F-polynomial formula from continued fractions

Rabideau, Michelle

doi:10.48550/arxiv.1612.06845

Cited by 1 publication

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“…Moreover, the numerator of that continued fraction is the number of perfect matchings of its associated snake graph. This relation to continued fractions was found in [CS4] and applications were given in [CS5,CLS,LS,R]. Therefore by Theorem 4.2, the numerator of the continued fraction associated to a Markov snake graph is that Markov number.…”

Section: Proof Of Conjectures 12 and 13supporting

confidence: 53%

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Continued fractions and orderings on the Markov numbers

Rabideau¹,

Schiffler²

2018

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Markov numbers are integers that appear in the solution triples of the Diophantine equation, x 2 + y 2 + z 2 = 3xyz, called the Markov equation. A classical topic in number theory, these numbers are related to many areas of mathematics such as combinatorics, hyperbolic geometry, approximation theory and cluster algebras.There is a natural map from the rational numbers between zero and one to the Markov numbers. In this paper, we prove two conjectures seen in Martin Aigner's book, Markov's theorem and 100 years of the uniqueness conjecture, that determine an ordering on subsets of the Markov numbers based on their corresponding rational.The proof uses the cluster algebra of the torus with one puncture and a resulting reformulation of the conjectures in terms of continued fractions. The key step is to analyze the difference in the numerator of a continued fraction when an operation is applied to its entries.

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Section: Proof Of Conjectures 12 and 13supporting

confidence: 53%

“…General snake graphs were introduced in [MSW] in order to give a combinatorial formula for cluster variables in terms of perfect matchings. These graphs were further studied in [CS,CS2,CS3,CS4,CS5,R]. The special case of Markov snake graphs already appeared in [P].…”

Section: Proof Of Conjectures 12 and 13mentioning

confidence: 99%