Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Why the 3D Writhe of Ideal Knots and Links Is Quantized and Additive   Besides mathematical interest, knots and knot theory have important
applications in physics, chemistry, and biology. Stasiak and colleagues devised
a constructive method for a knot "energy" using a Metropolis Monte Carlo
algorithm to minimize the length of rope needed to realize a given knot for a
rope of a fixed diameter. They called the resulting rendering of the knot the
"ideal" configuration. They have found (i) that the 3-dimensional (3D) writhe
of the ideal configuration of a knot appeared, empirically, to be quantized;
(ii) empirically, the 3D writhe of the ideal configuration was additive over
composition of knots; and (iii) the 3D writhe of ideal configuration of
alternating knots (which have a projection in which crossings alternate between
over and under as one transverses the knot in a fixed direction) seems to be
equal, to 1*T + 3/7*(W-B), where T is the "Tait number," and W and B are the
number of the white and black regions in the checkerboard coloring of the knot.
It is still not clear why properties i-iii might be true. Here I show that all
these empirical findings can be related to the fact that T and W-B can be
proven to be additive over composition of alternating knots. In particular, (1)
the 3D writhe of ideal configurations is likely only to be additive for
composition of alternating knots, and (2) remarkably, the geometrical structure
of all alternating (even prime) knots is determined by the three smallest
knots.

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The categories of this paper are: atom-ph
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The categories of this paper are: q-fin
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The categories of this paper are: nucl-th
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The categories of this paper are: hep-ex
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The categories of this paper are: acc-phys
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The categories of this paper are: econ
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The categories of this paper are: gr-qc
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The categories of this paper are: hep-th
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The categories of this paper are: stat
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The categories of this paper are: q-bio
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The categories of this paper are: quant-ph
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The categories of this paper are: cs
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The categories of this paper are: ao-sci
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The categories of this paper are: dg-ga
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The categories of this paper are: alg-geom
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The categories of this paper are: solv-int