and its…
Topological metrics of graphs provide a natural way to describe the prominent features of various types of networks. Graph metrics describe the structure and interplay of graph edges and have found applications in many scientific fields. In…
We find that the overall UV divergences of a renormalizable field theory with trivalent vertices fulfil a four-term relation. They thus come close to establish a weight system. This provides a first explanation of the recent successful…
Hyperfinite knots, or limits of equivalence classes of knots induced by a knot invariant taking values in a metric space, were introduced in a previous article by the author. In this article, we present new examples of hyperfinite knots…
We prove that the detection rate of n-crossing alternating links by many standard link invariants decays exponentially in n, implying that they detect alternating links with probability zero. This phenomenon applies broadly, in particular…
We give a fresh introduction to the Khovanov Homology theory for knots and links, with special emphasis on its extension to tangles, cobordisms and 2-knots. By staying within a world of topological pictures a little longer than in other…
We defined a grid homology theory for spatial graphs. We showed that the skein exact sequence of singular knots can be extended to our grid homology for spatial graphs.
We use a spanning tree model to prove a result of E. S. Lee on the support of Khovanov homology of alternating knots.
We consider an algebra of (classical or virtual) tangles over an ordered circuit operad and introduce Conway-type invariants of tangles which respect this algebraic structure. The resulting invariants contain both the coefficients of the…