Categories generated by a trivalent vertex

Abstract

This is the first paper in a general program to automate skein theoretic arguments. In this paper, we study skein theoretic invariants of planar trivalent graphs. Equivalently, we classify trivalent categories, which are nondegenerate pivotal tensor categories over generated by a symmetric self-dual simple object X and a rotationally invariant morphism $1 \rightarrow X \otimes X \otimes X$. Our main result is that the only trivalent categories with $\dim {\text {Hom}}(1 \rightarrow X^{\otimes n})$ bounded by 1, 0, 1, 1, 4, 11, 40 for $0 \le n \le 6$ are quantum SO(3), quantum $G_2$, a one-parameter family of free products of certain Temperley-Lieb categories (which we call ABA categories), and the H3 Haagerup fusion category. We also prove similar results where the map $1 \rightarrow X^{\otimes 3}$ is not rotationally invariant, and we give a complete classification of nondegenerate braided trivalent categories with dimensions of invariant spaces bounded by the sequence 1, 0, 1, 1, 4. Our main techniques are a new approach to finding skein relations which can be easily automated using Gröbner bases, and evaluation algorithms which use the discharging method developed in the proof of the 4-color theorem.

Appendices

Appendix 1: Skein theoretic invariants and pivotal categories

The goal of this section is to provide background so that this paper is accessible to knot theorists, graph theorists, and other readers unfamiliar with tensor categories or planar algebras.

Suppose we want to study certain numerical invariants f of planar trivalent graphs. Assume that f of the empty diagram is 1, that and are nonzero, and that f satisfies the following multiplicative conditions:

• (0)
• (1)
• (2)
• (3)

Thus, the invariant of any k-disconnected graph for $k \le 3$ is determined by the invariants of the pieces.

Example 9.1

An almost trivial example of a multiplicative invariant of graphs is $a^{\# V}$, for some number a, where $\# V$ denotes the number of trivalent vertices in the graph.

Example 9.2

An important example of a multiplicative invariant of graphs is the number of n-colorings of the faces of the graph, divided by n. (The division by n is a normalization factor ensuring that the empty graph is assigned 1 instead of n.) This example can be generalized by considering non-integer specializations of the chromatic polynomial.

Question

What examples are there of such multiplicative invariants of trivalent planar graphs?

While the question appears to be an elementary question about planar trivalent graphs, we discover that the examples are actually related to quite distant subjects in mathematics. In particular, we are able to identify each of the small examples we encounter with some surprising or exotic object coming from representation theory or the theory of subfactors!

In order to understand the main results of the paper in the language of graph invariants, we first want to extend this invariant of closed trivalent graphs to an invariant of planar graphs with boundary. That is, we extract a sequence of vector spaces, the ‘open graphs, modulo negligibles.’ We now describe how these vector spaces have the structure of a pivotal tensor category (or planar algebra).

Let denote the (infinite-dimensional) vector space with basis the planar trivalent graphs drawn in the disk, with n fixed boundary points, up to isotopy rel boundary. This vector space has a natural bilinear pairing, given by gluing two open graphs together (starting at a preferred boundary point), to obtain a closed planar graph, which we then evaluate to a number using our multiplicative invariant f. The kernel of this bilinear pairing is called ‘the negligible elements.’ Let denote the quotient vector space of by negligible elements.

One may assemble these vector spaces into a single algebraic structure, variously axiomatized as an (unshaded) planar algebra [23], a spider [26] or a pivotal tensor category [6]. We’ll only describe the last in any detail. The category, which we’ll call , has as objects the natural numbers. We’ll first describe a bigger category of trivalent graphs, which we call and which does not depend at all on our multiplicative invariant. In , the morphisms from n to m are simply the formal linear combinations of planar graphs drawn in a rectangle with n points along the bottom edge and m points along the top edge, i.e., the vector space . We can compose morphisms in the obvious way, by stacking rectangles. This category is a tensor category, with the tensor product given by drawing diagrams side by side. Finally, it is a pivotal category, with the evaluation and coevaluation maps given by caps and cups.

Inside , the negligible (with respect to f) elements form a planar ideal—if some (linear combination of) graphs pair with arbitrary other graphs to give zero, then glueing more graph to the boundary preserves this property. We thus define the category to be the quotient of by the negligible ideal. This “ideal” property says that we can treat the negligible elements as skein relations: they can be applied locally in any part of a graph. Furthermore, typically this ideal is finitely generated by a few particular skein relations.

Thus, in , the objects are still the natural numbers and the morphisms from n to m are just . The category is still a pivotal tensor category, and now it is evaluable (i.e., , and in fact may be identified with the ground field by sending the empty diagram to 1) and non-degenerate (i.e., for every morphism $x: a \rightarrow b$, there is another morphism $x': b \rightarrow a$ so ). Writing X for the generating object in (i.e., 1 in the natural numbers!), we see that X is a symmetrically self-dual object, with duality pairings and copairings given by the cap and cup diagrams. Moreover, the trivalent vertex is a rotationally symmetric map $1 \rightarrow X \otimes X \otimes X$.

Example 9.3

If the invariant is the normalized number of n-colorings described in Example 9.2, then a linear combinations of graphs is negligible if and only if for any coloring of the boundary faces the given linear combination of the numbers of ways of extending that coloring to the interior is zero. For example, the following element of is negligible:

In particular, this gives a skein relation in which says that you can remove a triangle and multiply by $(n-3)$. There are also other negligible elements; in fact after renormalizing the trivalent vertex, becomes equivalent to the pivotal category $SO(3)_q$ coming from quantum groups where q is a number satisfying $(q+q^{-1})^2=n$ (see Sect. 4 for a description of $SO(3)_q$).

Proposition 9.4

The construction of from f gives a bijective correspondence between trivalent categories and multiplicative invariants of planar graphs.

Proof

First, we prove that the category constructed from a multiplicative invariant f is trivalent. Consider . The empty diagram is not negligible, so we need only show that any closed diagram is a multiple of the empty diagram. If $\alpha$ is a closed diagram and $\beta$ is the empty diagram, then $\alpha - f(\alpha ) \beta$ is negligible, so in we have that $\alpha = f(\alpha ) \beta$. Now we look at . By multiplicativity, we have that any diagram with one boundary point is negligible, so . The remaining cases are similar.

Given a trivalent category , we need to construct a multiplicative invariant of planar graphs. The usual diagrammatic calculus for pivotal categories shows that any trivalent category gives an invariant of closed graphs just by interpreting the graphs as elements of and sending the empty diagram to 1.

We want to check that this invariant is multiplicative, in which case it is clear that it provides an inverse to . We first check that the loop and the theta are nonzero. The single strand in must be nonzero, because if it were zero, then all nonempty diagrams would be zero. Since , we see that any diagram in is a multiple of the single strand, hence nondegeneracy says that the inner product of the strand with itself is nonzero, hence the loop value is nonzero. Similarly, by considering we see that the theta value is nonzero. Next we want to prove the multiplicative properties. Each of these are similar, so we only prove (2). We have that

is some multiple of the single strand, so we see that (by pairing with the strand). Substituting this into the LHS of (2) gives the RHS.

Appendix 2: Polynomials appearing in determinants

This appendix contains some of the irreducible factors of determinants appearing in this paper. The other irreducible factors, which are very large, are contained in text files packaged with the arXiv source of this paper and described here. Each polynomial is named as $Q_{i,j}$, where i is the largest exponent of d and j is the largest exponent of t. Where two polynomials have the same pair of largest exponents, we name them with an additional character in the subscript, as in $Q_{2,4,a}$ and $Q_{2,4,b}$.

$$\begin{aligned} P_{SO(3)}&= d (t-1)-t+2 \\ P_{ABA}&= t^2-t-1 \\ P_{G_2}&= d^2 t^5+d \left( 2 t^5-4 t^4-t^3+6 t^2+4 t+1\right) \\&\qquad +t^5-4 t^4+t^3+7 t^2-2 \\ Q_{0,1}&= t+1 \\ Q_{1,1}&= d (t+1)+t \\ Q_{1,2}&= d \left( 2 t^2+2 t+1\right) +3 t^2-2 \\ Q_{2,3}&= d^2 \left( t^3+t^2-2 t-1\right) +d \left( 2 t^3-2 t^2+t\right) +t^3-3 t^2+t+4 \\ Q_{3,4}&= d^3 \left( t^4+3 t^3-t^2-3 t-1\right) +d^2 \left( 2 t^4+t^2+2 t+1\right) \\&\qquad + d \left( t^4-3 t^3+3 t^2+6 t+1\right) -t^2+2 t+2 \\ \end{aligned}$$

$$\begin{aligned} Q_{3,5}&= d^3 \left( 3 t^5+4 t^4-2 t^3-6 t^2-4 t-1\right) \\&\qquad +d^2 \left( 8 t^5+2 t^4-11 t^3-5 t^2+5 t+3\right) \\&\qquad + d \left( 7 t^5-6 t^4-6 t^3+7 t^2+3 t-1\right) \\&\qquad +2 t^5-4 t^4+t^3+5 t^2-2 t-2 \\ Q_{2,4,a}&= d^2 \left( t^4-t^3-4 t^2-3 t-1\right) +d \left( 2 t^4-6 t^3-7 t^2+t+3\right) \\&\qquad +t^4-5 t^3+t^2+2 t-2 \\ Q_{2,4,b}&= d^2 \left( t^4+2 t^3-t^2-2 t-1\right) +d \left( 2 t^4-2 t^3-2 t^2+3 t+4\right) \\&\qquad +t^4-4 t^3+5 t^2+2 t-4 \\ Q_{4,5}&= d^4 t^5+d^3 \left( 3 t^5-3 t^4-3 t^3+7 t^2+5 t+1\right) \\&\qquad +d^2 \left( 3 t^5-5 t^4-5 t^3+10 t^2+12 t+2\right) \\&\qquad + d \left( t^5-t^4-5 t^3+3 t^2+9 t+5\right) +t^4-3 t^3+4 t+1 \\ \end{aligned}$$

$$\begin{aligned} Q_{6,9}&= d^6 \left( 4 t^8+t^7-15 t^6-20 t^5-6 t^4+8 t^3+10 t^2+5 t+1\right) \\&\qquad + d^5 \left( 2 t^9+12 t^8-19 t^7-54 t^6-17 t^5+21 t^4-11 t^3-43 t^2-30 t-7\right) \\&\qquad + d^4 \left( 6 t^9-6 t^8-31 t^7+11 t^6-119 t^4-130 t^3-21 t^2+35 t+14\right) \\&\qquad + d^3 \left( 2 t^9-32 t^8+72 t^7+59 t^6-227 t^5-258 t^4+59 t^3+164 t^2+43 t-3\right) \\&\qquad + d^2 \left( -10 t^9+10 t^8+123 t^7-136 t^6-305 t^5+103 t^4+225 t^3+23 t^2-38 t-13\right) \\&\qquad + d \left( -12 t^9+56 t^8-9 t^7-149 t^6-16 t^5+175 t^4+46 t^3-89 t^2-17 t+16\right) \\&\qquad -4 t^9+28 t^8-49 t^7-4 t^6+69 t^5-54 t^4-9 t^3+54 t^2-14 t-20 \\ Q_{\omega ,9}&= d^9-7 d^8+15 d^7-2 d^6-14 d^5-16 d^4+41 d^3-23 d^2+d+5 \\ Q_{\omega ,60}&= d^{60}-42 d^{59}+825 d^{58}-10050 d^{57}+84827 d^{56}-524435 d^{55}+2444075 d^{54}\\&\qquad -8680920 d^{53}+23364055 d^{52}-46267136 d^{51}+62172868 d^{50}\\&\qquad -43026307 d^{49}-10724689 d^{48}+19327948 d^{47}+113757871 d^{46}\\&\qquad -289556454 d^{45}+161677043 d^{44}+403173198 d^{43}-822414523 d^{42}\\&\qquad +340360209 d^{41}+658154819 d^{40}-734499791 d^{39}-499750302 d^{38}\\&\qquad +1417408819 d^{37}-680996389 d^{36}-701113119 d^{35}+1161482902 d^{34}\\&\qquad -934417344 d^{33}+751648667 d^{32}-23523738 d^{31}-1359642298 d^{30}\\&\qquad +1528218917 d^{29}+342409869 d^{28}-1836361788 d^{27}+946900947 d^{26}\\&\qquad +763927401 d^{25}-1172104767 d^{24}+652553812 d^{23}-193252562 d^{22}\\&\qquad -352541742 d^{21}+857069723 d^{20}-561108191 d^{19}-289399926 d^{18}\\&\qquad +602082003 d^{17}-186224613 d^{16}-206339296 d^{15}+185432097 d^{14}\\&\qquad -10906225 d^{13}-54265030 d^{12}+26840191 d^{11}+547786 d^{10}\\&\qquad -5118901 d^9+1967134 d^8-218389 d^7-37050 d^6+47054 d^5\\&\qquad -35063 d^4+10325 d^3-903 d^2-49 d+7 \\ \end{aligned}$$

The other factors, $Q_{7,11}, Q_{8,12}, Q_{11,19}, Q_{21,33}, Q_{22,36}, Q_{51,69}, Q_{54,78}$, and $Q_{36,60}$ are available in

and Mathematica formats in the polynomials/ subdirectory of the arXiv source as files Q_i,j.tex and Q_i,j.m, and also in the Mathematica notebook code/GroebnerBasisCalculations.nb