REIDEMEISTER MOVES ON 1-MANIFOLDS IN R×D2

2002 ◽  
Vol 11 (01) ◽  
pp. 109-119
Author(s):  
BOBBY NEAL WINTERS
Keyword(s):  
Reidemeister Moves ◽  
Ambient Isotopy

In this paper we aim to set forth a notion of equivalence for 1-manifolds that are properly embedded in the space R×D2 and prove that this is the same as ambient isotopy.

AKUTZU-WADATI LINK POLYNOMIALS FROM FEYNMAN-KAUFFMAN DIAGRAMS

1989 ◽  
Vol 04 (13) ◽  
pp. 3351-3373 ◽  
Author(s):  
MO-LIN GE ◽  
LU-YU WANG ◽  
KANG XUE ◽  
YONG-SHI WU
Keyword(s):  
Knot Theory ◽  
Jones Polynomial ◽  
Feynman Diagrams ◽  
Cpt Symmetry ◽  
Group Representations ◽  
Reidemeister Moves ◽  
S Matrix ◽  
Ambient Isotopy ◽  
Definition Of ◽  
Bracket Polynomials

By employing techniques familiar to particle physicists, we develop Kauffman’s state model for the Jones polynomial, which uses diagrams looking like Feynman diagrams for scattering, into a systematic, diagrammatic approach to new link polynomials. We systematize the ansatz for S matrix by symmetry considerations and find a natural interpretation for CPT symmetry in the context of knot theory. The invariance under Reidemeister moves of type III, II and I can be imposed diagrammatically step by step, and one obtains successively braid group representations, regular isotopy and ambient isotopy invariants from Kauffman’s bracket polynomials. This procedure is explicitiy carried out for the N=3 and 4 cases. N being the number of particle labels (or charges). With appropriate symmetry ansatz and with annihilation and creation included in the S matrix, we have obtained link polynomials which generalize the definition of the Akutzu-Wadati polynomials from closed braids to any oriented knots or links with explicit invariance under Reidemeister moves.

6. Unknot or knot to be?

2019 ◽  
pp. 115-127
Author(s):  
Richard Earl
Keyword(s):  
Knot Theory ◽  
Simple Closed Curve ◽  
Closed Curve ◽  
Classification Theorem ◽  
Central Problem ◽  
Reidemeister Moves ◽  
3D Space ◽  
Jones Polynomials ◽  
Ambient Isotopy

‘Unknot or knot to be?’ explains that a knot is a smooth, simple, closed curve in 3D space. Being simple and closed means the curve does not cross itself except that its end returns to its start. All knots are topologically the same as a circle; what makes a circle knotted—or not—is how that circle has been placed into 3D space. The central problem of knot theory is a classification theorem: when is there an ambient isotopy between two knots or how do we show that no such isotopy exists? Key elements of knot theory are discussed, including the three Reidemeister moves, prime knots, adding knots, and the Alexander and Jones polynomials.

scholarly journals Biquasiles and dual graph diagrams

2017 ◽  
Vol 26 (08) ◽  
pp. 1750048 ◽  
Author(s):  
Deanna Needell ◽  
Sam Nelson
Keyword(s):  
Dual Graph ◽  
Algebraic Structures ◽  
Combinatorial Structures ◽  
Reidemeister Moves ◽  
Knots And Links ◽  
The Way

We introduce dual graph diagrams representing oriented knots and links. We use these combinatorial structures to define corresponding algebraic structures which we call biquasiles whose axioms are motivated by dual graph Reidemeister moves, generalizing the Dehn presentation of the knot group analogously to the way quandles and biquandles generalize the Wirtinger presentation. We use these structures to define invariants of oriented knots and links and provide examples.

Reidemeister moves. Knot arithmetics

Knot Theory ◽  
2018 ◽  
pp. 13-27
Author(s):  
Vassily Manturov
Keyword(s):  
Reidemeister Moves

Smoothing closed gridded surfaces embedded in ℝ4

2018 ◽  
Vol 27 (12) ◽  
pp. 1850065 ◽  
Author(s):  
Juan Pablo Díaz ◽  
Gabriela Hinojosa ◽  
Rogelio Valdez ◽  
Alberto Verjovsky
Keyword(s):  
Transverse Field ◽  
Ambient Isotopy

We say that a topological [Formula: see text]-manifold [Formula: see text] is a cubical [Formula: see text]-manifold if it is contained in the [Formula: see text]-skeleton of the canonical cubulation [Formula: see text] of [Formula: see text] ([Formula: see text]). In this paper, we prove that any closed, oriented cubical [Formula: see text]-manifold has a transverse field of 2-planes in the sense of Whitehead and therefore it is smoothable by a small ambient isotopy.

scholarly journals Making Curves Minimally Crossing by Reidemeister Moves

1997 ◽  
Vol 70 (1) ◽  
pp. 134-156 ◽  
Author(s):  
Maurits de Graaf ◽  
Alexander Schrijver
Keyword(s):  
Reidemeister Moves

GRAPH INVARIANTS AND THE TOPOLOGY OF RNA FOLDING

1994 ◽  
Vol 03 (03) ◽  
pp. 233-245 ◽  
Author(s):  
LOUIS H. KAUFFMAN ◽  
YURI MAGARSHAK
Keyword(s):  
Rna Folding ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Graph Embeddings ◽  
Graph Invariants ◽  
Vassiliev Invariants ◽  
Ambient Isotopy ◽  
Molecular Folding ◽  
General Method ◽  
Three Dimensional Space

A new general method is described for obtaining ambient isotopy or regular isotopy invariants of even valence rigid vertex graphs embedded in three-dimensional space. The paper concentrates on the case of 4-valent vertices and defines an RNA vertex in analogy to the structure of a folded molecule. Examples are given to show how these methods can discriminate graph embeddings that are indistinguishable via Vassiliev invariants. Applications to molecular folding are discussed.

scholarly journals Triple-crossing number and moves on triple-crossing link diagrams

2019 ◽  
Vol 28 (11) ◽  
pp. 1940001 ◽  
Author(s):  
Colin Adams ◽  
Jim Hoste ◽  
Martin Palmer
Keyword(s):  
Triple Point ◽  
Crossing Number ◽  
Reidemeister Moves ◽  
Link Diagrams ◽  
Triple Points ◽  
Definition Of ◽  
Hopf Link

Every link in the 3-sphere has a projection to the plane where the only singularities are pairwise transverse triple points. The associated diagram, with height information at each triple point, is a triple-crossing diagram of the link. We give a set of diagrammatic moves on triple-crossing diagrams analogous to the Reidemeister moves on ordinary diagrams. The existence of [Formula: see text]-crossing diagrams for every [Formula: see text] greater than one allows the definition of the [Formula: see text]-crossing number. We prove that for any nontrivial, nonsplit link, other than the Hopf link, its triple-crossing number is strictly greater than its quintuple-crossing number.

scholarly journals Quandle coloring quivers

2019 ◽  
Vol 28 (01) ◽  
pp. 1950001 ◽  
Author(s):  
Karina Cho ◽  
Sam Nelson
Keyword(s):  
Small Category ◽  
Link Diagram ◽  
Literal Sense ◽  
Link Invariants ◽  
Reidemeister Moves

We consider a quiver structure on the set of quandle colorings of an oriented knot or link diagram. This structure contains a wealth of knot and link invariants and provides a categorification of the quandle counting invariant in the most literal sense, i.e. giving the set of quandle colorings the structure of a small category which is unchanged by Reidemeister moves. We derive some new enhancements of the counting invariant from this quiver structure and show that the enhancements are proper with explicit examples.

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