65 research outputs found
Pitchfork Bifurcations of Invariant Manifolds
A pitchfork bifurcation of an -dimensional invariant submanifold of a
dynamical system in is defined analogous to that in
. Sufficient conditions for such a bifurcation to occur are stated
and existence of the bifurcated manifolds is proved under the stated
hypotheses. For discrete dynamical systems, the existence of locally attracting
manifolds and , after the bifurcation has taken place is proved by
constructing a diffeomorphism of the unstable manifold . For continuous
dynamical systems, the theorem is proved by transforming it to the discrete
case. Techniques used for proving the theorem involve differential topology and
analysis. The theorem is illustrated by means of a canonical example.Comment: 25 pages, 7 figure
Determinant density and biperiodic alternating links
Let L be any infinite biperiodic alternating link. We show that for any
sequence of finite links that Folner converges almost everywhere to L, their
determinant densities converge to the Mahler measure of the 2-variable
characteristic polynomial of the toroidal dimer model on an associated
biperiodic graph.Comment: 12 pages. V2: corrections in section
A note on quasi-alternating Montesinos links
Quasi-alternating links are a generalization of alternating links. They are
homologically thin for both Khovanov homology and knot Floer homology. Recent
work of Greene and joint work of the first author with Kofman resulted in the
classification of quasi-alternating pretzel links in terms of their integer
tassel parameters. Replacing tassels by rational tangles generalizes pretzel
links to Montesinos links. In this paper we establish conditions on the
rational parameters of a Montesinos link to be quasi-alternating. Using recent
results on left-orderable groups and Heegaard Floer L-spaces, we also establish
conditions on the rational parameters of a Montesinos link to be
non-quasi-alternating. We discuss examples which are not covered by the above
results.Comment: 12 pages, 8 figures, revised exposition, added section on
non-quasi-alternating Montesinos link
Stable Fixed Points of Card Trick Functions
The 21-card trick is a way of dealing cards in order to predict the card
selected by a volunteer. We give a mathematical explanation of why the
well-known 21-card trick works using a simple linear discrete function. The
function has a stable fixed point which corresponds to the position where the
selected card reaches at the end of the trick. We then generalize the 21(7 x
3)-card trick to a p x q - card trick where p and q are odd integers greater
than or equal to three, determine the fixed point and prove that it is also
stable
On the tail of Jones polynomials of closed braids with a full twist
For a closed n-braid L with a full positive twist and with k negative
crossings, 0\leq k \leq n, we determine the first n-k+1 terms of the Jones
polynomial V_L(t). We show that V_L(t) satisfies a braid index constraint,
which is a gap of length at least n-k between the first two nonzero
coefficients of (1-t^2)V_L(t). For a closed positive n-braid with a full
positive twist, we extend our results to the colored Jones polynomials. For
N>n-1, we determine the first n(N-1)+1 terms of the normalized N-th colored
Jones polynomial.Comment: 13 pages
A survey on the Turaev genus of knots
The Turaev genus of a knot is a topological measure of how far a given knot
is from being alternating. Recent work by several authors has focused attention
on this interesting invariant. We discuss how the Turaev genus is related to
other knot invariants, including the Jones polynomial, knot homology theories,
and ribbon-graph polynomial invariants.Comment: 18 page
Spanning trees and Khovanov homology
The Jones polynomial can be expressed in terms of spanning trees of the graph
obtained by checkerboard coloring a knot diagram. We show there exists a
complex generated by these spanning trees whose homology is the reduced
Khovanov homology. The spanning trees provide a filtration on the reduced
Khovanov complex and a spectral sequence that converges to its homology. For
alternating links, all differentials on the spanning tree complex are zero and
the reduced Khovanov homology is determined by the Jones polynomial and
signature. We prove some analogous theorems for (unreduced) Khovanov homology.Comment: Results in Version 2 have been split among two papers: The current
Version 3 and arXiv:0801.4937[math.GT
Mahler Measure and the Vol-Det Conjecture
The Vol-Det Conjecture relates the volume and the determinant of a hyperbolic
alternating link in . We use exact computations of Mahler measures of
two-variable polynomials to prove the Vol-Det Conjecture for many infinite
families of alternating links. We conjecture a new lower bound for the Mahler
measure of certain two-variable polynomials in terms of volumes of hyperbolic
regular ideal bipyramids. Associating each polynomial to a toroidal link using
the toroidal dimer model, we show that every polynomial which satisfies this
conjecture with a strict inequality gives rise to many infinite families of
alternating links satisfying the Vol-Det Conjecture. We prove this new
conjecture for six toroidal links by rigorously computing the Mahler measures
of their two-variable polynomials.Comment: 29 pages. V2: Minor changes, fixed typos, improved expositio
The 500 simplest hyperbolic knots
We identify all hyperbolic knots whose complements are in the census of
orientable one-cusped hyperbolic manifolds with eight ideal tetrahedra. We also
compute their Jones polynomials.Comment: Many knot diagrams simplified using Culler and Dunfield's Spherogram.
To appear in J. Knot Theory Ramifications. 29 page
A geometric method to compute some elementary integrals
An elementary, albeit higher dimensional, argument is used to compute the
area under the power function curve between 0 and 1.Comment: 24 pages, Mathematica code to rotate hyper-cubes included, color
illustrations are superior to their black and white rendering
- …