scholarly journals Reidemeister moves in Gauss diagrams

2021 ◽  
Vol 14 (3) ◽  
pp. 431-438
Author(s):  
Sandy Ganzell ◽  
Ellen Lehet ◽  
Cristina Lopez ◽  
Gilbert Magallon ◽  
Alyson Thompson
Keyword(s):  
Reidemeister Moves

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

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

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.

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.

On unknotting operations of rotation type

2015 ◽  
Vol 24 (10) ◽  
pp. 1540009
Author(s):  
Yongju Bae ◽  
Byeorhi Kim
Keyword(s):  
Finite Sequence ◽  
Reidemeister Moves

An unknotting operation is a local move on a knot diagram such that any knot diagram can be transformed into a diagram of the unknot by a finite sequence of the operations and Reidemeister moves. In this paper, we introduce a new local move H(T) on a knot diagram which is obtained by the rotation of a tangle diagram T and study their properties. As an application, we prove that the H(T)-move is an unknotting operation for any descending tangle diagram T.

scholarly journals A VOLUME FORM ON THE KHOVANOV INVARIANT

2012 ◽  
Vol 21 (04) ◽  
pp. 1250032
Author(s):  
JUAN ORTIZ-NAVARRO
Keyword(s):  
Chain Complex ◽  
Khovanov Homology ◽  
Volume Form ◽  
Reidemeister Torsion ◽  
Homology Groups ◽  
Numerical Invariant ◽  
Reidemeister Moves ◽  
Invariant Volume ◽  
Knots And Links

The Reidemeister torsion construction can be applied to the chain complex used to compute the Khovanov homology of a knot or a link. This defines a volume form on Khovanov homology. The volume form transforms correctly under Reidemeister moves to give an invariant volume on the Khovanov homology. In this paper, its construction and invariance under these moves is demonstrated. Also, some examples of the invariant are presented for particular choices for the bases of homology groups to obtain a numerical invariant of knots and links. In these examples, the algebraic torsion seen in the Khovanov chain complex when homology is computed over ℤ is recovered.

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