2,244 research outputs found
Grid diagram for singular links
In this paper, we define the set of singular grid diagrams
which provides a unified description for singular links, singular Legendrian
links, singular transverse links, and singular braids. We also classify the
complete set of all equivalence relations on which induce the
bijection onto each singular object. This is an extension of the known result
of Ng-Thurston for non-singular links and braids.Comment: 33 pages, 34 figure
Period and toroidal knot mosaics
Knot mosaic theory was introduced by Lomonaco and Kauffman in the paper on
`Quantum knots and mosaics' to give a precise and workable definition of
quantum knots, intended to represent an actual physical quantum system. A knot
(m,n)-mosaic is an matrix whose entries are eleven mosaic
tiles, representing a knot or a link by adjoining properly. In this paper we
introduce two variants of knot mosaics: period knot mosaics and toroidal knot
mosaics, which are common features in physics and mathematics. We present an
algorithm producing the exact enumeration of period knot (m,n)-mosaics for any
positive integers m and n, toroidal knot (m,n)-mosaics for co-prime integers m
and n, and furthermore toroidal knot (p,p)-mosaics for a prime number p. We
also analyze the asymptotics of the growth rates of their cardinality
Chaos in the segments from Korean traditional singing and western singing
We investigate the time series of the segments from a Korean traditional song
``Gwansanyungma'' and a western song ``La Mamma Morta'' using chaotic analysis
techniques.
It is found that the phase portrait in the reconstructed state space of the
time series of the segment from the Korean traditional song has a more complex
structure in comparison with the segment from the western songs. The segment
from the Korean traditional song has the correlation dimension 4.4 and two
positive Lyapunov exponents which show that the dynamic related to the Korean
traditional song is a high dimensional hyperchaotic process. On the other hand,
the segment from the western song with only one positive Lyapunov exponent and
the correlation dimension 2.5 exhibits low dimensional chaotic behavior.Comment: 23 pages including 10 eps figures, latex, to appear in J. Acoust.
Soc. A
A new intrinsically knotted graph with 22 edges
A graph is called intrinsically knotted if every embedding of the graph
contains a knotted cycle. Johnson, Kidwell and Michael showed that
intrinsically knotted graphs have at least 21 edges. Recently Lee, Kim, Lee and
Oh, and, independently, Barsotti and Mattman, showed that and the 13
graphs obtained from by moves are the only intrinsically
knotted graphs with 21 edges.
In this paper we present the following results: there are exactly three
triangle-free intrinsically knotted graphs with 22 edges having at least two
vertices of degree 5. Two are the cousins 94 and 110 of the family and
the third is a previously unknown graph named . These graphs are shown
in Figure 3 and 4. Furthermore, there is no triangle-free intrinsically knotted
graph with 22 edges that has a vertex with degree larger than 5
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