139 research outputs found
Homogeneous links, Seifert surfaces, digraphs and the reduced Alexander polynomial
We give a geometric proof of the following result of Juhasz. \emph{Let
be the leading coefficient of the Alexander polynomial of an alternating knot
. If then has a unique minimal genus Seifert surface.} In
doing so, we are able to generalise the result, replacing `minimal genus' with
`incompressible' and `alternating' with `homogeneous'. We also examine the
implications of our proof for alternating links in general.Comment: 37 pages, 28 figures; v2 Main results generalised from alternating
links to homogeneous links. Title change
On Upper Bounds for Toroidal Mosaic Numbers
In this paper, we work to construct mosaic representations of knots on the
torus, rather than in the plane. This consists of a particular choice of the
ambient group, as well as different definitions of contiguous and suitably
connected. We present conditions under which mosaic numbers might decrease by
this projection, and present a tool to measure this reduction. We show that the
order of edge identification in construction of the torus sometimes yields
different resultant knots from a given mosaic when reversed. Additionally, in
the Appendix we give the catalog of all 2 by 2 torus mosaics.Comment: 10 pages, 111 figure
A model for the initiation of translation at the late stage after infection by adenoviruses
The Alexander polynomial of planar even valence graphs
AbstractWe show how the Alexander/Conway link polynomial occurs in the context of planar even valence graphs, refining the notion of the number of their spanning trees. Then we apply knot theory to deduce several statements about this graph polynomial, in particular estimates for its coefficients and relations between congruences of the number of vertices and number of spanning trees of the graph
Knots, Braids and BPS States in M-Theory
In previous work we considered M-theory five branes wrapped on elliptic
Calabi-Yau threefold near the smooth part of the discriminant curve. In this
paper, we extend that work to compute the light states on the worldvolume of
five-branes wrapped on fibers near certain singular loci of the discriminant.
We regulate the singular behavior near these loci by deforming the discriminant
curve and expressing the singularity in terms of knots and their associated
braids. There braids allow us to compute the appropriate string junction
lattice for the singularity and,hence to determine the spectrum of light BPS
states. We find that these techniques are valid near singular points with N=2
supersymmetry.Comment: 38 page
Knots and Random Walks in Vibrated Granular Chains
We study experimentally statistical properties of the opening times of knots
in vertically vibrated granular chains. Our measurements are in good
qualitative and quantitative agreement with a theoretical model involving three
random walks interacting via hard core exclusion in one spatial dimension. In
particular, the knot survival probability follows a universal scaling function
which is independent of the chain length, with a corresponding diffusive
characteristic time scale. Both the large-exit-time and the small-exit-time
tails of the distribution are suppressed exponentially, and the corresponding
decay coefficients are in excellent agreement with the theoretical values.Comment: 4 pages, 5 figure
Searching for a Connection Between Matroid Theory and String Theory
We make a number of observations about matter-ghost string phase, which may
eventually lead to a formal connection between matroid theory and string
theory. In particular, in order to take advantage of the already established
connection between matroid theory and Chern-Simons theory, we propose a
generalization of string theory in terms of some kind of Kahler metric. We show
that this generalization is closely related to the Kahler-Chern-Simons action
due to Nair and Schiff. In addition, we discuss matroid/string connection via
matroid bundles and a Schild type action, and we add new information about the
relationship between matroid theory, D=11 supergravity and Chern-Simons
formalism.Comment: 28 pages, LaTex, section 6 and references adde
Braided racks, Hurwitz actions and Nichols algebras with many cubic relations
We classify Nichols algebras of irreducible Yetter-Drinfeld modules over
groups such that the underlying rack is braided and the homogeneous component
of degree three of the Nichols algebra satisfies a given inequality. This
assumption turns out to be equivalent to a factorization assumption on the
Hilbert series. Besides the known Nichols algebras we obtain a new example. Our
method is based on a combinatorial invariant of the Hurwitz orbits with respect
to the action of the braid group on three strands.Comment: v2: 35 pages, 6 tables, 14 figure
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