## Illinois Journal of Mathematics

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

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

Ian Agol and Vyacheslav Krushkal

#### 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

**Mathematical Reviews number (MathSciNet)**

MR3665181

**Zentralblatt MATH identifier**

1365.05137

**Subjects**

Primary: 05C31: Graph polynomials 57R56: Topological quantum field theories

Secondary: 57M15: Relations with graph theory [See also 05Cxx] 05C10: Planar graphs; geometric and topological aspects of graph theory [See also 57M15, 57M25]

#### Citation

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