1,537 research outputs found
Slice and Gordian numbers of track knots
We present a class of knots associated with labelled generic immersions of
intervals into the plane and compute their Gordian numbers and 4-dimensional
invariants. At least 10% of the knots in Rolfsen's table belong to this class
of knots. We call them track knots. They are contained in the class of
quasipositive knots. In this connection, we classify quasipositive knots and
strongly quasipositive knots up to 10 crossings.Comment: 16 pages, 17 figure
Gaussian distribution of LMOV numbers
Recent advances in knot polynomial calculus allowed us to obtain a huge
variety of LMOV integers counting degeneracy of the BPS spectrum of topological
theories on the resolved conifold and appearing in the genus expansion of the
plethystic logarithm of the Ooguri-Vafa partition functions. Already the very
first look at this data reveals that the LMOV numbers are randomly distributed
in genus (!) and are very well parameterized by just three parameters depending
on the representation, an integer and the knot. We present an accurate
formulation and evidence in support of this new puzzling observation about the
old puzzling quantities. It probably implies that the BPS states, counted by
the LMOV numbers can actually be composites made from some still more
elementary objects.Comment: 23 page
On the Topological Characterization of Near Force-Free Magnetic Fields, and the work of late-onset visually-impaired Topologists
The Giroux correspondence and the notion of a near force-free magnetic field
are used to topologically characterize near force-free magnetic fields which
describe a variety of physical processes, including plasma equilibrium. As a
byproduct, the topological characterization of force-free magnetic fields
associated with current-carrying links, as conjectured by Crager and Kotiuga,
is shown to be necessary and conditions for sufficiency are given. Along the
way a paradox is exposed: The seemingly unintuitive mathematical tools, often
associated to higher dimensional topology, have their origins in three
dimensional contexts but in the hands of late-onset visually impaired
topologists. This paradox was previously exposed in the context of algorithms
for the visualization of three-dimensional magnetic fields. For this reason,
the paper concludes by developing connections between mathematics and cognitive
science in this specific context.Comment: 20 pages, no figures, a paper which was presented at a conference in
honor of the 60th birthdays of Alberto Valli and Paolo Secci. The current
preprint is from December 2014; it has been submitted to an AIMS journa
Lens space surgeries on A'Campo's divide knots
It is proved that every knot in the major subfamilies of J. Berge's lens
space surgery (i.e., knots yielding a lens space by Dehn surgery) is presented
by an L-shaped (real) plane curve as a "divide knot" defined by N. A'Campo in
the context of singularity theory of complex curves. For each knot given by
Berge's parameters, the corresponding plane curve is constructed. The surgery
coefficients are also considered. Such presentations support us to study each
knot itself, and the relationship among the knots in the set of lens space
surgeries.Comment: 26 pages, 19 figures. The proofs of Theorem 1.3 and Lemma 3.5 are
written down by braid calculus. Section 4 (on the operation Adding squares)
is revised and improved the most. Section 5 is adde
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