Generalizations of a Conway algebra for oriented surface-links via marked graph diagrams

2018 ◽  
Vol 27 (13) ◽  
pp. 1842014
Author(s):  
Yongju Bae ◽  
Seonmi Choi ◽  
Seongjeong Kim
Keyword(s):  
Algebraic Structure ◽  
Polynomial Invariant ◽  
Oriented Surface ◽  
Binary Operations ◽  
Marked Graphs ◽  
Marked Graph

In 1987, Przytyski and Traczyk introduced an algebraic structure, called a Conway algebra, and constructed an invariant of oriented links, which is a generalization of the HOMFLY-PT polynomial invariant. In 2018, Kim generalized a Conway algebra, which is an algebraic structure with two skein relations, which is called a generalized Conway algebra. In 2017, Joung, Kamada, Kawauchi and Lee constructed a polynomial invariant of oriented surface-links by using marked graph diagrams. In this paper, we will introduce generalizations [Formula: see text] and [Formula: see text] of a Conway algebra and a generalized Conway algebra, which are called a marked Conway algebra and a generalized marked Conway algebra, respectively. We will construct invariants valued in [Formula: see text] and [Formula: see text] for oriented marked graphs and oriented surface-links by applying binary operations to classical crossings and marked vertices via marked graph diagrams. The polynomial invariant of oriented surface-links is obtained from the invariant valued in the marked Conway algebra with additional conditions.

scholarly journals On the generalization of Conway algebra

2018 ◽  
Vol 27 (02) ◽  
pp. 1850014 ◽  
Author(s):  
Seongjeong Kim
Keyword(s):  
Algebraic Structure ◽  
The Other ◽  
Second Step ◽  
Polynomial Invariant ◽  
Binary Operations ◽  
Other Hand ◽  
Skein Relation

In [Przytyski and Traczyk, Invariants of links of Conway type, Kobe J. Math. 4 (1989) 115–139], Przytyski and Traczyk introduced an algebraic structure, called a Conway algebra, and constructed an invariant of oriented links, which is a generalization of the Homflypt polynomial invariant. On the other hand, in [Kauffman and Lambropoulou, New invariants of links and their state sum models, arXiv:1703.03655v2 [math.GT] 15 Mar 2017], Kauffman and Lambropoulou introduced new 4-variable invariants of oriented links, which are obtained by two computational steps: in the first step, we apply a skein relation on every mixed crossing to produce unions of unlinked knots. In the second step, we apply another skein relation on crossings of the unions of unlinked knots, which introduces a new variable. In this paper, we will introduce a generalization of the Conway algebra [Formula: see text] with two binary operations and we construct an invariant valued in [Formula: see text] by applying those two binary operations to mixed crossings and pure crossing, respectively. The 4-variable invariant of Kauffman and Lambropoulou with a specific condition is derived from the invariant valued in [Formula: see text]. Moreover, the generalized Conway algebra gives us an invariant of oriented links, which satisfies nonlinear skein relations.

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.

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.

scholarly journals Lower and upper bounds of shortest paths in reachability graphs

2004 ◽  
Vol 2004 (57) ◽  
pp. 3023-3036 ◽  
Author(s):  
P. K. Mishra
Keyword(s):  
Petri Nets ◽  
Shortest Path ◽  
Petri Net ◽  
Free Choice ◽  
Shortest Paths ◽  
Upper Bounds ◽  
Lower And Upper Bounds ◽  
Marked Graphs ◽  
Marked Graph

We prove the following property for safe marked graphs, safe conflict-free Petri nets, and live and safe extended free-choice Petri nets. We prove the following three results. If the Petri net is a marked graph, then the length of the shortest path is at most(|T|−1)⋅|T|/2. If the Petri net is conflict free, then the length of the shortest path is at most(|T|+1)⋅|T|/2. If the petrinet is live and extended free choice, then the length of the shortest path is at most|T|⋅|T+1|⋅|T+2|/6, whereTis the set of transitions of the net.

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 Quandle coloring quivers of surface-links

2021 ◽  
Vol 30 (01) ◽  
pp. 2150002
Author(s):  
Jieon Kim ◽  
Sam Nelson ◽  
Minju Seo
Keyword(s):  
Directed Graph ◽  
Oriented Surface ◽  
Knot Invariant ◽  
Marked Graph ◽  
Knots And Links

Quandle coloring quivers are directed graph-valued invariants of oriented knots and links, defined using a choice of finite quandle [Formula: see text] and set [Formula: see text] of endomorphisms. From a quandle coloring quiver, a polynomial knot invariant known as the in-degree quiver polynomial is defined. We consider quandle coloring quiver invariants for oriented surface-links, represented by marked graph diagrams. We provide example computations for all oriented surface-links with ch-index up to 10 for choices of quandles and endomorphisms.

scholarly journals A POLYNOMIAL INVARIANT OF FINITE QUANDLES

2008 ◽  
Vol 07 (02) ◽  
pp. 263-273 ◽  
Author(s):  
SAM NELSON
Keyword(s):  
Algebraic Structure ◽  
Polynomial Invariant ◽  
Link Invariants

We define a two-variable polynomial invariant of finite quandles. In many cases this invariant completely determines the algebraic structure of the quandle up to isomorphism. We use this polynomial to define a family of link invariants which generalize the quandle counting invariant.

scholarly journals On Refined Neutrosophic R-module

2020 ◽  
pp. 87-96
Author(s):  
Necati Olgun ◽  
◽  
◽  
Ahmed Hatip
Keyword(s):  
Structural Properties ◽  
Algebraic Structure ◽  
Binary Operations ◽  
Module Homomorphism

Modules are one of the fundamental and rich algebraic structure concerning some binary operations in the study of algebra. In this paper, some basic structures of refined neutrosophic R-modules and refined neutrosophic submodules in algebra are generalized. Some properties of refined neutrosophic R-modules and refined neutrosophic submodules are presented. More precisely, classical modules and refined neutrosophic rings are utilized. Consequently, refinedneutrosophic R- modules that are completely different from the classical modular in the structural properties are introduced. Also, neutrosophic R-module homomorphism is explained and some definitions and theorems are presented.

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