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3-manifolds, tangles and persistent invariants

Published online by Cambridge University Press:  05 September 2005

JÓZEF H. PRZYTYCKI ,
DANIEL S. SILVER  and
SUSAN G. WILLIAMS
JÓZEF H. PRZYTYCKI
Affiliation:
Department of Mathematics, The George Washington University, Washington, DC 20052, U.S.A. e-mail: [email protected]
DANIEL S. SILVER
Affiliation:
Department of Mathematics and Statistics, University of South Alabama, Mobile, AL 36688-0002, U.S.A. e-mail: [email protected], [email protected]
SUSAN G. WILLIAMS
Affiliation:
Department of Mathematics and Statistics, University of South Alabama, Mobile, AL 36688-0002, U.S.A. e-mail: [email protected], [email protected]
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Abstract

Given a compact, connected, oriented 3-manifold $M$ with boundary, and epimorphism $\chi$ from $H_1M$ to a free abelian group $\Pi$, two invariants $\beta$, $\tau \in \bb {Z}\Pi$ are defined. If $M$ embeds in another such 3-manifold $N$ such that $\chi_N$ factors through $\chi$, then the product $\beta\tau$ divides $\Delta_0(H_1\tilde {N})$.

A theorem of D. Krebes concerning 4-tangles embedded in links arises as a special case. Algebraic and skein theoretic generalizations for $2n$-tangles provide invariants that persist in the corresponding invariants of links in which they embed. An example is given of a virtual 4-tangle for which Krebes's theorem does not hold.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 139 , Issue 2 , September 2005 , pp. 291 - 306
Copyright
2005 Cambridge Philosophical Society

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