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The Alexander module of a knotted theta-curve

Published online by Cambridge University Press:  24 October 2008

Rick Litherland
Rick Litherland
Affiliation:
Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803, U.S.A.
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Extract

Let K be a knotted theta-curve with exterior X, and let ∂_X be one of the two pieces into which ∂X is divided by the meridians of the edges of K. Let X be the universal abelian cover of X. Then is a module over the group ring of H1(X); i.e. over . We call this the Alexander module of K, and denote it by A(K). This, rather than H1(X), seems to be the analogue of the Alexander module of a classical knot; it is a torsion module of deficiency 0. Moreover, it is not an invariant of X alone.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 106 , Issue 1 , July 1989 , pp. 95 - 106
Copyright
Copyright © Cambridge Philosophical Society 1989

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References

REFERENCES

[1]Hempel, John. Intersection calculus on surfaces with applications to 3-manifolds. Mem. Amer. Math. Roc. vol. 43. no. 282 (American Mathematical Society. 1983).Google Scholar
[2]Wolcott, Keith. The knotting of theta-curves and other graphs in S 3. Ph.D. thesis. University of Towa (1985).Google Scholar