scholarly journals On generating sets of Yoshikawa moves for marked graph diagrams of surface-links

2015 ◽  
Vol 24 (04) ◽  
pp. 1550018 ◽  
Author(s):  
Jieon Kim ◽  
Yewon Joung ◽  
Sang Youl Lee
Keyword(s):  
Finite Sequence ◽  
The Other ◽  
Link Diagram ◽  
Oriented Surface ◽  
Generating Sets ◽  
Marked Graph

A marked graph diagram is a link diagram possibly with marked 4-valent vertices. S. J. Lomonaco, Jr. and K. Yoshikawa introduced a method of representing surface-links by marked graph diagrams. Specially, K. Yoshikawa suggested local moves on marked graph diagrams, nowadays called Yoshikawa moves. It is now known that two marked graph diagrams representing equivalent surface-links are related by a finite sequence of these Yoshikawa moves. In this paper, we provide some generating sets of Yoshikawa moves on marked graph diagrams representing unoriented surface-links, and also oriented surface-links. We also discuss independence of certain Yoshikawa moves from the other moves.

scholarly journals Biquandle cohomology and state-sum invariants of links and surface-links

2018 ◽  
Vol 27 (11) ◽  
pp. 1843016
Author(s):  
Seiichi Kamada ◽  
Akio Kawauchi ◽  
Jieon Kim ◽  
Sang Youl Lee
Keyword(s):  
Homology Theory ◽  
Oriented Surface ◽  
Marked Graph

In this paper, we discuss the (co)homology theory of biquandles, derived biquandle cocycle invariants for oriented surface-links using broken surface diagrams and how to compute the biquandle cocycle invariants from marked graph diagrams. We also develop the shadow (co)homology theory of biquandles and construct the shadow biquandle cocycle invariants for oriented surface-links.

scholarly journals Concatenation of Finite Sequences

2019 ◽  
Vol 27 (1) ◽  
pp. 1-13
Author(s):  
Rafał Ziobro
Keyword(s):  
Finite Sequence ◽  
The Other ◽  
Other Hand ◽  
Finite Sequences ◽  
Initial Object

Summary The coexistence of “classical” finite sequences [1] and their zero-based equivalents finite 0-sequences [6] in Mizar has been regarded as a disadvantage. However the suggested replacement of the former type with the latter [5] has not yet been implemented, despite of several advantages of this form, such as the identity of length and domain operators [4]. On the other hand the number of theorems formalized using finite sequence notation is much larger then of those based on finite 0-sequences, so such translation would require quite an effort. The paper addresses this problem with another solution, using the Mizar system [3], [2]. Instead of removing one notation it is possible to introduce operators which would concatenate sequences of various types, and in this way allow utilization of the whole range of formalized theorems. While the operation could replace existing FS2XFS, XFS2FS commands (by using empty sequences as initial elements) its universal notation (independent on sequences that are concatenated to the initial object) allows to “forget” about the type of sequences that are concatenated on further positions, and thus simplify the proofs.

scholarly journals Modifications and Cobounding Manifolds

1960 ◽  
Vol 12 ◽  
pp. 503-528 ◽  
Author(s):  
Andrew H. Wallace
Keyword(s):  
Algebraic Variety ◽  
Finite Sequence ◽  
Differentiable Manifold ◽  
The Other ◽  
Algebraic Varieties ◽  
The Given ◽  
Hyperplane Sections

The object of this paper is to establish a simple connection between Thorn's theory of cobounding manifolds and the theory of modifications. The former theory is given in detail in (8) and sketched in (3), while the latter is worked out in (1). In particular in (1) it is shown that the only modifications which can transform one differentiable manifold into another are what I call below spherical modifications, which consist in taking out a sphere from the given manifold and replacing it by another. The main result is that manifolds cobound if and only if each is obtainable from the other by a finite sequence of spherical modifications.The technique consists in approximating the manifolds by pieces of algebraic varieties. Thus if M1 and M2 form the boundary of M, the last is taken to be part of an algebraic variety such that M1 and M2 are two members of a pencil of hyperplane sections.

On the Alexander biquandles of oriented surface-links via marked graph diagrams

2014 ◽  
Vol 23 (07) ◽  
pp. 1460007 ◽  
Author(s):  
Jieon Kim ◽  
Yewon Joung ◽  
Sang Youl Lee
Keyword(s):  
Oriented Surface ◽  
Marked Graph

Carrell defined the fundamental biquandle of an oriented surface-link by a presentation obtained from its broken surface diagram, which is an invariant up to isomorphism of the fundamental biquandle. Ashihara gave a method to calculate the fundamental biquandle of an oriented surface-link from its marked graph diagram (ch-diagram). In this paper, we discuss the fundamental Alexander biquandles of oriented surface-links via marked graph diagrams, derived computable invariants and their applications to detect non-invertible oriented surface-links.

scholarly journals A BRACKET POLYNOMIAL FOR GRAPHS, III: VERTEX WEIGHTS

2011 ◽  
Vol 20 (03) ◽  
pp. 435-462 ◽  
Author(s):  
LORENZO TRALDI
Keyword(s):  
Link Diagram ◽  
Kauffman Bracket ◽  
Marked Graphs ◽  
Marked Graph ◽  
Bracket Polynomial

In earlier work the Kauffman bracket polynomial was extended to an invariant of marked graphs, i.e. looped graphs whose vertices have been partitioned into two classes (marked and not marked). The marked-graph bracket polynomial is readily modified to handle graphs with weighted vertices. We present formulas that simplify the computation of this weighted bracket for graphs that contain twin vertices or are constructed using graph composition, and we show that graph composition corresponds to the construction of a link diagram from tangles.

Linking invariants of even virtual links

2017 ◽  
Vol 26 (12) ◽  
pp. 1750072 ◽  
Author(s):  
Haruko A. Miyazawa ◽  
Kodai Wada ◽  
Akira Yasuhara
Keyword(s):  
Lower Bound ◽  
The Other ◽  
Virtual Link ◽  
Link Diagram ◽  
Linking Number ◽  
Reidemeister Moves ◽  
Link Diagrams ◽  
Classical Link ◽  
Virtual Links ◽  
The Difference

A virtual link diagram is even if the virtual crossings divide each component into an even number of arcs. The set of even virtual link diagrams is closed under classical and virtual Reidemeister moves, and it contains the set of classical link diagrams. For an even virtual link diagram, we define a certain linking invariant which is similar to the linking number. In contrast to the usual linking number, our linking invariant is not preserved under the forbidden moves. In particular, for two fused isotopic even virtual link diagrams, the difference between the linking invariants of them gives a lower bound of the minimal number of forbidden moves needed to deform one into the other. Moreover, we give an example which shows that the lower bound is best possible.

Clifford’s chain and its analogues in relation to the higher polytopes

1972 ◽  
Vol 330 (1583) ◽  
pp. 443-466 ◽  
Keyword(s):  
Infinite Number ◽  
Finite Sequence ◽  
Higher Dimensions ◽  
The Other ◽  
Four Dimensions ◽  
Symmetrical Configuration ◽  
Six Dimensions

Clifford’s chain of theorems on circles in a plane, Grace’s sequence of theorems on spheres, and Grace’s and Brown’s sequence on hyperspheres in four dimensions are unified and completed. This is achieved by demonstrating a correspondence between each configuration and one of Coxeter’s polytopes p gr in space of ( p + g + r + 1) dimensions. Thus Clifford’s configurations of points and circles in a plane correspond to the polytopes 1 1 , r r'. Grace’s figures of points and spheres correspond to polytopes l 2 r ; and Grace’s and Brown’s figures of points and hyperspheres correspond to polytopes l 3 r . As a result it is shown that to Grace’s sequence there may be added two more symmetrical configurations, one of 17280 points and 240 spheres, the other of an infinite number of points and spheres, before the sequence terminates. To Grace’s and Brown’s sequence may be added just one more symmetrical configuration of points and hyperspheres. Furthermore, it is shown that in space either of five or of six dimensions there exists a finite sequence of three configurations; in any space of seven or more dimensions there is a sequence of just two configurations each. A related chain of theorems due to Homersham Cox, more general than Clifford’s, is likewise shown to have analogues in spaces of higher dimensions.

scholarly journals ON GROUPS WHOSE GEODESIC GROWTH IS POLYNOMIAL

2012 ◽  
Vol 22 (05) ◽  
pp. 1250048 ◽  
Author(s):  
MARTIN R. BRIDSON ◽  
JOSÉ BURILLO ◽  
MURRAY ELDER ◽  
ZORAN ŠUNIĆ
Keyword(s):  
Nilpotent Group ◽  
Finite Index ◽  
The Other ◽  
Normal Closure ◽  
Finitely Generated ◽  
Generating Set ◽  
Finitely Generated Group ◽  
Cyclic Groups ◽  
Growth Functions ◽  
Generating Sets

This paper records some observations concerning geodesic growth functions. If a nilpotent group is not virtually cyclic then it has exponential geodesic growth with respect to all finite generating sets. On the other hand, if a finitely generated group G has an element whose normal closure is abelian and of finite index, then G has a finite generating set with respect to which the geodesic growth is polynomial (this includes all virtually cyclic groups).

CALCULATING THE FUNDAMENTAL BIQUANDLES OF SURFACE LINKS FROM THEIR CH–DIAGRAMS

2012 ◽  
Vol 21 (10) ◽  
pp. 1250102 ◽  
Author(s):  
SOSUKE ASHIHARA
Keyword(s):  
Double Point ◽  
Link Diagram ◽  
Oriented Surface ◽  
Generating Set

The fundamental biquandle is an invariant of an oriented surface link, which is defined by a presentation obtained from a surface diagram of the surface link: The generating set consists of labels of the semi-sheets and the relator set consists of relations defined at the double point curves. Any surface link can be presented by a link diagram with some markers which is called a ch-diagram. Using this fact, we give a method for calculating the fundamental biquandle of a surface link from its ch-diagram directly.

Uniqueness and characterization of prime models over sets for totally transcendental first-order theories

1972 ◽  
Vol 37 (1) ◽  
pp. 107-113 ◽  
Author(s):  
Saharon Shelah
Keyword(s):  
Simple Proof ◽  
Finite Sequence ◽  
The Other ◽  
Closed Field ◽  
Prime Model ◽  
First Order ◽  
Differential Field ◽  
Closed Fields ◽  
Transcendental Theory

If T is a complete first-order totally transcendental theory then over every T-structure A there is a prime model unique up to isomorphism over A. Moreover M is a prime model over A iff: (1) every finite sequence from M realizes an isolated type over A, and (2) there is no uncountable indiscernible set over A in M.The existence of prime models was proved by Morley [3] and their uniqueness for countable A by Vaught [9]. Sacks asked (see Chang and Keisler [1, question 25]) whether the prime model is unique. After proving this I heard Ressayre had proved that every two strictly prime models over any T-structure A are isomorphic, by a strikingly simple proof. From this followsIf T is totally transcendental, M a strictly prime model over A then every elementary permutation of A can be extended to an automorphism of M. (The existence of M follows by [3].)By our results this holds for any prime model. On the other hand Ressayre's result applies to more theories. For more information see [6, §0A]. A conclusion of our theorem is the uniqueness of the prime differentially closed field over a differential field. See Blum [8] for the total transcendency of the theory of differentially closed fields.We can note that the prime model M over A is minimal over A iff in M there is no indiscernible set over A (which is infinite).

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