TL;DR
This paper explores how large matrix integrals with quartic potentials can be used to count complex link and tangle structures, extending to higher genus, virtual, and colored variants, and analyzes their asymptotic behavior.
Contribution
It introduces a novel connection between matrix integrals and the enumeration of advanced link and tangle configurations, including virtual and colored types.
Findings
Matrix integrals can count alternating links and tangles.
Redundancies in counting are addressed via potential renormalizations.
Asymptotic behavior of tangles with many crossings is analyzed.
Abstract
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.
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