Virtual Knot Theory
TLDR
In this paper, the theory of virtual knots is introduced, dedicated to the memory of Francois Jaeger, who was a pioneer in the field of virtual knot theory and its applications.About:
This article is published in European Journal of Combinatorics.The article was published on 1999-10-01 and is currently open access. It has received 1045 citations till now. The article focuses on the topics: Virtual knot & Knot theory.read more
Citations
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What is a virtual link
TL;DR: In this paper, it was shown that every virtual link is uniquely represented by a link L SI in a thickened, compact, oriented surface S such that the link complement (SI)nL has no essential vertical cylinder.
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Stable equivalence of knots on surfaces and virtual knot cobordisms
TL;DR: An equivalence relation, called stable equivalence, is introduced on knot diagrams and closed generically immersed curves on surfaces and it is shown that Kauffman's example of a virtual knot diagram is not equivalent to a classical knot diagram.
Posted Content
Stable Equivalence of Knots on Surfaces and Virtual Knot Cobordisms
TL;DR: In this paper, the stable equivalence relation on knot diagrams and closed curves on surfaces was introduced, and the authors defined concordance and link homology for virtual links and proved that Kauffman's example of a virtual knot diagram is not equivalent to a classical knot diagram.
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Biquandles and virtual links
TL;DR: A birack and a biquandle is defined, generalizing the notion of a rack and a quandle, which gives rise to natural invariants of virtual knots and braids.
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Parity in knot theory
TL;DR: In this paper, it was shown that free knots are generally not invertible, and provided invariants which detect the invertibility of free knots using parity property arising from Gauss diagrams, and that even a gross simplification of the theory of virtual knots admits simple and highly nontrivial invariants.
References
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State Models and the Jones Polynomial
TL;DR: In this article, a state model for the Jones polynomial was constructed for the bracket polynomials, which is a normalization of a regular isotopy invariant of unoriented knots and links.
Book
Knots and physics
TL;DR: Physical Knots States and the Bracket Polynomial The Jones Polynominal and Its Generalizations Braids and Polynomials: Formal Feynman Diagrams, Bracket as Vacuum-Vacmum expectation and the Quantum Group SL(2)q Yang-Baxter Models for Specialization's of the Homfly Polymorphial The Alexander Polynomical Knot Crystals - Classical Knot Theory in Modem Guise The Kauffman PolynomIAL Three-Manifold Invariants from the Jones Polynials integral Heuristics and W
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Racks and links in codimension two
Roger Fenn,Colin Rourke +1 more
TL;DR: In this paper, it was shown that the fundamental rack is a complete invariant for irreducible framed links in a 3-manifold and for the 3 -manifolds itself.
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Finite Type Invariants of Classical and Virtual Knots
TL;DR: In this paper, the isotopy classification problem for virtual knots is reduced to an algebraic problem formulated in terms of an algebra of arrow diagrams, and a new notion of finite type invariant is introduced.
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The braid-permutation group
TL;DR: In this paper, the authors considered the subgroup of the automorphism group of the free group generated by the braid group and the permutation group, which is represented by generalised braids (braids in which some crossings are allowed to be "welded").