181 research outputs found
Knot theory and matrix integrals
The large size limit of matrix integrals with quartic potential may be used
to count alternating links and tangles. The removal of redundancies amounts to
renormalizations of the potential. This extends into two directions: higher
genus and the counting of "virtual" links and tangles; and the counting of
"coloured" alternating links and tangles. We discuss the asymptotic behavior of
the number of tangles as the number of crossings goes to infinity.Comment: chapter of the book Random Matrix Theory, Eds Akemann, Baik and Di
Francesc
The General O(n) Quartic Matrix Model and its application to Counting Tangles and Links
The counting of alternating tangles in terms of their crossing number, number
of external legs and connected components is presented here in a unified
framework using quantum field-theoretic methods applied to a matrix model of
colored links. The overcounting related to topological equivalence of diagrams
is removed by means of a renormalization scheme of the matrix model; the
corresponding ``renormalization equations'' are derived. Some particular cases
are studied in detail and solved exactly.Comment: 21 page
The Combinatorics of Alternating Tangles: from theory to computerized enumeration
We study the enumeration of alternating links and tangles, considered up to
topological (flype) equivalences. A weight is given to each connected
component, and in particular the limit yields information about
(alternating) knots. Using a finite renormalization scheme for an associated
matrix model, we first reduce the task to that of enumerating planar
tetravalent diagrams with two types of vertices (self-intersections and
tangencies), where now the subtle issue of topological equivalences has been
eliminated. The number of such diagrams with vertices scales as for
. We next show how to efficiently enumerate these diagrams (in time
) by using a transfer matrix method. We give results for various
generating functions up to 22 crossings. We then comment on their large-order
asymptotic behavior.Comment: proceedings European Summer School St-Petersburg 200
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