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An algebraic condition to determine whether a knot is prime

Published online by Cambridge University Press:  24 October 2008

Hayley Ryder
Hayley Ryder
Affiliation:
Liverpool University
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Extract

A quandle is the algebraic distillation of the second and third Reidemeister moves, on unframed links. Quandles have been studied by many people including Kauffman[K], Fenn and Rourke[F-R] and Joyce [J]. Joyce defines the fundamental quandle, which is a classifying invariant of irreducible, unframed links. Fenn and Rourke, who study generalised quandles or racks in [F—R], show that the natural map from the fundamental quandle of a knot K to the knot group is never injective if K is a connected sum. We now prove that the natural map from the fundamental quandle to the fundamental group of the complement of a link in S3 is injective if and only if the link is prime.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 120 , Issue 3 , October 1996 , pp. 385 - 389
Copyright
Copyright © Cambridge Philosophical Society 1996

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References

REFERENCES

[F-R]Fenn, R. and Rourke, C.. Racks and links in codimension two. J. Knot Theory and its Ramifications 1 (1992), 343–406.CrossRefGoogle Scholar
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[J]Joyce, D.. A classifying invariant of knots; the knot quandle. J. Pure Applied Algebra 23 (1982), 37–65.CrossRefGoogle Scholar
[R]Ryder, H. J.. The structure of racks. Ph.D. thesis (University of Warwick 1994).Google Scholar