8,599 research outputs found
Twisted Alexander Invariants of Twisted Links
Let L be an oriented (d+1)-component link in the 3-sphere, and let L(q) be
the d-component link in a homology 3-sphere that results from performing
1/q-surgery on the last component. Results about the Alexander polynomial and
twisted Alexander polynomials of L(q) corresponding to finite-image
representations are obtained. The behavior of the invariants as q increases
without bound is described.Comment: 21 pages, 6 figure
Twisted Alexander Polynomials and Representation Shifts
For any knot, the following are equivalent. (1) The infinite cyclic cover has
uncountably many finite covers; (2) there exists a finite-image representation
of the knot group for which the twisted Alexander polynomial vanishes; (3) the
knot group admits a finite-image representation such that the image of the
fundamental group of an incompressible Seifert surface is a proper subgroup of
the image of the commutator subgroup of the knot group.Comment: 7 pages, no figure
Mahler measure, links and homology growth
Let l be a link of d components. For every finite-index lattice in Z^d there
is an associated finite abelian cover of S^3 branched over l. We show that the
order of the torsion subgroup of the first homology of these covers has
exponential growth rate equal to the logarithmic Mahler measure of the
Alexander polynomial of l, provided this polynomial is nonzero. Our proof uses
a theorem of Lind, Schmidt and Ward on the growth rate of connected components
of periodic points for algebraic Z^d-actions.Comment: 13 pages, figures. Small corrections, references updated. To appear
in Topolog
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