SSKEIN THEORIES FOR VIRTUAL TANGLES
JOSHUA R. EDGE
Abstract.
In this paper, we use skein-theoretic techniques to classify all virtual knotpolynomials and trivalent graph invariants with certain smallness conditions. The firsthalf of the paper classifies all virtual knot polynomials giving non-trivial invariants strictlysmaller than the one given by the Higman-Sims spin model. In particular, we exhibit afamily of skein theories coming from Rep( O (2)) with an interesting braiding. In addition,all skein theories of oriented virtual tangles with some smallness conditions are classified.In the second half of the paper, we classify all non-trivial invariants of (perhaps non-planar)trivalent graphs coming from symmetric trivalent categories. For each of these categories,we also classify when the sub-category generated by only the trivalent vertex is braided. Aninteresting example of this arise from the tensor category Deligne’s S t . Introduction
Skein relations are tools that are used to study knots. In the late 1960s, Conway firstdemonstrated how the Alexander polynomial could be computed using skein relations. TheAlexander polynomial was the only known knot polynomial for almost 60 years, until Jonesexhibited a new knot polynomial using skein relations [Jon85]. This led to the discovery ofother knot polynomials using skein relations including the HOMFLY polynomial [FYH + strictly smaller than the one given by the Higman-Simsspin model. Skein theoretic classifications of this type can first be seen in [BJ00], [BJ03] andalso in [Kup94] among others. The result of this classification can be summarized as follows: Date : August, 10, 2020. a r X i v : . [ m a t h . QA ] A ug JOSHUA R. EDGE
Theorem 1.
A non-trivial virtual knot polynomial where the dimension of the 4-box spaceis less than 4 is equivalent to one of the following:i. The virtual Jones polynomialii. The Kauffman polynomial at d = − and a = ± equipped with a symmetric crossing = − (cid:32) + (cid:33) iii. A one-parameter family of knot polynomials where the crossing has formula = a − − a − a − − a a − + a Theorem 2. If V is a spherical quotient of the planar algebra of oriented virtual tangles suchthat the vector space of diagrams with endpoints oriented (+ , + , − , − ) is two-dimensional,then V is isomorphic to either Deligne’s GL t or Rep ( GL (1)) . In both cases the planar algebrais equipped with the following braiding: = a · In addition to the contributions skein-theoretic calculations have in knot theory, they alsoprovide invariants in graph theory. The planar algebras found in [MPS17], called trivalentcategories, give important invariants of planar trivalent graphs. One can think of the sym-metric version of these categories as simply removing the planarity condition on the graphs.The second part of this paper classifies all skein-theoretic invariants of trivalent graphs withsome smallness conditions. These graph-theoretic invariants are summarized in Theorem6.1.6. In the last portion of the paper, we prove the following result:
Theorem 3.
The non-degenerate quotient of Deligne’s S t has a non-symmetric braiding onthe subcategory generated by the trivalent vertex when t ∈ C − { , , , } and is given by theformula = ( q − · + q − − ( q + q − ) at t = q + 2 + q − . Those familiar with planar algebras will note that this is the formula for the braiding of( SO (3)) q . This connection is interesting and will be discussed in greater detail in this paper.1.1.
Source code.
When Mathematica is used to aid in computation, the source code forthat calculation is included in the ArXiv source code. The diagrams used in this paper areincluded under the “diagrams” subdirectory.1.2.
Acknowledgments.
The author would like to thank his advisor, Noah Snyder, for hisguidance through this project. He would also like to thank Pavel Etingof for explaining theorigin of the subbraiding on Rep( O (2)) and Dylan Thurston for his suggestion of the S action. In addition, he would like to acknowledge the authors of [MPS17]–Scott Morrison,Emily Peters, and Noah Snyder–for including the source code for the diagrams in theirpapers. Lastly, he would like to thank Maxime Scott for his helpful suggestions. The authorwas also supported through NSF grant DMS-1454767. KEIN THEORIES FOR VIRTUAL TANGLES 3 Planar algebra definitions
In this section, we record the formal definitions of a planar algebra similar to [Jon99]and [Pet09]. In addition, we also establish some conventions that we will use in subsequentsections.2.1.
Definition of a planar algebra.Definition 2.1.1.
A planar diagram is described as follows. The diagram is the unit disc D with a finite number of clockwise-numbered boundary points (possibly zero). In addition,there are a finite number of internal discs each with their own finite number of clockwise-numbered boundary points (possibly zero). The boundary points of all diagrams are pairedby smooth, non-crossing curves called the strings of the diagram. In addition, there mayalso be a finite number of closed strings not connecting to any discs. The strings lie betweenthe internal discs and D , and the connected components of the interior minus the strings arecalled the regions of the planar diagram. To indicate the first point on the boundary of theexternal or an internal disc, the region immediately preceding it is starred near the relevantboundary component.An example of a planar diagram is(1) ??? ? Because of the appearance of these diagrams, they are often called “spaghetti and meatball”diagrams. We can compose diagrams A and B if the number of strands connecting to theouter disc of B is equal to the number of strands connecting to one of the inner discs of A or vice versa. An example of this composition is below:(2) ?? ? ? ◦ ?? = ?? ? ? In this example, the planar diagram with inner disc 4 is inserted into inner disc 2 of theother planar diagram, giving the new planar diagram on the right.
Definition 2.1.2.
The planar operad, P , is the set of isotopy classes of planar diagramsunder composition as described above. Definition 2.1.3.
The standard involution of a planar diagram is performed in the followingsteps. Take any diameter of the outer disc through the starred region. Reflect the diagramover that diameter. Finally, relabel all inner and outer boundary points so that the originalstarred regions of each disc are preserved.
JOSHUA R. EDGE
As an example, consider ??? ? ∗ −→ ?? ? ? Taking an involution can be quite cumbersome for large planar diagrams. As a solution, wedraw the diagrams in what is often called “standard” form.
Definition 2.1.4.
A planar diagram T is drawn in standard form if the input and outputcircles are drawn as rectangles, with strings attached only to the top or bottom and thestarred region represented by the region bordering the left-hand side of the rectangle.By drawing planar diagrams in this way, an involution simply rotates the planar diagram180 degrees but takes the complex conjugate of any coefficients. In what follows, all planardiagrams are assumed to be in standard form. Definition 2.1.5.
A planar algebra is a family of k -vector spaces V = { V , V , V , . . . } , together with an action of the planar operad. This action assigns a multilinear map to every T ∈ P that respects the composition of planar diagrams. Moreover, if T is a planar diagramwith k inner discs, the corresponding multilinear map is T : V i ⊗ V i ⊗ · · · ⊗ V i k → V o so that i m is the number of strands connected to the boundary of the m -th inner boundarydisc, and o is the number of strands connected to the outer boundary disc. V is a planar ∗ -algebra if in addition the action of the planar operad also respects the involution on P . Put another way, a planar algebra is a graded vector space that respects the action ofthe planar operad. In this paper, we will set k = C . The dimension of a planar algebra isusually given by a sequence a , a , a , . . . where a i = dim( V i ) and V i is often called the i -boxspace.The planar operad can be thought of as the set of operations on a planar algebra. In (1),for example, a planar algebra would determine a multilinear map V ⊗ V ⊗ V → V Some of these operations appear often and are shown below for two elements of a planaralgebra X and Y : Multiplication of two diagrams: ...... X · ...... Y := ......... XY Capping off a diagram on the top: ...... X · := ...... X Rotating a diagram by “1 click:” ρ (cid:32) ...... X (cid:33) := ...... X KEIN THEORIES FOR VIRTUAL TANGLES 5
Of course, we might also want to cap a diagram off in some other way or rotate by morethan 1 click, both of which are defined similarly. Note that multiplication is only defined ifthe number of bottom endpoints for Y is the same as the number of top endpoints of X . Definition 2.1.6.
Let V be a planar algebra and f be the multilinear map associated withsome planar diagram T . Then V is said to have modulus d if for all T ∈ P inserting a closedstring into a region of T is equivalent to the map d · f, for some d ∈ k × . Alternatively, one can think of a planar algebra having modulus d if there exists a relationof the form: = d where the right-hand side of the equation is d times the diagram with no strings, called theempty diagram. For this reason, we often call d the circle parameter. Definition 2.1.7.
A planar algebra V is called evaluable if dim V = 1, or, equivalently, anyelement of V is equivalent to the empty diagram up to a scalar.Thus, for an evaluable planar algebra we can define a map from V → k by sending theempty diagram to 1. This map can be used to define an inner product on each V n . Definition 2.1.8.
Let V be an evaluable planar algebra. We define an inner product on V n as (cid:42) ...... X , ...... Y (cid:43) := ......... XY ... Of course, we could have also connected the strands on the left. If these quantities are thesame, the planar algebra is said to be spherical . Definition 2.1.9.
An non-zero element v n ∈ V n such that (cid:104) v n , w n (cid:105) = 0 for all w n ∈ V n iscalled a negligible element. Definition 2.1.10.
A planar algebra is non-degenerate if it has no negligible elements.How does one find all negligible elements of V n ? Given a basis for V n , { v , v , . . . , v k } , wecan take the matrix of inner products, M , where the ( i, j )-th entry of M is the inner productof v i and v j . The null space of this matrix would then give us the linear subspace of V n spanned by the negligible elements. Definition 2.1.11.
A rotational eigenvector is a diagram S such that if ρ is the planardiagram that rotates the diagram one click then ρ ( S ) = ζ · S, for some ζ ∈ C . If ρ has n output strands, we note that this immediately implies that ζ is an n -th root ofunity, as ζ n S = ρ n ( S ) = S . Because this space of relations is a Z /n Z representation, theaction is diagonalizable. Thus, given that ρ n ( S ) = S we know that ρ ( S ) = ζ · S. Definition 2.1.12.
Let V and W be planar algebras. A planar algebra map, Φ : V → W , isa collection of linear maps φ ± i : V ± i → W ± i intertwining the respective actions of the planaroperad. JOSHUA R. EDGE
By “intertwining” of the actions of the planar operad, we mean that if T ( v , . . . , v k ) = v o then T (Φ( v ) , . . . , Φ( v k )) = Φ( v o ) for all T ∈ P and v i ∈ V . Definition 2.1.13. W is a sub-planar algebra of V if W i ⊆ V i for all i and if the inclusionmaps { i k : W k → V k } intertwine the actions of the planar operad. Definition 2.1.14. Q is a quotient of a planar algebra V if Q i is a quotient of V i and if thequotient maps { π k : V k → Q k } intertwine the actions of the planar operad. Definition 2.1.15.
Two planar algebras V and W are isomorphic if there is a collectionof isomorphisms { φ k : V k → W k } that intertwine the actions of the planar operad. Whenthe planar algebras are ∗ -planar algebras, we also require that the isomorphism be a ∗ -isomorphism.2.1.1. Skein theories.
When one defines a ring or algebra, we often describe it using gen-erators and relations. Similarly, we can define when a set of diagrams generates a planaralgebra.
Definition 2.1.16.
A planar algebra, V , is generated by a set of elements, R , if for everyelement v ∈ V , there exists a planar diagram, T , such that T ( r , . . . , r k ) = v for some r i ∈ R . When one studies all quotients of a ring or algebra, one might instead classify all two-sidedideals. Similarly, when studying the quotients of a planar algebra, we instead classify all itsplanar ideals.
Definition 2.1.17.
Let V be a planar algebra. A planar ideal, I , is a subset of V such that I is the kernel of some quotient map of V . Equivalently, I is a planar ideal if it is closed under arbitrary composition with any elementof the planar algebra. That is, if we let T be any planar tangle, then T ( v , . . . , v k ) ∈ I ifat least one v i is an element of I for all possible choices of the other elements. Just like atwo-sided ideal in a ring, a planar ideal can also be written using relations, which we callskein relations. Definition 2.1.18.
Let V be a planar algebra. A skein relation is an element of the kernelof a quotient map from V . A skein theory is a set of skein relations that generates a planarideal.Thus, giving a skein theory would be an equivalent way of defining a quotient of a planaralgebra. If we wanted to classify all quotients of a planar algebra, then, we could insteadclassify all skein theories of that planar algebra. In Section 3, we will examine a number ofwell-known examples of skein theories.2.2. Oriented planar algebras.
Our definition of a planar algebra assumes that the stringsof planar diagrams are unoriented. One could imagine defining a planar algebra, though,by giving an action on planar diagrams with oriented strings, the collection of which isappropriately called the oriented planar operad. In this case, the orientation of the stringsgives orientations to the endpoints, either positive or negative. If we imagine every endpointof a diagram on the bottom, then an endpoint will be positively oriented if the strandconnecting to it is pointing away from the endpoint and negatively oriented if it is pointingtoward the endpoint.
KEIN THEORIES FOR VIRTUAL TANGLES 7
Definition 2.2.1.
An oriented planar algebra V is a collection of vector spaces V ( z ,...,z i ) where z j ∈ { + , −} together with an action of the oriented planar operad.Elements of an oriented planar algebra are also drawn diagrammatically with orientedstrands. The z i tell us the orientation of each endpoint. For example, elements of V (+ , + , − , − ) include any diagram that can be drawn in the following form: X Other concepts we have defined for unoriented planar algebras have oriented analogs, butthose definitions are omitted here.2.3.
Shaded planar algebras.
Suppose that we impose the additional requirement thatdiagrams in the planar operad be checkerboard shaded. As an example, consider(3) T := ∗ ∗ There are two main differences to note between shaded planar diagrams and their unshadedcounterparts. First, in order to achieve a checkerboard shading it is clear that every discmust have an even number of boundary points. Second, the starred region can now beshaded or unshaded. Both of these differences are reflected in the definition of a shadedplanar algebra.
Definition 2.3.1.
A shaded planar algebra is a collection of vector spaces V = { V ± i } ,i ∈ Z ≥ together with an action of the planar operad.Here, V ± i represents the vector spaces of diagrams with 2 i endpoints, which are alsotraditionally checkerboard-shaded. Thus, a shaded diagram with four endpoints is a 2-box, while an unshaded diagram with four endpoints is a 4-box. If the starred region isunshaded with 2 i endpoints, then it is an element of V + i . As an example, (3) defines themap T : V − → V +3 . Examples of shaded planar algebras appear in Section 3.
Definition 2.3.2.
Let V = { V ± i } i =0 , ,... be a shaded planar algebra. V even is the collectionof vector spaces { V +0 , V +2 , V +4 , . . . } (i.e. the 2 i -box spaces with unshaded starred region).The even portion of any shaded planar algebra is necessarily unshaded by assigning Thus, when we restrict to the even portion of maps of shaded planar algebras, we can oftenfind interesting maps of unshaded planar algebras. In Section 3.4.1, we will see an interestingexample of how one can obtain virtual knot and link invariants from a certain map of shadedplanar algebras. In that example, though, it turns out that the planar algebra in questionswill be too large for us to handle, so we will use the following method to make it moremanageable:
JOSHUA R. EDGE
Definition 2.3.3.
Let V be a shaded planar algebra and P be some projection in in V ± i . Acut-down of V is the planar algebra given by the map · · · · · ·· · · P when P is a + i -box and similarly when it is a − i -box.When P ∈ V ± i then V is an unshaded planar algebra, while it is still shaded if P ∈ V ± (2 i +1) .In this paper, we will only utilize projections in the 2-box space, and so all the resultingplanar algebras will be unshaded.2.3.1. Planar algebra of a bipartite graph.
Another class of shaded planar algebras is theplanar algebra of a bipartite graph (originally defined in [Jon00]). Although we define thisplanar algebra in full generality here, we will only make use of the planar algebra of a specialbipartite graph, ∗ n , which is also called the spin planar algebra in [Jon99].Let G be a locally-finite, connected bipartite graph (not necessarily simply-laced) withedge set E , vertex set U = U + ∪ U − , with | U + | = n , | U − | = n , and | U | = n + n = n .Further, there are no edges that connect U + or U − to itself. Because G is a bipartite graph,its adjacency matrix is of the form (cid:18) T (cid:19) where Λ is a n × n matrix and the ( i, j )-th entry is the number of edges connect v + i to v − j . In this project, we care about a particular class of graphs, which we call ∗ n : Definition 2.3.4. ∗ n is the complete bipartite graph with U − = { (cid:63) } and U + = S , a set of n even vertices.For each k ∈ Z ≥ we define the basis of L ± k to be loops in G of length k, with one vertexof the loop declared the “base.” Loops in L + k are based in U + and loops in L − k are basedin U − . We will denote such a loop with a pair of functions ( π, (cid:15) ) : { , , . . . , k − } → U ∪ E, where the π ( i ) is the ( i + 1)-th vertex of the loop and (cid:15) ( i ) is the ( i + 1)-th edge. The astutereader might note that given an edge, one can determine the corresponding vertices andrecreate the loop simply from (cid:15). In many of the examples that follow, however, the graphsare simply-laced and (cid:15) will be suppressed in favor of π. The following definition was given in [Jon00]:
Definition 2.3.5.
The planar algebra of a bipartite graph G , GPA( G ) , is a collection ofvector spaces { GP A ( G ) ± k } where GPA( G ) ± k = { f : L ± k → C } , the set of functionals ofbased loops in U ± of length k , together with an action of the planar operad.For any bipartite graph, G , it is true that L ± = U ± . Thus, a basis for GPA( G ) ± = { δ v | v ∈ U ± } , where δ v is the Kronecker delta(4) δ v ( w ) := (cid:26) w = v Proposition 2.3.6.
For any k , a basis for GPA ( G ) ± k is given by { δ l | l ∈ L ± k } . KEIN THEORIES FOR VIRTUAL TANGLES 9
Proof.
A functional f : X → C is a linear mapping of every element of X to a complexnumber. In our case, X = L k , so a functional is an assignment of every loop in L k to acomplex number. Then we can rewrite each functional as f = (cid:88) l ∈ L k δ l · f ( l )Thus, every such functional is a linear combination of the δ l . Additionally, since each of the δ l takes a value of 1 on a unique value, the set is clearly linearly independent. Thus, theymust form a basis, as desired. (cid:3) To form this set of vector space into a planar algebra, though, we must also define anaction of the planar operad. This is accomplished using “states”:
Definition 2.3.7.
A state σ on a planar diagram T is an assignment of vertices to regionsand edges to strings which is consistent with the graph structure of G . That is, given anassignment of vertices v and w to neighboring regions, the edge e assigned to the stringseparating those regions should have endpoints v and w .We note here that for a simply-laced graph, this labeling of strings can be omitted ifwe instead require that adjacent regions must be labeled by adjacent vertices. If σ is astate, then we define σ | i to be the based loop obtained by reading the states assigned to theregions of the i -th input disc with the starred region being the base, and σ | boundary circle = σ | o is defined similarly. Definition 2.3.8 (Action 1) . Let GPA( G ) be as described above, and let T be a planartangle. T acts on functionals f , . . . , f k by sending ( f , . . . , f k ) to the functional describedbelow:(5) T ( f , f , . . . , f k )( l ) = (cid:88) states on T such that σ | o = l k (cid:89) i =1 f i ( σ | i )As an example, consider the graph ∗ ? edcba and the planar tangle T := ∗ ∗ j i k where i, j, k are elements of U + . The shaded regions are labelled with (cid:63) but are omitted toavoid confusion with the starred regions. When our graph is ∗ n , we will write loops as aconcatenation of its vertices omitting (cid:63) . Thus, the loop ( (cid:63), a, (cid:63), b ) would be written as ab . By definition of Action 1 of the planar operad, when δ ab is inputted we obtain the functional T ( δ ab )( ijk ) = (cid:26) i = a and j = b (cid:88) k ∈ S δ abk One can extend this result linearly to get a functional for any choice of input. By consideringthe action of V on planar tangles with no input discs, we see that a b = (cid:26) a = b a (cid:54) = b and a b = 1for all a, b ∈ S. Thus, we know that(6) = n (cid:88) i =1 δ ii and(7) = n (cid:88) i,j =1 δ ij Let us now consider the following planar diagram: ? Note here that the center circle is not an input disc but instead a closed string. Using thestate sum formula, there are no functionals to input so for v ∈ U − ? ( v ) = deg( v )where deg( v ) is the number of neighbors of v in G . In our example, the only v ∈ U − is (cid:63) , which has degree 5. The planar diagram with the opposite shading would give 1 for allpossible choices of input in U + .The reader can verify that Action 1 respects the composition of planar tangles and, thus,gives an action of the planar operad on GPA( G ) . We also noted, however, that the value ofa closed string in a diagram was dependent upon the degree of the vertex under this action.When G = ∗ n , however, we have that = 1 = n In order for the circle to have a fixed value, d, we will redefine the action following theconstruction in [Jon00]. Before we do this, however, we need to define the Perron-Frobeniuseigenvector. Definition 2.3.9.
The Perron-Frobenius eigenvector of a graph is a map ω : V ( G ) → R such thati. ω ( v ) > v ∈ V ( G ) KEIN THEORIES FOR VIRTUAL TANGLES 11 ii. There is a d ∈ R ≥ such that (cid:88) w adj v ω ( w ) = d · ω ( v ).It is well-known that ω is an eigenvector of the adjacency matrix of G corresponding to thelargest eigenvalue d . It is also proven that such a d and function ω can be scaled to meet theabove requirements for any locally finite, connected graph. For, ∗ n , the largest eigenvalue is √ n with corresponding eigenvector ω = (cid:8) √ n, , , . . . , (cid:9) where (cid:63) represents the first coordinate of ω . We now define another action of the planaroperad on GPA( G ) using ω : Definition 2.3.10 (Action 2) . For a bipartite graph G , a planar diagram T acts on f , . . . , f k by(8) T ( f , f , . . . , f k )( l ) = (cid:88) states on T such that σ | o = l c ( T, σ ) k (cid:89) i =1 f i ( σ | i )where c ( T, σ ) is some constant depending on σ , T , and ω, the Perron-Frobenius eigenvectorof G .In [Jon00], the following theorem is proven: Theorem 2.3.11 ([Jon00]) . The definition of Action 2 makes GPA ( G ) into a planar algebra.This planar algebra has modulus d, the largest eigenvalue of the adjacency matrix of G .Further, if k = C then this action defines a planar C ∗ -algebra. Thus, any planar algebra defined using Action 1 can be re-normalized by using Action 2.2.3.2.
Spin Models.
The following definition is based on [Jon99]. More details about spinmodels can also be found in [Edg19]:
Definition 2.3.12.
Let V be a shaded planar algebra. A spin model for V is a map of planaralgebras Φ : V →
GPA( ∗ n ) for some n .Suppose V is generated by elements of V +2 . Recall that a basis for GPA( ∗ n ) k is thecollection of Kronecker deltas of loops, δ l , where l is any k -tuple of elements of S . If X is anelement of V +2 then, in general, Φ( X ) = (cid:88) a,b ∈ S c X ( a, b ) · δ ab , where c X ( a, b ) ∈ C . Let C X bethe n × n matrix where the ( a, b ) entry of C X is c X ( a, b ) . By the action of the planar operadon GPA( ∗ n ) , we know that the matrix corresponding to is the identity matrix; similarly,the matrix corresponding to is the matrix of all ones.To capture the fact that Φ is a map of planar algebras, we will write c X ( a, b ) := X a b for all a, b ∈ S and generators X . More generally we can give the following definition: Definition 2.3.13.
Suppose Φ gives a spin model for some shaded planar algebra, V , thatis generated by 2-boxes and let X ∈ V ± k . Then a state sum with respect to ( a , . . . , a k ) isthe sum over all possible labelings of the unshaded regions of X such that the i th exteriorregion of X (starting with the starred region and going clockwise) is a i . As an example, the state sum of P with respect to a is X b ∈ S P a b Just like for c X ( a, b ) for some generator X , one should think of the state sum of any diagram Y with respect to ( a , . . . , a i ) as the coefficient of δ a ··· a i of Φ( Y ) . Examples
We now discuss in detail a number of well-known planar algebras that will arise in ourclassifications.3.1.
Planar algebra of unoriented tangles.Definition 3.1.1.
The planar algebra of unoriented tangles is the planar algebra generatedby . Diagrams are considered up to isotopy. In addition, we will impose the followingrelations: Reidemeister II (R2): =Reidemeister III (R3): =We will frequently refer to these relations by R2 and R3 for simplicity. Reidemeister I,which is given by =is not required to be satisfied. In all of our following examples, however, we will require thatR1 be satisfied up to a scalar. A relation of this form is often called a twist relation:(9) = a · Note that (9) and R2 imply that = a − · Clearly, this planar algebra has trivial odd-box spaces. The even-box spaces, on theother hand, are infinite dimensional! This is evident even in the 0-box space, as everynon-equivalent knot and link are linearly independent. Rather than studying the planaralgebra of unoriented tangles itself, we will instead look at quotients of this planar algebraby studying their skein theories. A few well-known skein theories of the planar algebra ofunoriented tangles are the Jones and Kauffman polynomials, originally discussed in [Jon85]and [Kau90].
KEIN THEORIES FOR VIRTUAL TANGLES 13
Braided planar algebras.
Before we discuss these quotients, we give a general definitionof a braided planar algebra using the relations above.
Definition 3.1.2.
A planar algebra, V , is said to be braided if there exists an element of V that satisfies a twist relation, R2, and R3. In addition, the braiding must satisfy thefollowing naturality condition: ...... S = ...... S for any element S ∈ V . We will classify all such braidings of symmetric trivalent planar algebras in Section 6.While some planar algebras have elements that satisfy the Reidemeister relations, they donot always act naturally with every element in the planar algebra. In these instances,although the entire planar algebra is not braided, it is true that the planar algebra has asubplanar algebra that is braided.
Definition 3.1.3.
A planar algebra, V is sub-braided if it contains diagrams andsatisfying a twist relation, R2, and R3 and the naturality condition on some sub-planaralgebra W ⊆ V . Example 3.1.4 (Temperley-Lieb Jones polynomial/Kauffman bracket planar algebra) . TheJones polynomial planar algebra—also called the Temperley-Lieb-Jones (TLJ) planar algebra—is a quotient of the planar algebra of tangles with the following relations:= − ( A + A − ) = − A − = A + A − In the last relation, we will often call the identity while we denote the cupcap. Itcan easily be verified that this planar algebra is braided. It can also be shown that if V is a quotient of the planar algebra of tangles with initial dimensions 1 , , , , V is aspecialization of the TLJ planar algebra. Example 3.1.5 (The Kauffman/Dubrovnik polynomial planar algebra) . The Kauffman/-Dubrovnik polynomial planar algebra is a generalization of the TLJ planar algebra. Theskein relations are ± = z (cid:32) ± (cid:33) = d = a · The + case is called the Kauffman polynomial planar algebra and the − case is called theDubrovnik polynomial planar algebra. By capping off the first equation on the bottom, weget ± = z ± d · Thus, z (1 ± d ) = a − ± a, which gives a two-parameter family of relations in terms of a and d . If one sets a = − A − and d = − ( A − A − ) , note that one does not obtain the TLJplanar algebra. At these values of the parameters, the Kauffman polynomial planar algebrais still defined but is degenerate. See [BW89] and [MT90] for more information on this. Thequotient of that planar algebra by the negligible elements of this planar algebra, however, isisomorphic to the TLJ planar algebra.A fact that is proven in [MPS11] and that we will employ in Section 4 is recorded here: Theorem 3.1.6.
Let V be a quotient of the planar algebra of tangles such that dim V = 2 or . Then there exists a relation of the form ± = z (cid:32) ± (cid:33) Proof.
Since dim V = 2 or 3, we know there must be a relation amongst the two crossingsand the Temperley-Lieb diagrams which we can generically write as a · + a · = a · + a · By rotating the equation however, we see that a = ± a and a = ± a . If a = a = 0, thenthe cupcap is a multiple of the identity, which implies that dim V = 1 . Thus, a (cid:54) = 0, andso we can rewrite the equation as ± = a a · (cid:32) ± (cid:33) Setting z = a /a gives our desired formula. (cid:3) Planar algebra of unoriented virtual tangles.
Another planar algebra that wewill study is the planar algebra of unoriented virtual tangles. Virtual knot theory was firstdiscussed in [Kau99] while virtual tangles can be found in [SW99]. A virtual tangle is atangle that includes the following diagram as a possible crossing:in addition to the normal crossings. One can think of the planar algebra of tangles asa collection of planar tetravalent graphs, with crossings and boundary points serving asvertices. The inclusion of the virtual crossing merely drops the condition of planarity onthese graphs. The virtual crossing is neither an overcrossing nor an undercrossing, so onecould think of the strands as literally passing through each other. Alternatively, [Bro16]explains that one could imagine this non-planar graph as being embedded on some puncturedsurface in which the given diagram could be drawn without the need for self-intersection.
Definition 3.2.1.
A planar algebra, V , is said to be symmetric if there exists an element of V , , that satisfies the following relations:= = =Virtual Reidemeister I Virtual Reidemeister II Virtual Reidemeister III KEIN THEORIES FOR VIRTUAL TANGLES 15
In addition, the must satisfy the following naturality condition: ...... S = ...... S for any element S ∈ V . We will abbrieviate the virtual Reidemeister relations as vR1, vR2, and vR3, respectively.A classic example of a symmetric planar algebra generated by is Deligne’s O t , whichis also known as the Brauer category. One can find more information about the Brauercategory in [FY89] and [RT90] (Brauer originally defined the algebra in [Bra37].). Example 3.2.2 (Deligne’s O t ) . Deligne’s O t (found in [EGNO15](Section 12.9.3), [Del90],and [Del02]) is a symmetric planar algebra generated by with modulus t ∈ C × .A priori, it is not clear that the above relations are enough to reduce every closed diagramto a multiple of the empty diagram. This fact is proven in [RT90], however. Because O t isgenerated by just a symmetric crossing, we also get the following universal property: Proposition 3.2.3.
Let V be a symmetric planar algebra with modulus t . Then there existsa map O t (cid:44) → V Proof.
Let V be as described above, and X be the element of V satisfying the virtualReidemeister relations and naturality conditions. Then the map (cid:55)→ X is a map ofplanar algebras because X satisfies the virtual Reidemeister relations and V has modulus t. Since this map is clearly injective, there must exist a map O t (cid:44) → V , as desired. (cid:3) Definition 3.2.4.
The planar algebra of unoriented virtual tangles is the planar algebragenerated by andDiagrams are equivalent up to isotopy, a finite number of Reidemeister and virtual Reide-meister II and III moves, inaddition to the following relations:= ± =Virtual Twist Relation Mixed Reidemeister IIINote that a priori we do not require vR1 to be satisfied on the nose. By multiplying thevirtual crossing by itself and then capping, however, we see that vR1 must be satisfied upto a sign. Moreover, we have the following fact: Lemma 3.2.5.
The planar algebra of unoriented virtual tangles has an automorphism gen-erated by (cid:55)→ −
Proof.
Let Φ be the map of planar algebras generated by sending the virtual crossing toits negative. It is clear that this map is a map of planar algebras and that it is injective.Considering the map is also an idempotent, we see that the inverse map is also injective andthus the map is a planar algebra isomorphism. (cid:3)
Thus, any skein theory in which the virtual twist parameter is +1 can be transformed intoa skein theory in which it is − Definition 3.2.6.
A quotient of the planar algebra of unoriented virtual tangles is said togive a fully-flat invariant if the following relation holds:= ± Remark.
A planar algebra is flat if the overcrossing and undercrossing are equal to eachother. A planar algebra is fully-flat if the two crossings are equal and they are both equal tothe virtual crossing up to a sign. Thus, fully-flatness is a stronger condition than flatness.We will see examples of planar algebras that are flat yet not fully-flat (The virtual TLJplanar algebra in Example 3.2.13 at A = ± (cid:54) =The forbidden move implies that the virtual crossing is not natural with the actual crossingin the planar algebra of unoriented virtual tangles. In tensor category terms, [Bro16] explainsthat although there is a unique symmetric monoidal functor from the category of tangles tothe category of symmetric tangles, that functor is not braided. We explain this phenomenonin planar algebra terms in the following theorem: Theorem 3.2.7 ([Bro16]) . Let T virt be the planar algebra of unoriented virtual tangles, P bea symmetric planar algebra, and A be a braided sub-planar algebra of P . Then there existsa canonical map from T virt → P S n actions on elements generated by a virtual crossing. Suppose that V is a quotientof the planar algebra of virtual tangles that is generated by the virtual crossing. Then S n acts on V n by permutation of the endpoints. Two strings crossing is represented by .Moreover, this action forms an S n C -representation. An important example that will comeup in Section 4 is the following: Proposition 3.2.8.
Let V be a quotient of the planar algebra of virtual tangles generatedby . Then the permutation action of S on the diagrams of V forms a 15-dimensional C -representation of S . Moreover, this representation is a direct sum of the trivial represen-tation a 5-dimensional representation, and a 9-dimensional representation.Proof. Any quotient of the planar algebra of virtual tangles generated by the virtual crossinghas that dim V ≤
15. Thus, we obtain the following spanning set:
KEIN THEORIES FOR VIRTUAL TANGLES 17
It is clear the permuting the vertices of these diagrams gives an action of S on V and thatthe resulting representation of S is 15-dimensional. By examining the characters of thisrepresentation, we take the inner product of this representation with each of the irreduciblerepresentations of S . From the first orthogonality theorem, we see that the 15-dimensionalrepresentation is reducible as a direct sum of three irreducible representations: the trivialrepresentation, a 5-dimensional representation, and a 9-dimensional representation corre-sponding to the following Young diagrams:which completes our proof. (cid:3) Knowing how this action decomposes into irreducible representations is extremely helpfulin determine the possible dimensions of a box space. Let V be a quotient of the planaralgebra of virtual tangles that is generated by the virtual crossing and W be a quotient of V . Thus, S n acts on W n by a corresponding quotient representation. Since V and W aresemi-simple, any element appearing in the kernel of a surjective map between them mustbe equal to zero. In other words, the sub-space of relations that hold in W but not V isgiven by the kernel of a surjective map between them. Because V is semi-simple, W is alsoa sub-planar algebra of V . Moreover, the S n action on W must be a sub-representationof the S n action on V . Thus, if we can determine the invariant subspace of an irreduciblecomponent of the S n action on V , then we have found a space of possible relations for somequotient W . If we know how the latter representation decomposes as irreducibles, then weknow what the possible representations for the action on W n are and thus what the possiblebox-space dimensions are. Corollary 3.2.9.
Let V be a quotient of the planar algebra of virtual tangles generated bythe virtual crossing and suppose that dim V = 3 . Then dim V ∈ { , } .Proof. By Proposition 3.2.8, we know that S acts on V and forms a 15-dimensional represen-tation. Thus, S must act on any quotient of V . Moreover, the corresponding representationof S must be a quotient of that 15-dimensional representation. Since we have required thatdim V = 3, this implies that the cupcap, identity, and virtual crossing form a basis for V .By inspection, the trivial representation corresponds to the subspace of V generated by thesum of the 15 elements of V shown above. Thus, if the corresponding quotient representationof some sub-planar algebra did not include the trivial representation, this element would beequal to zero. Capping the resulting equation on the left, though, yields(11) 4 · + ( d + 3) + ( d + 2) = 0 Thus, since for all d ∈ C the cupcap, identity, and virtual crossing cannot simultaneouslyvanish, they must be linearly dependent, a contradiction. Thus, any quotient of V withdim V = 3 must have dim V ∈ { , , , } . When dim V = 1 this implies that everydiagram is a multiple of the identity, which implies that dim V = 1, a contradiction.Suppose that dim V = 6. Since the cupcap and identity are linearly independent, thisimplies a relation of the following form: = a · + a · + a · + a · + a · + a · By capping this off on the left, we obtain the equation: d · = ( da + a + a ) · + ( da + a + a ) · Since d (cid:54) = 0, this implies dim V <
3, a contradiction. Thus, dim V ∈ { , } , as desired. (cid:3) We can also imagine elements of S n as diagrams in this planar algebra: Definition 3.2.10.
Let σ be an element of S n . Then define ...... σ to be the diagram (in standard form) obtained by connecting the i -th point on the bottomof the diagram to the σ ( i )-th point on the top of the diagram. Any point at which strandsmust cross each other is represented by .As an example of this when σ = (123) for σ ∈ S we have the diagram. Definition 3.2.11.
Let V be the planar algebra of unoriented virtual tangles. We definethe element ∧ m +1 to be ...... ∧ m + 1 := (cid:88) σ ∈ S m +1 sgn( σ )( m + 1)! · ...... σ As the name suggests, this diagram is related to the exterior product, as arbitrary com-position of ∧ m +1 with the virtual crossing is the negative of ∧ m +1 . If we consider theseelements to be maps of representations (see Section 3.3.1 for more information on this), thisimplies that ∧ m +1 ( x ⊗ x ⊗ · · · ⊗ x m +1 ) = sgn( σ ) · ∧ m +1 ( x σ (1) ⊗ x σ (2) ⊗ · · · ⊗ x σ ( m +1) ) , forany permutation of the indices, σ. With this definition in hand, we are now ready to state the following result, which is aconsequence of the first fundamental theorem for orthogonal groups due to [Wey39]. (It isalso a consequence of the second fundamental theorem due to [LZ12]).
Theorem 3.2.12.
Let V be a quotient of the planar algebra of unoriented virtual tangles.Then V ∼ = Rep ( O ( m )) for m, n ∈ Z ifi. = m and KEIN THEORIES FOR VIRTUAL TANGLES 19 ii. ...... ∧ m + 1 = 0 ,where ∧ m +1 is as defined in Definition 3.2.11. Furthermore, these relations generate all otherrelations. Thus, in order to identify a quotient of the planar algebra of unoriented virtual tanglesas Rep( O ( t )) at t ∈ Z > we need only check that the planar algebra has d = t and that ∧ t +1 = 0 . The last statement in this theorem is also quite powerful, as any relation betweenany diagrams in any box space can be obtained from the above two relations.One well-known quotient of the planar algebra of unoriented virtual tangles is the virtualTLJ planar algebra (originally in [Kau99]).
Example 3.2.13 (The virtual TLJ planar algebra planar algebra) . The virtual TLJ planaralgebra is a quotient of the planar algebra of unoriented virtual tangles subject to the virtualReidemeister moves and the following relations:= − ( A + A − ) = − A − = A + A − Note that although this skein theory seems identical to the actual TLJ planar algebra,we now have the virtual crossing as a generator. To see that the planar algebra is evaluable(and thus gives a virtual link invariant), we use the following evaluation algorithm: Forany diagram of V , we start by using the crossing relation to remove any crossings from thediagram. This process leaves the diagram as a sum of diagrams with only virtual crossings.From the note in Example 3.2.2, we see it is then possible to reduce any such diagram to amultiple of the empty diagram.The similarities in the evaluation algorithms of O t and the virtual TLJ planar algebra arenot coincidental. It is well known that the non-virtual TLJ planar algebra is closely relatedto the quantum group SU (2) q . Moreover, by Theorem 3.2.7 we have an inclusion map SU (2) q (cid:44) → O t when t = q + 1 + q − . This relationship is intimately related to the underlying groupsthemselves.3.3.
Quotients of the planar algebra of oriented virtual tangles.Definition 3.3.1.
The planar algebra of oriented virtual tangles is the planar algebra gen-erated byIn addition, two diagrams will be equivalent if one can be obtained from the other via theoriented virtual Reidemeister moves. The action of the oriented planar operad is the usualinsertion action.Here the oriented virtual Reidemeister relations are the relations obtained from the usualvirtual Reidemeister relations with all possible orientations of the strands. Note that the generators of the planar algebra of oriented virtual tangles have two positively oriented end-points and two negatively oreinted endpoints. Thus, the number of positively and negativelyoriented endpoints must always be the same. Two examples of quotients of the planar algebraof oriented virtual tangles are Rep( S , a ) and Deligne’s GL t . Example 3.3.2.
Consider Rep( S , a ), where S is the circle group and a ∈ C × representsthe formula for the braiding below. More information about this planar algebra can be foundin [DGNO10]. It is given by the following relations: = 1 = 1= a · == = The reason this planar algebra includes Rep( S ) is detailed in Section 3.3.1.We leave it to the reader to check that the crossing satisfies the Reidemeister relations.Another example of a quotient of the planar algebra of oriented virtual tangles is Deligne’s GL t , found in [Del02]: Example 3.3.3 (Deligne’s GL t ) . This planar algebra is generated by the oriented symmetriccrossing and has modulus t . Because it is generated by the oriented symmetric crossing, everydiagram must have the same number of positive and negative endpoints. The planar algebra GL at represents the planar algebra GL t equipped with the braiding= a · for some a ∈ C × . Unlike in the unoriented case, one does not obtain one crossing from theother by rotating, so the planar algebra is not necessarily fully-flat. Note that at t = ± GL at is degenerate for all a. At t = 1, its non-degenerate quotient is Rep( GL (1) , a ) (cid:39) Rep( S , a ) . Planar algebras of the category Rep ( G ) . One class of planar algebras that is orientedin general is the one arising from the category Rep( G ) . We have already seen such an examplein Example 3.3.2.
Definition 3.3.4.
Let G be a group. The category Rep( G ) is a tensor category where ⊗ is defined as the tensor product of representations and is the trivial representation of G .Objects are tensor-generated by the irreducible, finite-dimensional representations of G andmorphisms are maps of representations.For more information on tensor categories, refer to [EGNO15]. We define the planaralgebra Rep( G ) as follows: Definition 3.3.5.
Let G be a group and X be a finite dimensional representation of G .Define X + = X and X − = X ∗ . Then the oriented planar algebra Rep(
G, X ) is a collectionof vector spaces Rep(
G, X ) ( z ,...,z i ) := Hom( , X z ⊗ · · · ⊗ X z i ) . KEIN THEORIES FOR VIRTUAL TANGLES 21
The reader might rightly object to our definition, as we are choosing a particular represen-tation. If the representation is faithful and irreducible, however, every irreducible representa-tion will occur as a summand of some tensor power of X and X ∗ . Thus, every representationwill appear as a projection in some box space of V , so we do not “lose” much information byonly considering maps involving the tensor powers of X and X ∗ . When G has a standardrepresentation, we will often omit X and write “Rep( G ).”Since we have chosen to work over C , Schur’s lemma guarantees for any simple objects X i and X j that Hom ( X i , X j ) ∼ = (cid:26) c · id X i i = j and c ∈ C c , we indicate c · id X by an oriented vertical strand and the c − · id X ∗ by a vertical strand with the opposite orientation. If the representation is self-dualthen we omit the orientation of the strands. Assume for brevity that X is self-dual. Thepairing and co-pairing of X and X ∗ can then be represented by a cap and cup, respectively.Thus, it is easy to check that the value of the circle, the map d : C → X ⊗ X → C is simplymultiplication by the dimension of X . The tensor product of maps of representations issimply concatenation in the planar algebra. Also note that because Rep( G, X ) is a symmetric tensor category (i.e. X ⊗ Y ∼ = Y ⊗ X for all X and Y ), we also have a map: X ⊗ X → X ⊗ X given by x ⊗ x (cid:55)→ x ⊗ x . The reader can check that this map is a map of representations.In order to give a presentation of Rep(
G, X ), we need “enough” maps to tensor-generateevery other map. That is, we need a set of maps such that any map from → X ⊗ n can beobtained by arbitrary composition of those maps. For example, Rep( O (2)) is generated by.Now that we have defined the planar algebra Rep( G ), we can prove the following corollaryto Proposition 3.2.3: Corollary 3.3.6.
Let G be a group and X be a faithful, irreducible, self-dual representationwith dim X = t . Then Rep ( G, X ) is a symmetric planar algebra.Proof. Since Rep(
G, X ) has a symmetric crossing that satisfies the naturality conditions ofDefinition 3.2.1, this proposition is a consequence of Proposition 3.2.3. (cid:3)
Examples of shaded planar algebras.Example 3.4.1 (GPA(Γ)) . The planar algebra GPA(Γ), described in Section 2.3.1 is ashaded planar algebra.Recall that for GPA( ∗ n ), GPA( ∗ n ) ± i is isomorphic as a vector space to linear combinationsof Kronecker deltas of loops of length i . If we fix an indexing of the vertices of S , then atwo-box in GPA( ∗ n ) +2 is a complex matrix where the ( i, j )-th entry is the the coefficientof δ ij . Suppose V is a planar algebra generated by a 2-box, P . Recall that a spin modelis a map from V to GPA( ∗ n ). Then giving a spin model for V is the same as specifying acomplex matrix as the image of P . This leads to the following definition: Definition 3.4.2.
Let V be a shaded planar algebra generated by a 2-box, P . Then a graphΓ gives a spin model for V if the image of P in the spin model is the adjacency matrix for Γ. See [Edg19] for more information on what types of graphs can appear for certain planaralgebras.3.4.1.
Group/Subgroup planar algebra.
An important sub-planar algebra of GPA( ∗ n ) is thegroup/subgroup planar algebra. Before we define it we make the following definition basedon the work of [Gup08]: Definition 3.4.3.
Let H ≤ G be finite groups with [ G : H ] = m and coset representatives { g i H } i =1 ...,m . The diagonal action of G on a k -tuple of cosets is defined by g · ( g i H · · · g i k H ) = ( gg i ) H · · · ( gg i k ) H Definition 3.4.4.
Let H ≤ G be finite groups with [ G : H ] = m . Suppose we have GPA( ∗ m )where the even vertices are labelled by coset representatives S = { g i H } i =1 ...,m . Then thegroup-subgroup planar algebra, PA( H ≤ G ) , is the sub-planar algebra of GPA( ∗ m ) that isinvariant under the diagonal action of G. That is, for any linear functional f ∈ V k f ( g i H · · · g i k H ) = f ( g ( g i H · · · g i k H )) = f (( gg i ) H · · · ( gg i k ) H )for all g ∈ G and g i H ∈ S .Because PA( H ≤ G ) is shaded, we will refer to diagrams with 2 i boundary points as i -boxes and PA( H ≤ G ) even as the collection of vector spaces { PA( H ≤ G ) +2 i } i = 0 , , . . . . Theorem 3.4.5 (Theorem 5.12 in [Gup08]) . Given a finite group G , a subgroup H suchthat [ G : H ] = n , and an outer action α of G on the hyperfinite II -factor R , the planaralgebra of the subgroup-subfactor R (cid:111) H ⊂ R (cid:111) G is isomorphic to the G-invariant planarsub-algebra of GPA ( ∗ n ) . Standard translation between subfactors and tensor categories gives us the following corol-lary:
Corollary 3.4.6. PA ( H ≤ G ) even ∼ = Rep ( G, X ) where X = Ind GH ( H ) , the induction of thetrivial representation from H to G. Another way to think about Ind GH ( H ) is that it is the permutation representation of G acting on the set of left cosets of H in G , so we will call it Perm GH for short. Let V begenerated by a 2-box. Suppose we have a spin model for Φ : V →
GPA( ∗ n ) given by agraph Γ. Γ is transitive if every pair of vertices is equivalent under some element of itsautomorphism group. Let G = Aut(Γ) and H be the subgroup of G fixing a special vertex x . Then if Γ is transitive the coset g i H can be thought of as the set of all automorphismssending x to some vertex g i ( x ) of Γ. In this way, we can identify all the vertices of Γ todistinct cosets. Thus, we see that necessarily [ G : H ] = n and that the action of G on theleft cosets of H is transitive. Hence, PA( H ≤ G ) sits inside GPA( ∗ n ). With this in mind,we can prove the following theorem: Theorem 3.4.7.
Let V be a planar algebra generated by a 2-box and Φ :
V →
GPA ( ∗ n ) be aspin model for V . Suppose that the matrix of weights for this map is given by the adjacencymatrix of some graph Γ . Let G = Aut (Γ) , the group of graph automorphisms of Γ , and H bethe subgroup of graph automorphisms fixing a chosen vertex x . If the graph Γ is transitivethen Φ( V ) ⊆ PA ( H ≤ G ) . KEIN THEORIES FOR VIRTUAL TANGLES 23
Proof.
Let V be generated by a single 2-box which we will call X . Suppose Γ is transitiveand gives a map Φ : V →
GPA( ∗ n ). Then by the note above PA( H ≤ G ) ⊆ GPA( ∗ n ) andwe will identify the vertices of Γ with the cosets g i H . To show Φ( V ) ⊆ PA( H ≤ G ) we needonly prove that Φ( X ) ∈ PA( H ≤ G ) . Since X (cid:55)→ (cid:88) i,j δ g i Hg j H , the set of Kronecker deltas ofedges in Γ, we see that the diagonal action of g on X is g (cid:88) i,j δ g i Hg j H = (cid:88) i,j g · δ g i Hg j H = (cid:88) i,j δ ( gg i ) H ( gg j ) H = (cid:88) k,l δ g k Hg l H = (cid:88) i,j δ g i Hg j H Thus Φ( X ) is invariant under the diagonal action, and so Φ( X ) ∈ PA( H ≤ G ) by definition.Since X generates V , Φ( V ) ∈ PA( H ≤ G ) , as desired. (cid:3) Unfortunately, Perm GH is never irreducible when H (cid:54) = G , which means that the box spacesof Rep( G, Perm GH ) can grow very quickly. To make the planar algebra more manageable tostudy, we will consider a cut-down (See Definition 2.3.3) of it instead.When P ∈ V ± i then V is an unshaded planar algebra, while it is still shaded if P ∈ V ± (2 i +1) .In this paper, we will only utilize projections in the 2-box space, and so all the resultingplanar algebras will be unshaded. If V is an unoriented planar algebra generated by a 2-boxwith a spin model given by a transitive graph Γ, then we know that V is a sub-planar algebraof PA( H ≤ G ) , which is isomorphic to Rep( G, Perm GH ). If we cut down V by an appropriateprojection in the 2-box space, however, the even portion of the cut-down is then isomorphicto Rep( G, X ) , where X is some summand of Perm GH .Let Γ be a transitive graph, G = Aut(Γ) , and H be the automorphism group fixing achosen point. Then in this case we can describe the decomposition of Perm GH into irreduciblerepresentations. This exercise is equivalent to finding the projections in the 2-box space ofPA( H ≤ G ). Recall from the definition of PA( H ≤ G ) that every element of the 2-box spaceis of the form (cid:88) g ∈ G g · δ Hg i H where the g i are specified coset representatives. Thus, δ Hg i H appears as a summand in everyelement of the 2-box space, and we can think of describing the action of G on the 2-boxspace as sending Hg i H (cid:55)→ Hg j H. Further, we can think of these Hg i H as representing therelationship between every point of Γ and our chosen fixed point (i.e. adjacent, non-adjacent,or equal), by identifying x with the left coset 1 · H = H . Since G is the automorphismgroup of Γ, every element of G must preserve the above relationships. Thus for all g ∈ G,g · ( HH ) = HH , and for g i (cid:54) = 1 , g · ( Hg i H ) = Hg j H, where g j H is adjacent to H if andonly if g i H is. Thus, in this way we see that there are at most three invariant subspaces ofthe 2-box space of PA( H ≤ G ) . Moreover, by the transitivity of Γ these invariant subspacescannot contain any non-trivial, proper invariant subspaces.Suppose Γ is a non-complete graph with at least one edge. Translating the above infor-mation to
Rep ( G, Perm GH ) , using Corollary 3.4.6, we know when Γ is transitive thatPerm GH = G ⊕ X k ⊕ X n − k − with dim X k = k and dim X n − k − = n − k − k being the number of neighbors ofour chosen vertex. The number of non-neighbors is thus n − k − . Note that transitivityimplies that Γ is regular, and hence the above decomposition is independent of the choiceof vertex. In the case where Γ is the unique graph with 0 or 1 vertex, G = H = { } , so Perm GH = G . When Γ is a complete or empty graph of at least two vertices, we note thatPerm GH = Perm S n S n − = S n ⊕ X n − , where X n − is the standard representation for S n . Proposition 3.4.8.
Suppose V is a planar algebra that has a spin model given by a transitivegraph, Γ . Let G = Aut (Γ) and H be the automorphism group fixing a chosen point. Further,assume that V even is braided. Then Rep ( G, X ) takes a map from the planar algebra of unori-ented virtual tangles, where X is an irreducible summand of the permutation representationof G induced from H appearing exactly once.Proof. Suppose a planar algebra, V , has a spin model given by the adjacency matrix ofsome graph Γ and suppose that Γ is transitive. Then we know from Corollary 3.4.7 thatthe even portion of this planar algebra is a sub-planar algebra of PA( H ≤ G ) , where G =Aut(Γ) and H is the subgroup of G fixing some vertex x. By Corollary 3.4.6 we know thatPA( H ≤ G ) (cid:39) Rep( G, Perm GH ). Because Γ is a transitive graph, Perm is the sum of at mostthree distinct representations, including G . Let X be an irreducible summand of Perm GH appearing exactly once. Since the characters of Perm GH are integral, Perm GH is necessarilyself-dual. Moreover, since this irreducible representation of Perm GH appears exactly once, weknow it is also self-dual.Let W be the cut-down of V by the appropriate projection of V +2 such that W even (cid:39) Rep(
G, X ). It is clear that V even being braided implies that W even is braided also. Moreover,because X is self-dual, Rep( G, X ) is an unoriented planar algebra, and thus a symmetricplanar algebra by Corollary 3.3.6. Since W even is braided, we know that Rep( G, X ) has asub-braiding. Thus by Theorem 3.2.7, Rep(
G, X ) must take a map from the planar algebraof unoriented virtual tangles, as desired. (cid:3)
Kuperberg [Kup97] reformulates the classification of spin models for the Kauffman poly-nomial [Jae95] in terms of the combinatorial B spider. In particular, he proved that Γ gavea spin model for the B spider if and only if it gave a spin model for a Kauffman polyno-mial planar algebra. The B spider is generated by two types of strands, which are denotedby single and doubled strands. If one takes the sub-planar algebra generated by only thedoubled strands, she would obtain a Kauffman polynomial planar algebra, which is clearlybraided. Moreover, it is the cut-down of the Kauffman polynomial planar algebra in [Jae95]with spin model given by Γ. Since every graph that gives a spin model for the Kauffmanpolynomial is necessarily transitive, the even portion of this sub-planar algebra of the B spider is a sub-planar algebra of Rep(Aut(Γ) , X ) , where X is now some irreducible repre-sentation given by the cut-down. As any Kauffman polynomial planar algebra is braided,Rep(Aut(Γ) , X ) must take a map from the planar algebra of unoriented virtual tangles byProposition 3.4.8. Two examples of this are given below: Example 3.4.9 (The pentagon spin model for the Kauffman polynomial) . The pentagongives a spin model for the B spider when the value of Q in [Kup97] is equal to e iπ/ .The pentagon has automorphism group D , the dihedral group of 10 elements, so the evenportion of the sub-planar algebra of the B spider given by the doubled strands at thisvalue of Q is a sub-planar algebra of Rep( D , X ) , where X is one of the two 2-dimensionalrepresentations of D . In this case, the choice of X is irrelevant as they both appear assummands of Perm GH . Moreover, the dimensions of Rep( D , X ) and B even2 show that the twoare actually isomorphic. By Corollary 3.3.6 and the fact that Rep( D ) is non-degenerate, KEIN THEORIES FOR VIRTUAL TANGLES 25 we know that there exists the following map:Rep( O (2)) (cid:44) → Rep( D , X )We note here that if H (cid:44) → G then Rep( G ) (cid:44) → Rep( H ) . Because, B even2 is braided, weknow by Proposition 3.4.8 that Rep( D ) is a quotient of the planar algebra of unorientedvirtual tangles. Thus, we can give a skein-theoretic description of Rep( D ) by determiningwhich quotient it corresponds to. As we will see in Section 4, Rep( D ) is isomorphicto Rep( O (2) , e iπ/ ) , the quotient of the planar algebra of unoriented virtual tangles withdim V = dim V = 1 and dim V = 3, subject to the following relations:= Q − − Q · + Q − − Q · + 1 + √ · = 2where Q = e iπ/ . Example 3.4.10 (The Higman-Sims spin model for the Kauffman polynomial) . Now con-sider the specialization of the B spider with spin model given by the Higman-Sims graph,which occurs at Q = τ , the square of the golden ratio. The automorphism group of thisgraph is called HS. . Because the Higman-Sims graph is transitive, we know that the imageof the even portion of the sub-planar algebra of the B spider generated by the doubled strandat Q = τ is contained in Rep( HS. , X k ), where the summand of Perm GH is 22-dimensional.Like with the pentagon we obtain a mapRep( O (22)) (cid:44) → Rep(
HS. , X k )Unlike with the pentagon, however, the even portion of the sub-planar algebra of B gen-erated by the doubled strand is not isomorphic to Rep( HS. , X k ) , which can again be seenby comparing the dimensions of the planar algebras. In particular, Rep( HS.
2) contains anadditional 5-box. As such, a skein theoretic description of Rep(
HS.
2) is not possible usingthe same methods as in the case of the pentagon. For more spin models arising from theKaufman polynomial planar algebra, please see [Edg19].Since B even2 is braided, we know Rep( HS.
2) takes a map from the planar algebra of unori-ented virtual tangles by Proposition 3.4.8. By inspection, the image of this planar algebrahas a 4-dimensional 4-box space. Further, no skein-theoretic description of this sub-planaralgebra has yet been given. The dimension bounds in Section 4 were chosen to include allquotients of the planar algebra of unoriented virtual tangles with 4-box space strictly smallerthan Rep(
HS. V = 4, so thatwe can include this sub-planar algebra of Rep( HS.
Examples of symmetric trivalent planar algebras.
For all previous planar alge-bras that have been presented, it has been true that the odd-box spaces have been trivial.Now suppose that we consider the unshaded planar algebra generated by a trivalent vertexwhich is rotationally invariant. All non-degenerate quotients of the planar algebra of planar trivalent graphs with initial box space dimension bounds 1 , , , , , ,
41 were classified in [MPS17], which the authors call a trivalent category , as these planar algebras arise fromtensor categories in the same manner as the planar algebra Rep( G ) does. Theorem 3.5.1 ([MPS17]) . The following table is an exhaustive list of non-degenerate triva-lent planar algebras with initial box space dimension bounds , , , , , , :Dimension bounds Name , , , , , . . . SO (3) ζ , , , , , . . . SO (3) q or OSp (1 | , , , , , , . . . ABA , , , , , , . . . ( G ) ζ , , , , , , . . . ( G ) q , , , , , , , . . . H where ABA is a sub-planar algebra of the free product of
T L ( √ dt − ) ∗ T L ( t ) and H is thefusion category found in [GS12] . If we also include as a generator and impose the virtual Reidemeister moves, we canreally think of this planar algebra as being the set of graphs with only trivalent and univalentvertices. In this case, the virtual crossing would simply represent two edges intersecting. Thisleads us to our next definition:
Definition 3.5.2.
The planar algebra of trivalent graphs is a collection of vector spaces V = { V , V , . . . } where V n consists of simply-laced graphs (i.e. no multiple edges) with n numbered univalent vertices and some number of trivalent vertices. Two graphs are consid-ered equivalent if there exists a graph isomorphism between them that respects the numberingof the univalent vertices.Note that the graph isomorphism property implies that the planar algebra is symmetric.We will study quotients of this planar algebra with dimension bounds 1 , , , , , , . Definition 3.5.3.
A symmetric trivalent planar algebra is a quotient of the planar algebraof trivalent graphs with dim V = 0 and dim V = dim V = dim V = 1 that satisfies thevirtual Reidemeister moves and the following additional relations:= d = t − c · = c · = = Remark.
In [MPS17], a trivalent planar algebra is generated by only the trivalent vertex. Asymmetric trivalent planar algebra, though, is generated by both the trivalent vertex and thesymmetric crossing. Thus, while every symmetric planar algebra appearing in [MPS17] is asymmetric trivalent planar algebra, not every symmetric trivalent planar algebra appears in[MPS17].
KEIN THEORIES FOR VIRTUAL TANGLES 27
We will first classify all symmetric trivalent planar algebras and then classify all theirsub-braidings. Some examples of planar algebras that do satisfy this property are givenpresently:
Example 3.5.4 (Rep( SO (3))) . Here the chosen faithful, irreducible representation is thestandard representation of SO (3). Thus, d = 3. It has initial box space dimensions 1 , , , , − ·− = − The latter relation is usually referred to as an “I=H” relation. The fact that the first relationis actually rotationally invariant is implied by the I=H relation.
Example 3.5.5 (Rep(
OSp (1 | . In this case the irreducible representation is the standardrepresentation of
OSp (1 | − − − · − + 2 ·− = − (cid:34) − (cid:35) Again, the fact that the formula for the virtual crossing is actually rotationally invariant isimplied by the given I=H relation.
Example 3.5.6 (Rep( G )) . In this case the representation is the standard, 7-dimensionalrepresentation of the compact real Lie group G . Thus, the circle parameter is 7. It does nothave an I=H relation but does have the following relation for the virtual crossing:= 12 (cid:34) + (cid:35) − (cid:34) + (cid:35) Example 3.5.7 (Rep( S )) . This planar algebra is generated by the faithful two-dimensionalrepresentation of S , the permutation group on 3 letters. Thus, it has circle parameter 2 andthe following relations: = + − = − Note that while the planar algebra Rep( SO (3)) and Rep( S ) have the same I=H relation,they have different formulas for the virtual crossing, and so they are non-isomorphic planaralgebras. The previous example gives rise to a general class of planar algebras called Deligne’s S t found in [EGNO15] (Section 9.12.1) and originally in [Del07]. When t is a non-negativeinteger, S t is degenerate and has Rep( S t ) as its only non-degenerate quotient, which is provenin the following theorem from [CO11] and [Del07]: Theorem 3.5.8 ([CO11] and [Del07]) . At generic values of t ∈ C , the planar algebra S t has initial box space dimensions , , , , , , and is non-degenerate. When t is a non-negative integer, S t with the aforementioned dimensions exists but is a degenerate planaralgebra. The only non-degenerate quotient of that planar algebra is isomorphic to the planaralgebra Rep ( S t ) with n -box space dimensions equal to dim Hom ( , X ⊗ n ) , where X is thestandard t − dimensional representation of S t . We should note here that at t = 0 , the non-degenerate quotient of S is isomorphic toRep( OSp (1 | . The definition given below of Deligne’s S t from [EGNO15]Section 9.12.1using the standard representation and not the permutation representation (See [DO14] formore information). This is why the circle value is t − t . Example 3.5.9. (Deligne’s S t ) Deligne’s S t is a symmetric trivalent planar algebras subjectto the additional I=H relation − = 1 t − (cid:32) − (cid:33) It turns out that this planar algebra has an interesting sub-braiding on it given by= ( q − · + q − − ( q + q − )with t = q + 2 + q − . We will prove that this is a sub-braiding and classify all such sub-braidings of symmetric trivalent planar algebras in Section 6. Those familiar with the quan-tum group SO (3) q –also called the second-colored TLJ planar algebra (See [CFS95] and[KL94])–will recognize this formula as the standard formula for the braiding on that planaralgebra. The fact that it appears as a braiding for S t arises from the fact that there is existsa map SO (3) q (cid:44) → S t when t = q + 2 + q − . Given this relationship, S t is often called the virtual second-coloredTLJ planar algebra.4. Classification of simple virtual skein theories for unoriented virtualtangles
In this section, we would like to classify certain quotients of the planar algebra of unori-ented virtual tangles, described in Section 3.2. Let V := { V i } i =0 , ... be such a quotient. Then V inherits the relations from the planar algebra of unoriented virtual tangles. That is, V must satisfy the Reidemeister relations and their virtual counterparts.Moreover, we will require that dim V = dim V = 1 and that dim V ≤
3. This impliesthat V has the following relations:= d = a · =circle parameter twist relation virtual Reidemeister I KEIN THEORIES FOR VIRTUAL TANGLES 29 where d, a ∈ C × . Given that dim V ≤
3, then there must be at least one relation amongNotice that Examples 3.2.2 and 3.2.13 are examples of such a quotient while a notable non-example is Example 3.4.10. Our goal for this section will be to prove the following theorem:
Theorem 4.1.1.
Let V be a non-fully-flat quotient of the planar algebra of unoriented virtualtangles such that dim V = dim V = 1 and dim V ≤ . Then V is isomorphic to one of thefollowing:i. The virtual TLJ planar algebraii. The Kauffman polynomial planar algebra at a = ± and d = − equipped with thevirtual crossing: = − (cid:32) + (cid:33) iii. Rep ( O (2) , a ) , where a ∈ C × denotes that the planar algebra is equipped with a braidinggiven by the formula (12) = a − − a − a − − a a − + a Lemma 4.1.2.
Let V be a quotient of the planar algebra of unoriented virtual tangles. Ifthe cupcap and identity are linearly dependent in V , then the planar algebra is fully-flat.Proof. Suppose that(13) = z · By multiplying both sides of the equation by the virtual crossing, we obtain a relationrelating the virtual crossing to a multiple of the cupcap (after using vR1). Performing asimilar operation with the actual crossing, we obtain a similar relation with the crossing.This implies that the actual and virtual crossings are multiples of each other, which impliesthat the planar algebra is fully-flat, as desired. (cid:3)
Lemma 4.1.3.
Let V be a quotient of the planar algebra of unoriented virtual tangles inwhich the cupcap and the identity are linearly independent. If there is a relation between thevirtual crossing, cupcap, and identity, it must be of the form (14) = − (cid:32) + (cid:33) Moreover, in this instance the circle parameter is − .Proof. Because the virtual crossing is rotationally invariant, any relation must be of the form(15) = z (cid:32) + (cid:33) where z ∈ C × . Squaring both sides of (15) and applying vR2 to the left-hand side gives theequation = z · + z · (2 + d ) · Since the identity and the cupcap were assumed to be linearly independent, this tells us that z = 1 and that z · (2 + d ) = 0. Since z = 1 (cid:54) = 0 , it must be the case that d = −
2. Usingthis value of d and capping off (15) at the top, we see that z = − , as desired. (cid:3) The above two lemmas allow us to prove the following result:
Proposition 4.1.4.
All non-fully-flat quotients of the planar algebra of unoriented virtualtangles with dim V less than or equal to such that there is a linear dependence amongthe virtual crossing, cupcap and the identity are isomorphic to either the virtual TLJ planaralgebra at A = ± or the Kauffman polynomial planar algebra at d = − and a = ± equipped with a virtual crossing given by the formula = − (cid:32) + (cid:33) Proof.
Let V be such a quotient. Then Lemma 4.1.2 tells us that the cupcap and identitymust be linearly independent since V is not fullly-flat. By Lemma 4.1.3, if there is a relationamong the identity, cupcap, virtual crossing, it must be (14) and the circle parameter mustbe − V = 2 then the cupcap and identity form a basis for V . This implies a relation of the form= ± (cid:32) + (cid:33) which implies that V is fully-flat. Thus dim V (cid:54) = 2 . Suppose dim V = 3 . Then by Theorem 3.1.6 we have a relation of the form(16) ± = z · (cid:34) ± (cid:35) Capping (16) on the top tells us that a − − a = z in the − case and that a − − a = − z in the+ case. Since V is a quotient of the planar algebra of unoriented virtual tangles, we knowthat it must satisfy the mixed Reidemeister III relation(17) =By expanding the virtual crossing on both sides of this formula using (14), we obtain thefollowing relation:(18) − + − + − = a · " − KEIN THEORIES FOR VIRTUAL TANGLES 31
By solving the mixed Reidemeister move with the other crossing, one obtains a similarrelation:(19) − + − + − = a − · " − If we rotate and rearrange (18), however, we obtain the negative of the left-hand side of (19).This implies that a = a − or that a = ±
1. In the − case, z = 0 and the two crossings areequal. Thus, the actual crossing is symmetric or anti-symmetric. A quick calculation showsthat in this skein theory, the forbidden move is actually satisfied. Because the forbiddenmove is satisfied, this planar algebra is isomorphic to the virtual TLJ planar algebra at A = ± (cid:55)→ and (cid:55)→ . In the + case, we have that d = − . We obtain the relation z = ∓ . This gives thefollowing relation: + = ± (cid:32) + ‘ (cid:33) This planar algebra is isomorphic to the Kauffman polynomial planar algebra at a = ± d = −
2. It is degenerate, however, as ± (cid:32) + ‘ (cid:33) is in the null space of the inner-product matrix of V .Thus, the only non-fully flat quotient of the planar algebra of unoriented virtual tangleswith a dependence between the virtual crossing, cupcap, and the identity is the virtual TLJplanar algebra at A = ± d = − a = ± , as desired. (cid:3) If there is a relation between the virtual crossing, identity, and cupcap, then we knowthat the associated planar algebra is either a specialization of the virtual TLJ planar algebraat A = ± a = ± d = − A = ± Lemma 4.1.5.
Suppose that V is a quotient of the planar algebra of unoriented virtualtangles with initial box space dimensions , , , , . Then if V has a relation of the form = x · + y · then V is isomorphic to the virtual TLJ planar algebra.Proof. By rotating the above equation and squaring we see that y = x − and that thecircle parameter d = − ( x + x − ) . By capping off the above equation we see that the twistparameter a = − x − . Clearly, these are the exact skein relations of the virtual TLJ planaralgebra described in Example 3.2.13, and so the two planar algebras must be isomorphic. (cid:3)
Proposition 4.1.6.
Let V be any quotient of the planar algebra of unoriented virtual tangleswith initial box space dimensions , , , , with the following relation (20) = x · − x · + z · where x and z are not simultaneously 0. Then dim V ∈ { , } . Furthermore, when dim V = 15 , the planar algebra is fully-flat. When dim V = 10 , we have that ∧ = 0 . Proof.
Let V be as described above. By multiplying (20) with its 1-click rotation we obtain,we obtain the equation = ( z − x ) · + (2 x − dx ) · Because the cupcap and the identity are linearly independent by assumption, we see that z − x = 1 and x (2 − d ) = 0. Thus, either x = 0 or d = 2. If x = 0, then z = ± V is fully-flat.Suppose now that d = 2 and let us consider V . Since dim V = 3, we know that anybasis of V can be written using only the virtual crossing, the cupcap, and the identity.In addition, vR3 tells us that both 6-box diagrams with three virtual crossings are equal.Hence, we obtain the spanning set listed in Proposition 3.2.8. Suppose that this spanningset were a basis. This would imply that dim V = 15. By expanding R3 using (20) we obtain = x + (cid:2) − x − xz (cid:3) + (cid:2) − x + xz (cid:3) + x + (cid:2) x − x z (cid:3) + x z − x z − x z + 2 x z − x z − xz − xz + xz + z = (cid:2) − x − xz (cid:3) + (cid:2) − x − xz (cid:3) + (cid:2) x + xz (cid:3) + (cid:2) − x − x z (cid:3) − x − x z − x z +2 x z − x z − x z − xz − xz + xz + z Since these diagrams are all linearly independent, setting the right-hand sides of both equa-tions above equal to each other gives a system of 15 equations. By inspection, the only
KEIN THEORIES FOR VIRTUAL TANGLES 33 solutions for ( x, z ) are { (0 , , (0 , − , (0 , } . Since x and z are not simultaneously 0, then x = 0 and z = ± V is fully-flat.Suppose that the spanning set is not a basis. By Corollary 3.2.9, we know that thedimension of V is 10 as dim V = 3. In this case, the invariant subspace of the 5-dimensionalirreducible representation of S gives us new relations in V . By inspection, that subspacegives us the following rotational eigenvector (see Definition 2.1.11):(21) = − + − + − + By multiplying both sides of (21) by certain 6-boxes, we can obtain relations simplifying thefollowing diagrams to terms with fewer crossings:In particular, we obtain the relation " − − + + + − = 0Since the left-hand side is ∧ by Definition 3.2.11, we see that ∧ = 0 in this case. Thiscompletes our proof. (cid:3) We note here that the results of Proposition 4.1.6 followed from the work done in [CH15],but we give a quicker, diagrammatic proof of the result in our case. Nevertheless, thisproposition tells us that if we have a planar algebra with the above crossing formula, it iseither fully-flat or has the special property that ∧ = 0 . This last fact will be key in provingthe following proposition:
Proposition 4.1.7.
Let V be a quotient of the planar algebra of unoriented virtual tangleswith dim V = 3 subject to the following relations: = 2 , = a (22) = a − − a − a − − a a − + a Then V is non-degenerate and is isomorphic to Rep ( O (2) , a ) , where a denotes the abovebraiding formula. The proof of this proposition will appear in the next section. In this particular instancethis family of possible skein theories has another strange property. The forbidden move (seeSection 3.2 for the definition) is satisfied in this skein theory if and only if R2 and R3 are, which can be seen by using the so-called “Kauffman trick:” = 2 a − + a − a − − a a − + a ) + a − − a a − + a )= 2 a − + a − a − − a a − + a ) + a − − a a − + a )= Proof of Proposition 4.1.7 and Theorem 4.1.1.
In order to show that this planaralgebra is isomorphic to Rep( O (2) , a ), we will make an identification of the unoriented di-agrams in this family of skein theories with oriented ones (Thanks to Pavel Etingof for hissuggestion that led us to this trick.). Proposition 4.2.1.
Rep ( S , a ) in Example 3.3.2 contains Rep ( O (2)) as a sub-planar algebra.Proof. Let V be Rep( O (2)) and f : V →
Rep( S , a ) be the map generated by (cid:55)→ +Better put, f is the map that sends any unoriented diagram to the sum of all possibleorientations of the strands. For example, the overcrossing would map to (cid:55)→ + + +We now show that this map is a map of planar algebras. In order to do this, we check thatall relations in V are preserved under f . Since by Theorem 3.2.12 the circle parameter andthe ∧ = 0 relation generate all the others, we need only check those relations still hold inthe image of f . The circle relation is clearly preserved under f . By using a computer to aidin the simplification process (See wedge3 . nb for the details of this computation), it is easyto see that the image of(23) " − − + + + − is 0 under f . By looking at the image of the identity, cupcap, and virtual crossing, it is clearthat f is an injective map. Thus, f ( V ) is isomorphic to Rep( O (2)) by Theorem 3.2.12, asdesired. (cid:3) Thus, we see that Rep( S , a ) contains Rep( O (2)) as a sub-planar algebra. This relationshipis rather surprising. Recall that O (2) ∼ = S (cid:111) α Z / Z , where α is the automorphism that sends e iθ to e − iθ . Thus, any quotient of S that is invariant under switching the direction of thearrows must contain O (2), which explains this exotic braiding. Proof of Proposition 4.1.7.
Let V be the planar algebra in Proposition 4.1.7 and a ∈ C × . Let f be as in Proposition 4.2.1, in which it was shown that f is an injective map whose image isisomorphic to Rep( O (2)). In order to show that this image is isomorphic to Rep( O (2) , a ), we KEIN THEORIES FOR VIRTUAL TANGLES 35 also need to show that the formula for the crossing is preserved under f . Using the relationson these diagrams from the definition of Rep( S , a ), we note that (cid:55)→ + + += a − + a + a − + a = a − (cid:34) + (cid:35) + a (cid:34) + (cid:35) (24)Simplifying the right hand side gives a − − a − a − − a a − + a (cid:55)→ a − · (cid:34) (cid:35) + a · (cid:34) (cid:35) = a − · (cid:34) + (cid:35) + a · (cid:34) + (cid:35) (25)Hence, (24) and (25) are equal and so the formula for the crossing is preserved under f .Thus, since f is injective V is isomorphic to a subplanar algebra of Rep( S , a ). ByProposition 4.2.1 and the preservation of the crossing formula under f , we know that V ∼ = Rep( O (2) , a ) , as desired. (cid:3) Now that we have shown that Rep( O (2) , a ) is a sub-quotient of the planar algebra ofunoriented virtual tangles, we can easily prove the following theorem: Theorem 4.2.2.
Let V with dim V = 3 be a non-fully flat quotient of the planar algebraof unoriented virtual tangles such that the virtual crossing, identity, and cupcap are linearlyindependent. Then V is isomorphic to one of the following:i. The virtual TLJ planar algebraii. Rep ( O (2) , a ) .Proof. Assume that the virtual crossing, the cupcap, and the identity are linearly indepen-dent and dim V = 3. Since these diagrams now form a basis, we know that there must be arelation of the form(26) = x + y + z If x = y = 0 then V is fully-flat. If z = 0 then by Lemma 4.1.5, we know that our quotientis isomorphic to the virtual TLJ planar algebra. Thus, let us assume that z (cid:54) = 0 and thatone of x or y is non-zero. By rotating (26) and multiplying the two equations together weobtain the following equation:= ( xy + z ) · + ( x + y + dxy + yz + xz ) · + ( xz + yz ) · Since we assumed that these were linearly independent, we have the following equations: xy + z = 1 xz + yz = 0 x + y + dxy + yz + xz = 0Since z (cid:54) = 0, we know that x = − y from the second equation. From the first equation,we know that z − x = 1. Additionally, the last equation tells us that x + x − dx = x · (2 − d ) = 0 . Since x = 0 implies y = 0, we see that d = 2.Using this information, we now have the simpler equation(27) = x − x + z with z = 1 + x . By capping (27) on the top, we see that − x + z = a − , and by capping (27)on the left we obtain the relation x + z = a. This, gives a one parameter family of possibleskein theories in terms of a where (27) can be re-parameterized to(28) = a − − a − a − − a a − + a ∧ = 0 in V . Thus, by Proposition 4.1.7 V is isomorphicto Rep( O (2) , a ) . As we have exhausted all possible relations, we know that V must beisomorphic to the virtual TLJ planar algebra or Rep( O (2) , a ) , as desired. (cid:3) We can now prove the results of Theorem 4.1.1:
Proof of Theorem 4.1.1.
Let V be a non-fully-flat quotient of the planar algebra of unorientedvirtual tangles with dim V = dim V = 1 and dim V ≤
3. Then Proposition 4.1.2 statesthat the cupcap and identity must be linearly independent. If the cupcap and identity arelinearly independent but the virtual crossing, identity and cupcap are linearly dependent,Corollary 4.1.4 tells us that V is isomorphic to the virtual TLJ planar algebra at A = ± d = − a = ± x + y + z as we chose dim V ≤ . Thus, Theorem 4.2.2 tells us that the planar algebra is either thevirtual TLJ planar algebra or Rep( O (2) , a ) . Since we have exhausted all possibilities, we seethat, up to isomorphism, V must be the Kauffman polynomial planar algebra at d = − a = ±
1, the virtual TLJ planar algebra, or or Rep( O (2) , a ) , which completes our proof. (cid:3) Link invariants.
Since all of the above planar algebras are evaluable, they also providevirtual link invariants. The Kauffman polynomial planar algebra and virtual TLJ planaralgebra are well-understood and the invariants they give can be found in [Kau90] and [Kau99].The Rep( O , a ) case could potentially give an interesting invariant. In fact, though, theinvariant is rather boring after we normalize away the writhe: Proposition 4.3.1.
Let K be a virtual knot and wr ( K ) be the writhe of some fixed ori-entation of K . Then the knot invariant given by Rep ( O (2) , a ) assigns a wr ( K ) + a − wr ( K ) to K . KEIN THEORIES FOR VIRTUAL TANGLES 37
Proof.
Let K be a virtual knot. Thus it is an element of V of Rep( O (2) , a ) . Since Rep( O (2) , a )is isomorphic to a subplanar algebra of Rep( S , a ), K can be thought of as the sum over allpossible orientations of it. For knots, note that there are only two possible orientations onthe strands. Moreover, the writhes of these knots are negatives of each other.Fix an orientation on K and let wr( K ) be its writhe. Then by using the relations ofRep( S , a ) , we perform the following algorithm. First, we resolve all virtual crossings byusing the relation =Note that this does not involve any scalars. Next, we resolve all actual crossings by usingthe relations = a · = a − · The resulting diagram is some number of unknots multiplied by a wr( K ) . Since the circleparameters are 1 in Rep( S , a ) , this diagram is equivalent to a wr( K ) times the empty diagram.If we were to take the other orientation, we would obtain a − wr( K ) times the empty diagram.Thus the knot K is equivalent to a wr( K ) + a − wr( K ) , as desired. (cid:3) After normalizing away the writhe, we see that the invariant is trivial for all virtual knots.5.
Classification of small quotients of the planar algebra of orientedvirtual tangles
We will now classify all spherical quotients of the planar algebra of oriented virtual tangles, V , such that dim V = 1 , the dimension of all vector spaces with 2 endpoints is 1, and thedimension of all vector spaces with 4 endpoints is 1 or 2. Like for the unoriented case, theseconditions immediately imply some relations. The fact that V and any vector space withtwo endpoints are one-dimensional implies the following two sets of relations= d = d = a · = a · Circle parameters Twist RelationsFor an arbitrary planar algebra, these parameters can all be different. Since we have assumedthe quotient is spherical, however, this implies that d = d and a = a . Thus, we will usethe letters d and a to represent them. GL at and Rep( S , a ) from Examples 3.3.2 and 3.3.3are examples of such a quotient. Theorem 5.1.1.
Let V be a spherical quotient of the planar algebra of oriented virtualtangles with dim V = 1 , the dimension of all non-empty vector spaces with 2 endpoints is1, and the dimension of all non-empty vector spaces with 4 endpoints is 1 or 2. Then V isisomorphic to one of the following:i. Deligne’s GL at where = a · ii. The non-degenerate quotient of GL a ± , where the virtual crossing is equal to plus orminus the identity. Proof.
Let V be described as above and fix an orientation on the vertices. Without loss ofgenerality assume that orientation is (+ , + , − , − ) . Now suppose that there exists a relationbetween the virtual crossing and the identity. Because vR2 is satisfied, this implies that(30) = ± By capping off on the left, we see that d = ± z · By capping off on the left, we see that z = ± a depending on whether d = ± . It is trivialto check that this formula satisfies the remaining Reidemeister relations. Since these areexactly the skein relations of the non-degenerate quotient of GL a ± , any spherical quotient ofthe planar algebra of oriented virtual tangles with a dependence between the virtual crossingand the identity must be isomorphic to one of these planar algebras.Now, suppose that the identity and the virtual crossing are linearly independent. Thus,we must have the following relations:(31) = x · + z · (32) = x · + z · By capping off (31) and (32) above on the left we obtain the equations a = dx + z and a = dx + z . Since a (cid:54) = 0, we can always reparameterize the x i and z i for any value of d, sowithout loss of generality assume that a = 1. Thus we have 1 = dx + z and 1 = dx + z . By multiplying (31) and (32) together, we see that x x + z z = 1 and x z + x z = 0 . If we expand R3 with the above formula, we see that x = 0 or z = 0 by the independenceof the identity and virtual crossing. Solving this system of equations gives us 2 possible cases:The case where x = x = 0 and z = z = 1 or the case where d = ± . By mR3, the lattercase would imply a relation between the virtual crossing and the identity, a contradictionto their independence. The former case is exactly the skein theory of GL t . Rememberingthat we made a choice of parameterization for a , we see that when the identity and virtualcrossing are independent we obtain GL at , where a represents the case where the crossing isequal to a times the virtual crossing. This completes our proof. (cid:3) Classification of symmetric trivalent categories
In [MPS17], a classification of trivalent planar algebras, V , with dim V ≤ V ≤
4. Someof the relations that will explicitly appear later are listed below:= d = t − c · = c · circle parameter lollipop relation bigon relation triangle relation KEIN THEORIES FOR VIRTUAL TANGLES 39
We note that there is a choice of normalization that allows us to rescale c appropriately tomake c any non-zero constant. Thus, we will set c = 1.In order to use the classification of [MPS17], we will also require that our planar algebrabe nondegenerate. There are some degenerate symmetric trivalent planar algebras that fallwithin our dimension bounds. For example, S t is degenerate by Theorem 3.5.8 at every t ∈ Z ≥ . Instead, we will consider its nondegenerate quotient at those values of t , which byTheorem 3.5.8 is Rep( S t ) . Classification of symmetric trivalent planar algebras.
Since we have imposeddim V ≤
4, it is necessarily true that we have a relation of the form(33) a + a · + a · + a · + a · = 0with at least two of the a i ∈ C not 0. We have already seen many examples in Section3.5 of symmetric trivalent planar algebras. A priori, the above relation could give manydifferent skein relations. As the next two lemmas show, however, we can easily classify allskein theories with dim V = 1 or 2 . Lemma 6.1.1.
In any symmetric trivalent planar algebra, the cupcap and identity mustbe linearly independent. In particular, any symmetric trivalent planar algebra must have dim V ≥ .Proof. If the identity and cupcap were linearly dependent then,= (cid:32) (cid:33) · (cid:32) (cid:33) = (cid:32) (cid:33) · (cid:32) x · (cid:33) = 0which contradicts our assumption that dim V = 1. Thus, the cupcap and identity mustbe linearly independent, which implies that dim V ≥ (cid:3) Lemma 6.1.2.
There are no symmetric trivalent planar algebras with dim V = 2 . Inaddition, the cupcap, identity, and I must be linearly independent.Proof. Assume that we have a symmetric trivalent planar algebra with dim V = 2. ByLemma 6.1.1, we know that the cupcap and the identity must be linearly independent andthus span V . Hence we must a relation of the form:= A + B The solution to this equation is well-known and yields on two possible relations:= − ± √ (cid:3) Thus, there are no symmetric trivalent planar algebras with dim V = 1 or 2 . This leavesus only with the cases where the dim V = 3 or 4 . Lemma 6.1.3.
There is no symmetric trivalent planar algebra with a relation of the form: = z · Proof.
If there were such a relation, capping the top of both sides would imply by nondegen-eracy that either dim V = 0 (if z = 0) or that dim V = 0, both of which are contradictionsto our dimension constraints. Thus, there can be no such dependence. (cid:3) We will also make use of the following fact from [MPS17]:
Proposition 6.1.4 (Corollary 8.9 in [MPS17]) . The only trivalent planar algebras with asymmetric braiding are Rep ( SO (3)) , Rep ( S ) , Rep ( G ) , and Rep ( OSp (1 | and these arethe only braidings on those planar algebras. We also have the following fact:
Lemma 6.1.5.
Suppose we have a symmetric trivalent planar algebra with t (cid:54) = 0 or . Thena planar algebra has a relation of the form + = z · (cid:34) + (cid:35) if and only if it has a relation of the form − = z (cid:48) · (cid:34) − (cid:35) with z = 1 t and z (cid:48) = 1 t − . Proof.
Assume that the first relations holds. By capping off on the top, we see that z = 1 t . By using the above I=H relation to simplify the triangle, we see that c = − t − t . Bymultiplying the first relation by an H we get the following relation: − t − t · + = 1 t · + 1 t · Since the square diagram above is rotationally invariant, by taking the difference of thisequation with its rotation we obtain the following relation: − = 1 t − · (cid:34) − (cid:35) , which exists when t (cid:54) = 2. If we have a relation of the form − = z (cid:48) · (cid:34) − (cid:35) , KEIN THEORIES FOR VIRTUAL TANGLES 41
We see by capping that z (cid:48) = 1 t − . Again using this I=H relation to simplify the triangle,we see that c = t − t − . Multiplying the I=H relation by an H on top tells us that t − t − · − = 1 t − · − t − · If we again take the difference of this relation with its rotation we get the new relation+ = 1 t · (cid:34) + (cid:35) when t (cid:54) = 0, which completes our proof. (cid:3) We are now ready to classify all symmetric trivalent planar algebras:
Theorem 6.1.6.
Let V be a symmetric trivalent planar algebra. Then V is one of thefollowing: Dimensions Name , , , , , . . . Rep ( SO (3)) , Rep ( OSp (1 | , and Rep ( S )1 , , , , , , . . . Rep ( G )1 , , , , , , . . . Rep ( S )1 , , , , , , . . . S t for t (cid:54)∈ Z ≥ , , , , , , . . . Rep ( S t ) for t ∈ Z ≥ − { , , , } Proof.
By Lemmas 6.1.1 and 6.1.2, we know that dim V ≥ V = 4. This implies that we have a relation of the following form:= x + y + z By rotating the above equation and solving for I again, we obtain two possible relations: − = 1 t − (cid:34) − (cid:35) or + = 1 t (cid:34) + (cid:35) By Lemma 6.1.5, however, we know that any skein theory has one relation if and only if ithas the other when t (cid:54) = 0 or 2 . At t = 0 , note that the second relation is undefined. At t = 2,the first relation is undefined and the second relation implies that I and H are equal, whichis impossible by Lemma 6.1.3. Thus t (cid:54) = 2 and so we will be justified in assuming the toprelation holds.The planar algebra with this I=H relation is exactly the skein theory of Deligne’s S t , described in Example 3.5.9. However, at t ∈ Z ≥ the planar algebra is degenerate. Taking a non-degenerate quotient at these values gives Rep( S t ). When t = 0 this planar algebra isisomorphic to OSp (1 | (cid:3) Sub-braidings of symmetric trivalent planar algebras.
We wish to classify allsub-braidings (See Definition 3.1.3) on these planar algebras. In particular, we want to findall relations of the form(34) = x + y + z + w When looking for sub-braidings, however, we again note that while there may be an elementof the 4-box space which satisfies all of the Reidemeister moves, it may not meet the natu-rality condition described in Definition 3.1.2. We could impose those requirements outright.Namely,(35) = We will call this a Type 2.5 Reidemeister move. If a planar algebra satisfies R2.5, then wewill say it is fully sub-braided. If we instead asked that(36) = be satisfied but not necessarily R2.5, we call this an even sub-braiding. Note that every fullsub-braiding gives rise to an even sub-braiding as evidenced by the following lemma: Lemma 6.2.1.
Every full sub-braiding of a symmetric trivalent planar algebra is equivalentto an even sub-braiding.Proof.
Suppose that we have a full sub-braiding denoted by . Then it satisfies (35). Wecreate a new element of the planar algebra called with = − . Clearly,satisfies the virtual Reidemeister moves because we assumed that did. By substitutingfor in (35), we see that the left-hand side is negative, while the right-hand sideremains positive. Hence, is an even sub-braiding and every full sub-braiding correspondsto an even sub-braiding, as desired. (cid:3)
A priori, there could be even sub-braidings that have no corresponding full sub-braiding.By inspection, however, this is not the case. To begin to classify the full sub-braidings of ourplanar algebras, we note that Theorem 6.1.4 tells us the only full sub-braidings (which arealso actual braidings) of Rep( SO (3)) , Rep(
OSp (1 | , Rep( G ) , and Rep( S ) are fully-flat.The only remaining sub-braidings to classify then are the ones for S t . KEIN THEORIES FOR VIRTUAL TANGLES 43
Proposition 6.2.2.
The only non-fully-flat sub-braidings of S t for generic t and Rep ( S t ) for t ∈ Z ≥ − { , , , } is the sub-braiding inherited from SO (3) q : = ( q − · + q − − ( q + q − ) for t = q + 2 + q − . At t = 0 and all sub-braidings are fully-flat.Proof. At t = 1 and 2, S t is not a symmetric trivalent planar algebra, so they are excludedfrom our classification. Because of the laborious nature of determining all possible sub-braidings on the other values of t , a Mathematica program was written to find all suchsub-braidings. The Mathematica code used for this is included in the materials uploaded tothe Arxiv but follows this algorithm:i. We begin by expanding both sides of the relations implied by the Reidemeister movesand R2.5 using (34). For more detailed examples of this type of calculation see Section4.ii. After removing the crossings, we can now think of these diagrams as graphs withonly trivalent and univalent vertices. Because of the naturality relations we assumedbetween the virtual crossing and trivalent crossing, we can think of the virtual crossingas a crossing of edges on our graph.iii. Next, we can use the I=H relation to simplify complicated diagrams into our chosenbasis according to the following process:(a) Remove all faces from any diagram.(b) Turn any tree (in the graph-theoretic sense) of n univalent vertices into a speci-fied tree of the same number of univalent vertices.iv. Now that all of our diagrams are in terms of our chosen basis, we can solve theequations given by the Reidemeister moves (including R2.5) for generic values of t .v. At t = 0 , , , or 5, there is a dimension drop in the 6-box space or lower, and so onemust calculate the null space of the inner product matrix of the appropriate n -boxspaces to transform our chosen generic spanning set into the specific basis at thosevalues.The results of the calculation are the ones given above. Except when t = 0 or 3, theonly non-trivial sub-braidings were the ( SO (3)) q sub-braidings. At t = 0 or 3 , dim V = 3,so there is a relation giving I as a linear combination of the identity, cupcap, and virtualcrossing. When one makes this substitution, the other terms cancel and we see that thecrossing is equal to the symmetric braiding, giving a fully-flat braiding. (cid:3) The work from [MPS17] and the above proposition give us the following theorem:
Corollary 6.2.3.
The only symmetric trivalent planar algebras exhibiting a non-fully-flatsub-braiding are S t for generic t and Rep ( S t ) with t ∈ Z ≥ . Proof.
By Theorem 6.1.4, the only planar algebra that could possibly exhibit a non-trivialsub-braiding is S t . The classification of such sub-braidings is proven in Proposition 6.2.2,where we see the only such sub-braiding is the one inherited from ( SO (3)) q . (cid:3) By the note in Example 3.5.9, when S t is equipped with this braiding it is also called thesecond-colored Jones polynomial planar algebra. Corollary 6.2.4.
Every even sub-braiding of a symmetric trivalent planar algebra corre-sponds to a full sub-braiding.
Proof.
When all possible even sub-braidings were classified in Mathematica, the formulafor the crossing for every even sub-braiding could be obtained from a full sub-braiding viathe method described in Lemma 6.2.1. Thus, by exhaustion no other even sub-braidingsexist. (cid:3)
References [BJ00] Dietmar Bisch and Vaughan F. R. Jones. Singly-generated planar algebras of small dimension.
Duke Math. J. , 101(1):41–75, 2000.[BJ03] Dietmar Bisch and Vaughan F. R. Jones. Singly-generated planar algebras of small dimension,part ii.
Adv. Math. , 175(2):297–318, 2003.[Bra37] Richard Brauer. On algebras which are connected with the semisimple continuous groups.
Ann.of Math. (2) , 38(4):857–872, 1937.[Bro16] Adrien Brochier. Virtual tangles and fiber functors.
ArXiv e-prints , February 2016.[BW89] Joan S. Birman and Hans Wenzl. Braids, link polynomials and a new algebra.
Trans. Amer.Math. Soc. , 313(1):249–273, 1989.[CFS95] J. Scott Carter, Daniel E. Flath, and Masahico Saito.
The classical and quantum 6 j -symbols ,volume 43 of Mathematical Notes . Princeton University Press, Princeton, NJ, 1995.[CH15] Jonathan Comes and Thorsten Heidersdorf. Thick ideals in Deligne’s category Rep( O δ ). ArXive-prints , July 2015.[CO11] Jonathan Comes and Victor Ostrik. On blocks of Deligne’s category Rep( S t ). Adv. Math. ,226(2):1331–1377, 2011.[Del90] Pierre Deligne. Cat´egories tannakiennes. In
The Grothendieck Festschrift, Vol. II , volume 87 of
Progr. Math. , pages 111–195. Birkh¨auser Boston, Boston, MA, 1990.[Del02] Pierre Deligne. Cat´egories tensorielles.
Mosc. Math. J. , 2(2):227–248, 2002. Dedicated to Yuri I.Manin on the occasion of his 65th birthday.[Del07] Pierre Deligne. La cat´egorie des repr´esentations du groupe sym´etrique S t , lorsque t n’est pas unentier naturel. In Algebraic groups and homogeneous spaces , volume 19 of
Tata Inst. Fund. Res.Stud. Math. , pages 209–273. Tata Inst. Fund. Res., Mumbai, 2007.[DGNO10] Vladimir Drinfeld, Shlomo Gelaki, Dmitri Nikshych, and Victor Ostrik. On braided fusion cate-gories. I.
Selecta Math. (N.S.) , 16(1):1–119, 2010.[DO14] Zajj Daugherty and Rosa Orellana. The quasi-partition algebra.
J. Algebra , 404:124–151, 2014.[Edg19] Joshua R. Edge. Classification of spin models for Yang-Baxter planar algebras. arXiv e-prints ,page arXiv:1902.08984, Feb 2019.[EGNO15] Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych, and Victor Ostrik.
Tensor categories , volume205 of
Mathematical Surveys and Monographs . American Mathematical Society, Providence, RI,2015.[FY89] Peter J. Freyd and David N. Yetter. Braided compact closed categories with applications tolow-dimensional topology.
Adv. Math. , 77(2):156–182, 1989.[FYH +
85] Peter Freyd, David Yetter, Jim Hoste, William B. R. Lickorish, Kenneth Millett, and Adrian Oc-neanu. A new polynomial invariant of knots and links.
Bull. Amer. Math. Soc. (N.S.) , 12(2):239–246, 1985.[GS12] Pinhas Grossman and Noah Snyder. Quantum Subgroups of the Haagerup Fusion Categories.
Communications in Mathematical Physics , 311:617–643, May 2012.[Gup08] Ved Prakash Gupta. Planar algebra of the subgroup-subfactor.
Proc. Indian Acad. Sci. Math.Sci. , 118(4):583–612, 2008.[Jae95] Fran¸cois Jaeger. Spin models for link invariants. In
Surveys in combinatorics, 1995 (Stirling) ,volume 218 of
London Math. Soc. Lecture Note Ser. , pages 71–101. Cambridge Univ. Press,Cambridge, 1995.[Jon85] Vaughan F. R. Jones. A polynomial invariant for knots via von Neumann algebras.
Bull. Amer.Math. Soc. (N.S.) , 12(1):103–111, 1985.[Jon99] Vaughan F. R. Jones. Planar algebras, I.
ArXiv Mathematics e-prints , September 1999.[Jon00] Vaughan F. R. Jones. The planar algebra of a bipartite graph. In
Knots in Hellas ’98 (Delphi) ,volume 24 of
Ser. Knots Everything , pages 94–117. World Sci. Publ., River Edge, NJ, 2000.
KEIN THEORIES FOR VIRTUAL TANGLES 45 [Kau87] Louis H. Kauffman. State models and the Jones polynomial.
Topology , 26(3):395–407, 1987.[Kau90] Louis Kauffman. An invariant of regular isotopy.
Tansactions of the American MathematicalSociety , 1990.[Kau99] Louis H. Kauffman. Virtual knot theory.
European Journal of Combinatorics , 1999.[KL94] Louis H. Kauffman and S´ostenes L. Lins.
Temperley-Lieb recoupling theory and invariants of -manifolds , volume 134 of Annals of Mathematics Studies . Princeton University Press, Princeton,NJ, 1994.[Kup94] Greg Kuperberg. The quantum G link invariant. Internat. J. Math. , 5(1):61–85, 1994.[Kup97] Greg Kuperberg. Jaeger’s Higman-Sims state model and the B spider. J. Algebra , 195(2):487–500, 1997.[LZ12] Gustav Lehrer and Ruibin Zhang. The second fundamental theorem of invariant theory for theorthogonal group.
Ann. of Math. (2) , 176(3):2031–2054, 2012.[MPS11] Scott Morrison, Emily Peters, and Noah Snyder. Knot polynomial identities and quantum groupcoincidences.
Quantum Topol. , 2(2):101–156, 2011.[MPS17] Scott Morrison, Emily Peters, and Noah Snyder. Categories generated by a trivalent vertex.
Selecta Math. (N.S.) , 23(2):817–868, 2017.[MT90] Hugh R. Morton and Pawel Traczyk. Knots and algebras. In E. Martin-Peinador and A. RodezUsan, editors,
Contribuciones Matematicas en homenaje al profesor D. Antonio Plans Sanz deBremond , pages 201–220. University of Zaragoza, 1990.[Pet09] Emily E. Peters.
A planar algebra construction of the Haagerup subfactor . ProQuest LLC, AnnArbor, MI, 2009. Thesis (Ph.D.)–University of California, Berkeley.[RT90] Nicolai Yu. Reshetikhin and Vladimir G. Turaev. Ribbon graphs and their invariants derivedfrom quantum groups.
Comm. Math. Phys. , 127(1):1–26, 1990.[SW99] Daniel S. Silver and Susan G. Williams. Virtual tangles and a theorem of Krebes.
J. Knot TheoryRamifications , 8(7):941–945, 1999.[Wey39] Hermann Weyl.
The Classical Groups. Their Invariants and Representations . Princeton Univer-sity Press, Princeton, N.J., 1939.
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