scholarly journals On a uniformly random chord diagram and its intersection graph

2017 ◽  
Vol 340 (8) ◽  
pp. 1967-1985 ◽  
Author(s):  
Hüseyin Acan
Keyword(s):  
Intersection Graph ◽  
Chord Diagram

scholarly journals THE INTERSECTION GRAPH CONJECTURE FOR LOOP DIAGRAMS

2000 ◽  
Vol 09 (02) ◽  
pp. 187-211 ◽  
Author(s):  
BLAKE MELLOR
Keyword(s):  
Equivalence Class ◽  
Intersection Graph ◽  
Chord Diagram ◽  
Single Loop ◽  
Vassiliev Invariants ◽  
Chord Diagrams

The study of Vassiliev invariants for knots can be reduced to the study of the algebra of chord diagrams modulo certain relations (as done by Bar-Natan). Chmutov, Duzhin and Lando defined the idea of the intersection graph of a chord diagram, and conjectured that these graphs determine the equivalence class of the chord diagrams. They proved this conjecture in the case when the intersection graph is a tree. This paper extends their proof to the case when the graph contains a single loop, and determines the size of the subalgebra generated by the associated "loop diagrams." While the conjecture is known to be false in general, the extent to which it fails is still unclear, and this result helps to answer that question.

scholarly journals WEIGHT SYSTEMS FOR MILNOR INVARIANTS

2008 ◽  
Vol 17 (02) ◽  
pp. 213-230 ◽  
Author(s):  
BLAKE MELLOR
Keyword(s):  
Finite Type ◽  
Spanning Tree ◽  
Recursive Relation ◽  
Intersection Graph ◽  
Chord Diagram ◽  
Link Invariants ◽  
Finite Type Invariants ◽  
Milnor Invariants ◽  
Skein Relation

We use Polyak's skein relation to give a new proof that Milnor's string link invariants μ12⋯n are finite type invariants, and to develop a recursive relation for their associated weight systems. We show that the obstruction to the triviality of these weight systems is the presence of a certain kind of spanning tree in the intersection graph of a chord diagram.

scholarly journals TREE DIAGRAMS FOR STRING LINKS

2006 ◽  
Vol 15 (10) ◽  
pp. 1303-1318 ◽  
Author(s):  
BLAKE MELLOR
Keyword(s):  
Finite Type ◽  
Intersection Graph ◽  
Chord Diagram ◽  
Intersection Graphs ◽  
Finite Type Invariants ◽  
Link Diagrams ◽  
String Links

In previous work, the author defined the intersection graph of a chord diagram associated with a string link (as in the theory of finite type invariants). In this paper, we classify the trees which can be obtained as intersection graphs of string link diagrams.

scholarly journals TREE DIAGRAMS FOR STRING LINKS II: DETERMINING CHORD DIAGRAMS

2008 ◽  
Vol 17 (06) ◽  
pp. 649-664
Author(s):  
BLAKE MELLOR
Keyword(s):  
Finite Type ◽  
Intersection Graph ◽  
Chord Diagram ◽  
Finite Type Invariants ◽  
Chord Diagrams ◽  
String Links

In previous work [7], we defined the intersection graph of a chord diagram associated with a string link (as in the theory of finite type invariants). In this paper, we look at the case when this graph is a tree, and we show that in many cases these trees determine the chord diagram (modulo the usual 1-term and 4-term relations).

scholarly journals Study on the Identification Method of Pyrogenetic Rock Based on N -DP Intersection Graph

2021 ◽  
Vol 804 (2) ◽  
pp. 022020
Author(s):  
Gang Hu ◽  
Zhiqiang Mao ◽  
Ping Yan ◽  
Qian Yin ◽  
Xiyu Wang ◽  
...  
Keyword(s):  
Intersection Graph ◽  
Identification Method

scholarly journals Turán-type results for intersection graphs of boxes

2021 ◽  
pp. 1-6
Author(s):  
István Tomon ◽  
Dmitriy Zakharov
Keyword(s):  
Short Note ◽  
Intersection Graph ◽  
Intersection Graphs ◽  
Poset Dimension ◽  
Axis Parallel ◽  
Turán Theorem ◽  
Separation Dimension

Abstract In this short note, we prove the following analog of the Kővári–Sós–Turán theorem for intersection graphs of boxes. If G is the intersection graph of n axis-parallel boxes in $${{\mathbb{R}}^d}$$ such that G contains no copy of K t,t , then G has at most ctn( log n)2d+3 edges, where c = c(d)>0 only depends on d. Our proof is based on exploring connections between boxicity, separation dimension and poset dimension. Using this approach, we also show that a construction of Basit, Chernikov, Starchenko, Tao and Tran of K2,2-free incidence graphs of points and rectangles in the plane can be used to disprove a conjecture of Alon, Basavaraju, Chandran, Mathew and Rajendraprasad. We show that there exist graphs of separation dimension 4 having superlinear number of edges.

scholarly journals The intersection graph of a finite simple group has diameter at most 5

2021 ◽  
Author(s):  
Saul D. Freedman
Keyword(s):  
Simple Group ◽  
Upper Bound ◽  
Finite Simple Group ◽  
Unitary Groups ◽  
Intersection Graph ◽  
Monster Group ◽  
Prime Dimension

AbstractLet G be a non-abelian finite simple group. In addition, let $$\Delta _G$$ Δ G be the intersection graph of G, whose vertices are the proper non-trivial subgroups of G, with distinct subgroups joined by an edge if and only if they intersect non-trivially. We prove that the diameter of $$\Delta _G$$ Δ G has a tight upper bound of 5, thereby resolving a question posed by Shen (Czechoslov Math J 60(4):945–950, 2010). Furthermore, a diameter of 5 is achieved only by the baby monster group and certain unitary groups of odd prime dimension.

A Krasnosel’skii-type theorem for certain orthogonal polytopes starshaped via k-staircase paths

2020 ◽  
Vol 20 (2) ◽  
pp. 169-177 ◽  
Author(s):  
Marilyn Breen
Keyword(s):  
Type Theorem ◽  
Common Point ◽  
Finite Family ◽  
Intersection Graph ◽  
Axis Parallel ◽  
Orthogonal Polytopes

AbstractLet 𝒞 be a finite family of distinct axis-parallel boxes in ℝd whose intersection graph is a tree, and let S = ⋃{C : C in 𝒞}. If every two points of S see a common point of S via k-staircase paths, then S is starshaped via k-staircase paths. Moreover, the k-staircase kernel of S will be convex via k-staircases.

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