## Illinois Journal of Mathematics

### Tutte relations, TQFT, and planarity of cubic graphs

#### Abstract

It has been known since the work of Tutte that the value of the chromatic polynomial of planar triangulations at $(3+\sqrt{5})/2$ has a number of remarkable properties. We investigate to what extent Tutte’s relations characterize planar graphs. A version of the Tutte linear relation for the flow polynomial at $(3-\sqrt{5})/2$ is shown to give a planarity criterion for $3$-connected cubic (trivalent) graphs. A conjecture is formulated that the golden identity for the flow polynomial characterizes planarity of cubic graphs as well. In addition, Tutte’s upper bound on the chromatic polynomial of planar triangulations at $(3+\sqrt{5})/2$ is generalized to other Beraha numbers, and an exponential lower bound is given for the value at $(3-\sqrt{5})/2$. The proofs of these results rely on the structure of the Temperley–Lieb algebra and more generally on methods of topological quantum field theory.

#### Article information

Source
Illinois J. Math., Volume 60, Number 1 (2016), 273-288.

Dates
Received: 22 December 2015
Revised: 3 November 2016
First available in Project Euclid: 21 June 2017

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1498032033

Digital Object Identifier
doi:10.1215/ijm/1498032033

Mathematical Reviews number (MathSciNet)
MR3665181

Zentralblatt MATH identifier
1365.05137

#### Citation

Agol, Ian; Krushkal, Vyacheslav. Tutte relations, TQFT, and planarity of cubic graphs. Illinois J. Math. 60 (2016), no. 1, 273--288. doi:10.1215/ijm/1498032033. https://projecteuclid.org/euclid.ijm/1498032033