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 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 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 Intrinsically knotted and 4-linked directed graphs

2018 ◽  
Vol 27 (06) ◽  
pp. 1850037 ◽  
Author(s):  
Thomas Fleming ◽  
Joel Foisy
Keyword(s):  
Directed Graph ◽  
Undirected Graph ◽  
Directed Graphs ◽  
Edge Contraction ◽  
Special Cases ◽  
Linkless Embedding ◽  
The Relationship ◽  
Knots And Links

We consider intrinsic linking and knotting in the context of directed graphs. We construct an example of a directed graph that contains a consistently oriented knotted cycle in every embedding. We also construct examples of intrinsically 3-linked and 4-linked directed graphs. We introduce two operations, consistent edge contraction and H-cyclic subcontraction, as special cases of minors for digraphs, and show that the property of having a linkless embedding is closed under these operations. We analyze the relationship between the number of distinct knots and links in an undirected graph [Formula: see text] and its corresponding symmetric digraph [Formula: see text]. Finally, we note that the maximum number of edges for a graph that is not intrinsically linked is [Formula: see text] in the undirected case, but [Formula: see text] for directed graphs.

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.

Embedding knots and links in an open book II. Bounds on arc index

1996 ◽  
Vol 119 (2) ◽  
pp. 309-319 ◽  
Author(s):  
Peter R. Cromwell ◽  
Ian J. Nutt
Keyword(s):  
Finite Number ◽  
Polar Coordinates ◽  
Open Book ◽  
Knot Invariant ◽  
Braid Index ◽  
Satellite Links ◽  
Minimum Number ◽  
Arc Index ◽  
Knots And Links ◽  
Simple Arc

There is an open-book decomposition of the 3-sphere which has open discs as pages and an unknotted circle as the binding. We can think of the 3-sphere as ℝ3 ∪ {∞} and of the circle as the z–axis ∪ {∞}. The pages are then half-planes Hθ at angle θ when the x–y plane has polar coordinates. In their investigation of the braid index of satellite links, Birman and Menasco [B–M] embed the companion knot in finitely many such half-planes so that the knot meets each half-plane in a single simple arc, and therefore meets the axis in a finite number of points. At the end of their paper they mention that the minimum number of planes required to present a given knot in this manner is a knot invariant and that it seems to have escaped attention. (Jósef Przytycki has since pointed out to us that the phenomenon is evident in a one-hundred-year-old paper by H. Brunn[Br].) We call this invariant the arc index of a link and denote it by α(L).

Knots and Links

2004 ◽  
Author(s):  
Peter R. Cromwell
Keyword(s):  
Knots And Links
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