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Invariants of welded virtual knots via crossed module invariants of knotted surfaces

Part of: PL-topology Low-dimensional topology

Published online by Cambridge University Press:  01 July 2008

Louis H. Kauffman  and
João Faria Martins
Louis H. Kauffman
Affiliation:
University of Illinois at Chicago, 851 South Morgan St., Chicago, IL 60607-7045, USA (email: kauffman@uic.edu)
João Faria Martins
Affiliation:
Departamento de Matemática, Instituto Superior Técnico, (Universidade Técnica de Lisboa), Av. Rovisco Pais, 1049-001 Lisboa, Portugal (email: jmartins@math.ist.utl.pt) Also at Departamento de Matemática, Universidade Lusófona de Humanidades e Tecnologia, Av. do Campo Grande 376, 1749-024, Lisboa, Portugal
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Abstract

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We define an invariant of welded virtual knots from each finite crossed module by considering crossed module invariants of ribbon knotted surfaces which are naturally associated with them. We elucidate that the invariants obtained are non-trivial by calculating explicit examples. We define welded virtual graphs and consider invariants of them defined in a similar way.

Keywords

welded virtual knotsknotted surfacescrossed modulequandle invariantsAlexander module

MSC classification

Secondary: 57M25: Knots and links in $S^3$ 57Q45: Knots and links (in high dimensions)
Type
Research Article
Information
Compositio Mathematica , Volume 144 , Issue 4 , July 2008 , pp. 1046 - 1080
Copyright
Copyright © Foundation Compositio Mathematica 2008