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On the Head and the Tail of the Colored Jones Polynomial
The colored Jones polynomial is a series of one variable Laurent polynomials
J(K,n) associated with a knot K in 3-space. We will show that for an
alternating knot K the absolute values of the first and the last three leading
coefficients of J(K,n) are independent of n when n is sufficiently large.
Computation of sample knots indicates that this should be true for any fixed
leading coefficient of the colored Jones polynomial for alternating knots. As a
corollary we get a Volume-ish Theorem for the colored Jones Polynomial.Comment: 14 pages, 6 figure
Spectral Theory of Discrete Processes
We offer a spectral analysis for a class of transfer operators. These
transfer operators arise for a wide range of stochastic processes, ranging from
random walks on infinite graphs to the processes that govern signals and
recursive wavelet algorithms; even spectral theory for fractal measures. In
each case, there is an associated class of harmonic functions which we study.
And in addition, we study three questions in depth:
In specific applications, and for a specific stochastic process, how do we
realize the transfer operator as an operator in a suitable Hilbert space?
And how to spectral analyze once the right Hilbert space has
been selected? Finally we characterize the stochastic processes that are
governed by a single transfer operator.
In our applications, the particular stochastic process will live on an
infinite path-space which is realized in turn on a state space . In the case
of random walk on graphs , will be the set of vertices of . The
Hilbert space on which the transfer operator acts will then
be an space on , or a Hilbert space defined from an energy-quadratic
form.
This circle of problems is both interesting and non-trivial as it turns out
that may often be an unbounded linear operator in ; but even
if it is bounded, it is a non-normal operator, so its spectral theory is not
amenable to an analysis with the use of von Neumann's spectral theorem. While
we offer a number of applications, we believe that our spectral analysis will
have intrinsic interest for the theory of operators in Hilbert space.Comment: 34 pages with figures removed, for the full version with all the
figures please go to http://www.siue.edu/~msong/Research/spectrum.pd
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