scholarly journals Expansions of a Chord Diagram and Alternating Permutations

10.37236/5120 ◽  
2016 ◽  
Vol 23 (1) ◽  
Author(s):  
Tomoki Nakamigawa
Keyword(s):  
Finite Sequence ◽  
Chord Diagram ◽  
Alternating Permutations ◽  
Chord Diagrams ◽  
One To One

A chord diagram is a set of chords of a circle such that no pair of chords has a common endvertex. A chord diagram $E$ with $n$ chords is called an $n$-crossing if all chords of $E$ are mutually crossing. A chord diagram $E$ is called nonintersecting if $E$ contains no $2$-crossing. For a chord diagram $E$ having a $2$-crossing $S = \{ x_1 x_3, x_2 x_4 \}$, the expansion of $E$ with respect to $S$ is to replace $E$ with $E_1 = (E \setminus S) \cup \{ x_2 x_3, x_4 x_1 \}$ or $E_2 = (E \setminus S) \cup \{ x_1 x_2, x_3 x_4 \}$. It is shown that there is a one-to-one correspondence between the multiset of all nonintersecting chord diagrams generated from an $n$-crossing with a finite sequence of expansions and the set of alternating permutations of order $n+1$.

scholarly journals Genus Ranges of Chord Diagrams

2015 ◽  
Vol 24 (04) ◽  
pp. 1550022 ◽  
Author(s):  
Jonathan Burns ◽  
Nataša Jonoska ◽  
Masahico Saito
Keyword(s):  
Outer Boundary ◽  
Orientable Surface ◽  
Fixed Number ◽  
Chord Diagram ◽  
Boundary Circle ◽  
Line Segments ◽  
Chord Diagrams ◽  
Computer Calculations

A chord diagram consists of a circle, called the backbone, with line segments, called chords, whose endpoints are attached to distinct points on the circle. The genus of a chord diagram is the genus of the orientable surface obtained by thickening the backbone to an annulus and attaching bands to the inner boundary circle at the ends of each chord. Variations of this construction are considered here, where bands are possibly attached to the outer boundary circle of the annulus. The genus range of a chord diagram is the genus values over all such variations of surfaces thus obtained from a given chord diagram. Genus ranges of chord diagrams for a fixed number of chords are studied. Integer intervals that can be, and those that cannot be, realized as genus ranges are investigated. Computer calculations are presented, and play a key role in discovering and proving the properties of genus ranges.

scholarly journals Ribbon graphs and bialgebra of Lagrangian subspaces

2016 ◽  
Vol 25 (12) ◽  
pp. 1642006 ◽  
Author(s):  
Victor Kleptsyn ◽  
Evgeny Smirnov
Keyword(s):  
Vector Space ◽  
Symplectic Form ◽  
Chord Diagram ◽  
Ribbon Graph ◽  
Dimensional Vector Space ◽  
Ribbon Graphs ◽  
Chord Diagrams ◽  
Intersection Matrix ◽  
Lagrangian Subspace ◽  
Lagrangian Subspaces

To each ribbon graph we assign a so-called [Formula: see text]-space, which is a Lagrangian subspace in an even-dimensional vector space with the standard symplectic form. This invariant generalizes the notion of the intersection matrix of a chord diagram. Moreover, the actions of Morse perestroikas (or taking a partial dual) and Vassiliev moves on ribbon graphs are reinterpreted nicely in the language of [Formula: see text]-spaces, becoming changes of bases in this vector space. Finally, we define a bialgebra structure on the span of [Formula: see text]-spaces, which is analogous to the 4-bialgebra structure on chord diagrams.

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 Alternating Permutations with Restrictions and Standard Young Tableaux

10.37236/2306 ◽  
2012 ◽  
Vol 19 (2) ◽  
Author(s):  
Yuexiao Xu ◽  
Sherry H. F. Yan
Keyword(s):  
Intermediate Structure ◽  
Young Tableaux ◽  
Alternating Permutations ◽  
One To One ◽  
Standard Young Tableaux

In this paper, we establish bijections between the set of 4123-avoiding down-up alternating permutations of length $2n$ and the set of standard Young tableaux of shape $(n,n,n)$, and between the set of 4123-avoiding down-up alternating permutations of length $2n-1$ and the set of shifted  standard Young tableaux of shape $(n+1, n, n-1)$ via an intermediate structure of Yamanouchi words. Moreover, we show that 4123-avoiding up-down alternating permutations of length $2n+1$  are in one-to-one correspondence with  standard Young tableaux of shape $(n+1,n,n-1)$, and 4123-avoiding up-down alternating permutations of length $2n$ are in bijection with shifted standard Young tableaux of shape $(n+2,n,n-2)$.

BRAIDED CHORD DIAGRAMS

1998 ◽  
Vol 07 (01) ◽  
pp. 1-22 ◽  
Author(s):  
Joan S. Birman ◽  
Rolland Trapp
Keyword(s):  
Finite Type ◽  
Equivalence Relation ◽  
Chord Diagram ◽  
Finite Type Invariants ◽  
Chord Diagrams ◽  
Braid Index

The notion of a braided chord diagram is introduced and studied. An equivalence relation is given which identifies all braidings of a fixed chord diagram. It is shown that finite-type invariants are stratified by braid index for knots which can be represented as closed 3-braids. Partial results are obtained about spanning sets for the algebra of chord diagrams of braid index 3.

scholarly journals Multistring based matrices

2020 ◽  
Vol 29 (06) ◽  
pp. 2050038
Author(s):  
David R. Freund
Keyword(s):  
Homotopy Class ◽  
Open Problem ◽  
Chord Diagram ◽  
Homotopy Classes ◽  
Chord Diagrams ◽  
Text String ◽  
Reidemeister Moves

A virtual[Formula: see text]-string is a chord diagram with [Formula: see text] core circles and a collection of arrows between core circles. We consider virtual [Formula: see text]-strings up to virtual homotopy, compositions of flat virtual Reidemeister moves on chord diagrams. Given a virtual 1-string [Formula: see text], Turaev associated a based matrix that encodes invariants of the virtual homotopy class of [Formula: see text]. We generalize Turaev’s method to associate a multistring based matrix to a virtual [Formula: see text]-string, addressing an open problem of Turaev and constructing similar invariants for virtual homotopy classes of virtual [Formula: see text]-strings.

scholarly journals Linear Chord Diagrams with Long Chords

10.37236/6809 ◽  
2017 ◽  
Vol 24 (4) ◽  
Author(s):  
Everett Sullivan
Keyword(s):  
Chord Diagram ◽  
Chord Diagrams ◽  
Geometric Sequence

A linear chord diagram of size $n$ is a partition of the set $\{1,2,\dots,2n\}$ into sets of size two, called chords. From a table showing the number of linear chord diagrams of degree $n$ such that every chord has length at least $k$, we observe that if we proceed far enough along the diagonals, they are given by a geometric sequence. We prove that this holds for all diagonals, and identify when the effect starts.

Supplement to a chord diagram of a ribbon surface-link

2017 ◽  
Vol 26 (05) ◽  
pp. 1750033 ◽  
Author(s):  
Akio Kawauchi
Keyword(s):  
Chord Diagram ◽  
Chord Diagrams ◽  
Ribbon Surface

A revised proof is given to an assertion on chord diagrams of a ribbon surface-link.

On better-quasi-ordering transfinite sequences

1968 ◽  
Vol 64 (2) ◽  
pp. 273-290 ◽  
Author(s):  
C. St. J. A. Nash-Williams
Keyword(s):  
Ordered Set ◽  
Ordinal Number ◽  
Finite Sequence ◽  
Ordered Sets ◽  
Transitive Relation ◽  
Ordinal Numbers ◽  
One To One ◽  
Quasi Ordering

AbstractLet Q be a quasi-ordered set, i.e. a set on which a reflexive and transitive relation ≤ is defined. If, for every finite sequence q1, q2, … of elements of Q, there exist i and j such that i < j and qi ≤ qj then we call Q well-quasi-ordered. For any ordinal number α the set of all ordinal numbers less than α is called an initial set. A function from an initial set into Q is called a transfinite sequence on Q. If ƒ: I1 → Q, g: I2 → Q are transfinite sequences on Q, the statement ƒ ≤ g means that there is a one-to-one order-preserving function ø:I1 → I2 such that f(α) ≤ g(ø(α)) for every α ∈ I1. Milner has conjectured in (3) that, if Q is well ordered, then any set of transfinite sequences on Q is well-quasi-ordered under the quasi-ordering just defined. In this paper, we define so-called ‘better-quasi-ordered sets’, which are well-quasi-ordered sets of a particularly ‘well-behaved’ kind, and we prove that any set of transfinite sequences on a better-quasi-ordered set is better-quasi-ordered. Milner's conjecture follows a fortiori, since every well ordered set is better-quasi-ordered and every better-quasi-ordered set is well-quasi-ordered.

A chord diagram of a ribbon surface-link

2015 ◽  
Vol 24 (10) ◽  
pp. 1540002 ◽  
Author(s):  
Akio Kawauchi
Keyword(s):  
Fundamental Group ◽  
Cyclic Group ◽  
Chord Diagram ◽  
Infinite Cyclic Group ◽  
Chord Diagrams ◽  
Ribbon Surface ◽  
Virtual Links

A ribbon chord diagram, or simply a chord diagram, of a ribbon surface-link in the 4-space is introduced. Links, virtual links and welded virtual links can be described naturally by chord diagrams with the corresponding moves, respectively. Some moves on chord diagrams are introduced by overseeing these special moves. Then the faithful equivalence on ribbon surface-links is stated in terms of the moves on chord diagrams. This answers questions by Nakanishi and Marumoto affirmatively. The faithful TOP-equivalence on ribbon surface-links derives the same result. By combining a previous result on TOP-triviality of a surface-knot, a ribbon surface-knot is DIFF-trivial if and only if the fundamental group is an infinite cyclic group. This corrects an erroneous proof in Yanagawa's old paper.

Sign in / Sign up

Export Citation Format

Share Document