Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the Characterization Problem of Alexander Polynomials of Closed
  3-Manifolds   We give a characterization for the Alexander Polynomials of closed orientable
3-manifolds M with first Betti number 1, as well as some partial results for
the characterization problem for M having first Betti number > 1. We first
prove an analogue of a theorem of Levine: that the product of an Alexander
polynomial of M with a symmetric polynomial in the same number of variables
having non 0 trace, is again an Alexander polynomial of a closed orientable
3-manifold. Using the fact that there exists M with Alexander polynomial = 1
for M with first Betti number 1, 2 or 3, we conclude that symmetric polynomials
of non 0 trace in 1, 2 or 3 variables are Alexander polynomials of closed
orientable 3-manifolds. When the first Betti number of M is 1 we prove that non
0 trace symmetric polynomials are the only ones that can arise. Finally, for M
with first Betti number > 3 we prove that the Alexander polynomial can not be
1, implying that for such manifolds not all symmetric polynomials having non 0
trace will occur.

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The categories of this paper are: nucl-ex
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The categories of this paper are: stat
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The categories of this paper are: adap-org
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The categories of this paper are: hep-th
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The categories of this paper are: mtrl-th
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The categories of this paper are: physics
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The categories of this paper are: quant-ph
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The categories of this paper are: atom-ph
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The categories of this paper are: dg-ga
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The categories of this paper are: bayes-an
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The categories of this paper are: patt-sol
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The categories of this paper are: alg-geom
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The categories of this paper are: hep-ex
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The categories of this paper are: acc-phys
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The categories of this paper are: math-ph
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The categories of this paper are: cs