Study explores properties of bipartite knots.
problem None explicitly stated; focuses on properties of bipartite knots.
method Exploration of combinatorial structure.
result Rich combinatorial structure of bipartite knots.
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48 results for “bipartite knots”
New research finds six bipartite intrinsically knotted graphs with 23 edges.
problem Identifying intrinsically knotted bipartite graphs with 23 edges.
method Analyzing embeddings and graph minors to find minimal intrinsically knotted graphs.
result No minor minimal intrinsically knotted bipartite graph exists with 23 edges.
New method extends knot theory to non-bipartite knots, revealing PDs.
problem Extending knot theory to non-bipartite knots.
method Developed a new positive decomposition (PD) for HOMFLY polynomials of non-bipartite knots.
result PD exists for non-bipartite knots, not just bipartite ones.
A graph is intrinsically knotted if every embedding contains a knotted cycle. It is known that intrinsically knotted graphs have at least 21 edges and that the KS graphs, and the 13 graphs obtained from by moves, are the only minor minimal intrinsically knotted graphs with 21 edges. This set incl…
We present evidence in support of a conjecture that a bipartite graph with at least five vertices in each part and |E(G)| \geq 4 |V(G)| - 17 is intrinsically knotted. We prove the conjecture for graphs that have exactly five or exactly six vertices in one part. We also show that there is a constant C_n such that a bipa…
Simplified Khovanov-Rozansky calculus for bipartite knots.
problem Complexity in calculating superpolynomials for knots.
method Bipartite calculus generalizes Khovanov-Rozansky calculus for a restricted class of knots.
result Simplification of Khovanov-Rozansky polynomials for bipartite knots.
A quantum N-dimer modelmath.QA
Quantum model for knotted graphs from knot theory.
problem Constructing an isotopy invariant polynomial for knotted bipartite ribbon graphs.
method Applying quantum topology to construct an isotopy invariant polynomial.
result Computed the expected number of loops in the double dimer model.
We give an algorithmic computation for the height of Kauffman's clock lattice obtained from a knot diagram with two adjacent regions starred and without crossing information specified. We show that this lattice is more familiarly the graph of perfect matchings of a bipartite graph obtained from the knot diagram by over…
Planar decomposition simplifies HOMFLY polynomial calculation for certain knots and links.
problem Calculating HOMFLY polynomial for specific types of knots and links.
method Planar decomposition of bipartite diagrams, lifting from sl(2) to sl(N).
result HOMFLY polynomials of many knots and links have planar decompositions.
We generalize the construction of the Heegaard Floer homology for a singular knot to that for a balanced bipartite graph. For a given graph, we provide a combinatorial description of the Euler characteristic of its Heegaard Floer homology by using the "Kauffman states" on a graph diagram.
The paper calculates a specific weight system for chord diagrams with a particular graph structure.
problem Calculating a specific weight system for chord diagrams with a complete bipartite graph structure.
method Using a Lie algebra and its weight system, the authors derive a function on chord diagrams.
result The authors compute the weight system for chord diagrams with a complete bipartite graph structure.
Bipartite knotsmath.GT
We give a solution to a part of Problem 1.60 in Kirby's list of open problems in topology thus answering in the positive the 1987 conjecture by J.Przytycki concerning the existence of knots without matched diagrams.
In recent work the author investigates perfect matchings of a bipartite graph obtained from a knot diagram and demonstrates that these correspond to discrete Morse functions on a 2-complex for the 2-sphere. This relationship is expounded below for the opposite audience: those who may be unfamiliar with knots.
This paper presents an algorithm to construct a weighted adjacency matrix of a plane bipartite graph obtained from a pretzel knot diagram. The determinant of this matrix after evaluation is shown to be the Jones polynomial of the pretzel knot by way of perfect matchings (or dimers) of this graph. The weights are Tutte'…
The interior polynomial is an invariant of (signed) bipartite graphs, and the interior polynomial of a plane bipartite graph is equal to a part of the HOMFLY polynomial of a naturally associated link. The HOMFLY polynomial is a famous link invariant with many known properties. For example, the HOMFLY polynom…
New method for calculating HOMFLY polynomials in symmetric representations.
problem Calculating HOMFLY polynomials for symmetric representations.
method Planar decomposition and projection to symmetric representations.
result Restoration of planarity and new insights into HOMFLY polynomials.
This paper improves upper bounds on ribbonlength for certain alternating links.
problem Finding tighter bounds on the ribbonlength of alternating links.
method Proved a new upper bound for alternating links with bipartite dual graphs.
result Improved upper bounds on ribbonlength for specific links.
Circle graph complexes reveal link properties via Khovanov homology.
problem Understanding topological properties of links using circle graph complexes.
method Analyzing homotopy types and computing Khovanov homology of specific knots.
result Real extreme Khovanov homology of a 4-strand pretzel knot computed.
Formula for weight system on complete bipartite graphs.
problem Computing values of weight system for chord diagrams.
method Chmutov-Varchenko recurrence relation, Hopf algebra projections.
result Computed values for chord diagrams with complete bipartite intersection graphs.
Automates machine learning of correlations between knot invariants.
problem Discovering and validating new relationships between knot invariants.
method Trained a neural network on 200,000 sets of knot invariants to predict an output invariant.
result Found novel correlations not explained by known results in knot theory.
Simplified Khovanov polynomials for bipartite links.
problem Computing Khovanov polynomials for bipartite links.
method Reduced Khovanov-Rozansky technique to Kauffman-Khovanov cycle calculus.
result Consistency demonstrated between reduced technique and bipartite Khovanov polynomials.
Bipartite networks are a common type of network data in which there are two types of vertices, and only vertices of different types can be connected. While bipartite networks exhibit community structure like their unipartite counterparts, existing approaches to bipartite community detection have drawbacks, including im…
In this work we present a complete (no misses, no duplicates) census for closed, connected, orientable and prime 3-manifolds induced by plane graphs with a bipartition of its edge set (blinks) up to edges. Blinks form a universal encoding for such manifolds. In fact, each such a manifold is a subtle class of blin…
New model for detecting communities in weighted bipartite networks.
problem Lack of models for weighted bipartite networks.
method Introducing Bipartite Distribution-Free model and its extension.
result Spectral algorithms for consistent estimation of node labels.
This article is an exposition of a body of existing results, together with an announcement of recent results. We discuss a theory of polytopes associated to bipartite graphs and trinities, developed by Kálmán, Postnikov and others. This theory exhibits a variety of interesting duality and triality relations, and extend…
Proves Khovanov homology has no torsion for bipartite circle graphs.
problem Proving properties of Khovanov homology for bipartite circle graphs.
method Proved homotopy equivalence of independence complexes to wedges of spheres.
result Extreme Khovanov homology has no torsion.
Improved bipartite link prediction using 2-hop paths.
problem Link prediction in bipartite networks without node attributes.
method Multiply reconstructed adjacency matrix with symmetrically normalized training adjacency matrix to form 2-hop paths.
result 2-hop paths improve link prediction performance.
New model for detecting communities in weighted bipartite networks.
problem No model for community detection in overlapping bipartite weighted networks.
method Introduces BiMMDF model allowing any distribution with block structure.
result Efficient algorithm with theoretical guarantee of consistent estimation.
Bipartite graphs have been used to represent data relationships in many data-mining applications such as in E-commerce recommendation systems. Since learning in graph space is more complicated than in Euclidian space, recent studies have extensively utilized neural nets to effectively and efficiently embed a graph's no…
A new method calculates HOMFLY-PT polynomials for bipartite links.
problem Computing HOMFLY-PT polynomials for bipartite links efficiently.
method Generalizes Goeritz matrix method for bipartite links.
result Reduces HOMFLY-PT polynomial calculation to matrix algebra.
We define integral odd Khovanov homology of principally unimodular bipartite graph-links.
PAC learning simplified as bipartite matching.
problem Efficiently solving PAC learning problems.
method Transductive learning and one-inclusion graphs.
result PAC learning can be reduced to bipartite matching.
We present a simple combinatorial model for quasipositive surfaces and positive braids, based on embedded bipartite graphs. As a first application, we extend the well-known duality on standard diagrams of torus links to twisted torus links. We then introduce a combinatorial notion of adjacency for bipartite graph links…
Develops a new variational estimator for node popularity in bipartite networks.
problem Estimating node popularity in bipartite networks with varying patterns.
method Variational Expectation-Maximization (VEM) framework for the Two-Way Node Popularity Model (TNPM).
result The proposed method achieves superior estimation accuracy across different types of networks.
Improved text summarization using belief propagation on weighted bipartite graphs.
problem Text summarization from a graph theory perspective.
method Generalized belief propagation algorithm for weighted bipartite graphs.
result Our algorithm outperforms greedy methods in text summarization tasks.
Bipartite Riemann-Finsler geometries with complementary Finsler structures are constructed. Calculable examples are presented based on a bilinear-form coefficient for explicit Lorentz violation.
Incorrect parity-based descriptions of realizable Gauss diagrams found, but bipartite graphs provide a valid approach.
problem Incorrect descriptions of realizable Gauss diagrams using parity conditions.
method Used bipartite graphs to describe realizable Gauss diagrams.
result Realizable Gauss diagrams can be accurately described using bipartite graphs.
Common Structure Discovery in Collections of Bipartite Networks: Application to Pollination Systemsstat.ML
A new model detects common patterns in pollination networks.
problem Comparing organization of bipartite networks to understand community structure.
method colBiSBM, a family of probabilistic models for collections of bipartite networks.
result The method uncovers shared ecological roles and partitions networks.
We characterize which automorphisms of an arbitrary complete bipartite graph can be induced by a homeomorphism of some embedding of the graph in .
Neural execution solves complex graph problems like bipartite matching.
problem Solving complex graph algorithms like maximum bipartite matching.
method Reduces bipartite matching to a flow problem and uses Ford-Fulkerson for maximum flow.
result Neural network achieves optimal matching almost 100% of the time.
The symmetries of complex molecular structures can be modeled by the {\em topological symmetry group} of the underlying embedded graph. It is therefore important to understand which topological symmetry groups can be realized by particular abstract graphs. This question has been answered for complete graphs; it is natu…
New tests detect communities in dense bipartite graphs with high accuracy.
problem Detecting communities in dense bipartite graphs with high accuracy.
method Non-asymptotic upper and lower bounds, novel minimax-optimal tests, hard-thresholded nonlinear statistics.
result Non-asymptotic upper and lower bounds match for any configuration of graph sizes.
We study the Thurston-Bennequin number of complete and complete bipartite Legendrian graphs. We define a new invariant called the total Thurston-Bennequin number of the graph. We show that this invariant is determined by the Thurston-Bennequin numbers of 3-cycles for complete graphs and by the Thurston-Bennequin number…
In this paper we analyse the bipartite Colombian firms-products network, throughout a period of five years, from 2010 to 2014. Our analysis depicts a strongly modular system, with several groups of firms specializing in the export of specific categories of products. These clusters have been detected by running the bipa…
Positive Thompson links are arborescent tangles.
problem Understanding the structure of Thompson group elements.
method Analyzing closures of bipartite arborescent tangles.
result Positive Thompson links correspond to arborescent tangles.
In bipartite networks, community structures are restricted to being disassortative, in that nodes of one type are grouped according to common patterns of connection with nodes of the other type. This makes the stochastic block model (SBM), a highly flexible generative model for networks with block structure, an intuiti…
There exists a simplified Bar-Natan Khovanov complex for open 2-braids. The Khovanov cohomology of a knot diagram made by gluing tangles of this type is therefore often amenable to calculation. We lift this idea to the level of the Lipshitz-Sarkar stable homotopy type and use it to make new computations. Similarly, the…
The paper connects knot theory and cluster algebras via dimer face polynomials.
problem Understanding the relationship between knot theory and cluster algebras.
method Analyzing dimer face polynomials and their connections to Alexander polynomials and cluster algebras.
result Dimer face polynomials are multivariate generalizations of Alexander polynomials and -polynomials in cluster algebras.