# Invariants of links in thickened surfaces

@article{Carter2014InvariantsOL,
title={Invariants of links in thickened surfaces},
author={J. Scott Carter and Daniel S. Silver and Susan G. Williams},
journal={Algebraic \& Geometric Topology},
year={2014},
volume={14},
pages={1377-1394}
}
• Published 17 April 2013
• Mathematics
• Algebraic & Geometric Topology
A group invariant for links in thickened closed orientable surfaces is studied. Associated polynomial invariants are dened. The group

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## References

SHOWING 1-10 OF 30 REFERENCES
• Mathematics
• 2003
Properties of polynomial invariants Δi for oriented virtual links are established. The effects of taking mirror images and reversing orientation of the link diagram are described. The relationship
Several authors have recently studied virtual knots and links because they admit invariants arising from R-matrices. We prove that every virtual link is uniquely represented by a link L SI in a
On Alexander-Conway Polynomials for Virtual Knots and Links
A polynomial invariant of virtual links, arising from an invariant of links in thickened surfaces introduced by Jaeger, Kauffman, and Saleur, is defined and its properties are investigated. Examples
Introduction Codimension one and other matters The fundamental group Three-dimensional PL geometry Seifert surfaces Finite cyclic coverings and the torsion invariants Infinite cyclic coverings and
Virtual Knot Theory
This paper is an introduction to the theory of virtual knots. It is dedicated to the memory of Francois Jaeger.
ABSTRACT LINK DIAGRAMS AND VIRTUAL KNOTS
• Computer Science
• 2000
It is prove that there is a bijection from the equivalence classes of virtual link diagrams to those of abstract link diagrams, and a generalization to higher dimensional cases is introduced, and the state-sum invariants are treated.
STABLE EQUIVALENCE OF KNOTS ON SURFACES AND VIRTUAL KNOT COBORDISMS
• Mathematics, Computer Science
• 2000
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.
Algebraic Topology
The focus of this paper is a proof of the Nielsen-Schreier Theorem, stating that every subgroup of a free group is free, using tools from algebraic topology.
Minimal surface representations of virtual knots and links
• Computer Science
• 2005
Methods that may determine whether a virtual knot diagram is non-classical (and hence non-trivial) are developed and it is shown that, except for special cases, link dia- grams with a single virtualization and link diagrams with asingle virtual crossing are non- classical.
A Course in the Theory of Groups
This is a detailed introduction to the theory of groups: finite and infinite; commutative and non-commutative. Presupposing only a basic knowledge of modern algebra, it introduces the reader to the