edges and one green edge meet. We call this colouring preferred. It should be noted that it is not a priori clear why this procedure does not run into ambiguities. But this will never happen, see Lemma 3.12.8. The same lemma ensures that the result at the end does not depend on the choices involved.
The algorithm works as follows. First colour all 0-nested components wc0j by the following procedure.
(1) At the initial stage, colour all edges of the external face ofw0cj red.
(2) At thei-th stage we complete the colouring of the edges incident to the vertices ofwc0jwith at least one edge already coloured from the(i−1)-th stage.
(3) At a given vertex, if only one edgeei−1 had already been coloured and it is green, then we colour the two remaining edgesei, e′i red (if one is already red, then colour the remaining one also red). If two edgesei−1, e′i−1had already been coloured and they are red, then we colour the remaining edgeeigreen.
ei−1 ei e′i
e′i−1 ei−1
ei
(4) At a given vertex, if only one edgeei−1 had already been coloured and it is is red, then we colour one of the other incident edges in the following fashion.
• Green forei, if the type of the corresponding face ofei is already green.
• The remaining edgese′iincident to this vertex should be red.
Below we have summarised the convention in a picture.
ei−1 e′i ei
ei−2 ei−2 ei−1 e′i ei
Now, if allk-nested components are coloured, then colour exactly one edge of eachk + 1-nested by first choosing a line that cuts though all the components at least once (e.g. the cut line). Then at least one edge of allk+ 1-nested component is neighbouring an already coloured edge. Choose one such edge for allk+ 1-nested component and colour it in the following way and then repeat the procedure from before.
→ → → →
Or in words, use the same colour, iff the orientation of the line is reversed. The algorithm stops if every edge is coloured.
Lemma 3.12.8. The 2-colouring ofwabove is well-defined and it does not depend on the choices involved.
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Proof. First we prove the second statement, i.e. that the choices do not matter. In fact this can be easily seen by locally alternating the line that cuts though the components as illustrated below.
Fg Fg Fg
Note that all three choices above give the same result since the corresponding face is of green type.
We leave it to the reader to check the other possible colourings. Hence, since choosing a different edge can also be seen as rearranging the line, we get the second statement.
To see that the colouring is well-defined, we proceed by induction on the numbermof faces of w. It can be easily checked that it is well-defined for a circle or a theta web.
Moreover, one easily checks that every closedsl3webwhas at least one circle, digon or square face. Fix one such faceF and remove it from the web as explained in the subsection before. The resulting web w′ is, by induction, colourable without ambiguities. If the faceF is a circle, then one easily sees that the colouring ofw′ can be extended without ambiguities towby applying the procedure in Definition 3.12.7.
IfF is either a digon or a square, then one can extend the colouring in the following way.
7→
←[ w w′
7→
←[ w w′
7→
←[ w w′
in the case thatF is a digon and ifF is a square, then we use the following.
7→
←[
w w′
7→
←[
w w′
7→
←[
w w′
It should be noted that the only ambiguous seeming case, i.e. the case where the result of the square removal ofwhas two alternating coloured edges inw′, does not occur. This is because the two strings inw′ have a different orientation.
Hence, if they are part of the same connected component, then they have the same colour, because the colours and the orientations always alternate at vertices. If they are not part of the same connected component, then we have to use the cut procedure explained before to see that
they have the same colour. This proves the first statement.
Remark 3.12.9. Note that the Definition 3.12.7 can also be made for half webs, i.e. websuwith a given sign stringS at the boundary. An analogue of Lemma 3.12.8 can be shown as above.
Moreover, it is easy to show that, given a fixed sign string S and two webs u, v such that S = ∂u = ∂v, then the preferred colourings of u and v match at the boundary and the pre-ferred colourings of u∗v andv∗u are given by glueing the preferred colourings of u, v together.
This is already implicit in Lemma 3.12.8 because the way how to cut a web does not affect the result, i.e. we can take the cut line for non-elliptic webs.
Example 3.12.10. An easy example of how the colouring works is shown below.
Definition 3.12.11. (Preferred flow on closed webs) Consider the subgraph ofwgiven by the red edges. It is easy to see that this subgraph consists of a disjoint set of closed cycles. By orienting these edges so that the flows around the parts of the web are oriented counterclockwise and all other cycles clockwise we obtain a flow onw. We call this flow the preferred flow and writewpfor wwith the preferred flow.
Note that, by construction, the preferred flowwp always consists of flows inside faces or flows around the web. That is, they will never cross through edges, i.e. the case below will never occur.
A good example is the flow pictured in Example 3.12.18.
The following definition of the preferred face removal is given for a connected web w. For an arbitrary web, the whole process should be repeated for any connected component.
Definition 3.12.12. (The preferred face removing) Given a closed web with a flow denoted as wf. At each stage of the face removing algorithm let the order of the faces be determined in the following way.
It should be noted that the preferred flow from Definition 3.12.11 implies that any faceF ofwp
is of the following type because every face has an even number of edges and the closed circles of the preferred flowpare always oriented clockwise except the flows around the web. We note that, by abuse of notation, we denote a face around a connected component of the web asFext, although this face is not the external face for nested parts.
• FacesF1, such that the preferred flow has a component ofpinsideF1.
• FacesF2, such that the preferred flowphas the same orientation as the edges ofF2.
• FacesF3, such that the preferred flowpis against the orientation of the edges ofF3. To summarise see the figure below. A face of typekshould be labelledk.
F1
F3
F2
F1
F3 F2
Fext
F3
F2
Fext
F3 F2
At each stage of the algorithm we continue to call the web with flow wf. We removewf by the following algorithm.
(1) Remove all circles inwf using the local circle rules of Definition 3.12.12, in any order. If there are no faces remaining, the algorithm stops. If there are faces remaining and some of
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them are digons, then proceed to step 2. If no remaining faces are digons, then proceed to step 3.
(2) Remove all digons with the smallest label using the local digon rules of Definition 3.12.12, in any order. Go back to step 1.
(3) Remove all square faces with the smallest label using the local square rules of Defini-tion 3.12.12, in any order. Go back to step 1.
We call the above process of obtaining a foamFwf ∈ F(w)the preferred face removal of wf or short preferred resolution ofwf.
Remark 3.12.13. Notice that it is relatively easy to calculate the order of face removing since the preferred colouring of the web w′ (and therefore the preferred flow) after removing a particular face ofwcan be computed directly as indicated before in the proof of Lemma 3.12.8.
It is straightforward to check that this recursive procedure gives the same answer as if one calculates the preferred colouring ofw′as in Definition 3.12.7.
Lemma 3.12.14. The preferred resolution ofwf is well-defined.
Proof. We note that the labelling of the faces is in such a way that neighbouring faces never have the same label, i.e. they define a colouring of the faces ofwwith three colours. Hence, we only need to show that the consecutive resolution of two non-neighbouring faces in two different orders results in isotopic foams.
A simple illustration shows that the consecutive removing of the two faces in two different orders yields isotopic foams.
=
The above only illustrates the effect of removing both square faces in one particular way, but similar arguments demonstrate the claim for different removing of the square faces as well as removing of
digon faces.
Theorem 3.12.15. Given a webw=u∗v, the face removing algorithm 3.12.3 together with the pre-ferred face removal 3.12.12 gives an isotopy invariant, homogeneous basis ofF(w)parametrised by flows onw. Ifwis symmetric, then the basis contains the identity given by the canonical flow.
Proof. This is a direct consequence of Theorem 3.12.6, Proposition 3.12.4 and the fact that the procedures explained in the Definitions 3.12.3 and 3.12.12 work in an isotopy invariant way.
From Theorem 3.12.15 we get the following corollary since for any fixed sign string S the algebraKS was defined as
KS = M
u,v∈BS uKv.
Corollary 3.12.16. Let S be a fixed sign string. The set of flows on all webs w with S = ∂w parametrises an isotopy invariant, homogeneous basis for KS, via the preferred face removing algorithm.
Example 3.12.17. Consider the theta webwfrom Example 3.12.5 again. We know that the graded dimension ofF(w)is given by the Kuperberg bracket and is therefore[2][3] =q−3+2q−1+2q+q3. The six possible flows (ordered by weight) onware illustrated below.
wt=3
wt=−1
wt=1
wt=−1
wt=1
wt=−3
Notice that in this case the canonical flow is also the preferred flow. Hence, the order of removal of the faces is from right to left. This gives the following basis elements (ordered byq-degree) for the corresponding flows.
q−deg=0
q−deg=4
q−deg=2
q−deg=4
q−deg=2
q−deg=6
Example 3.12.18. The counterexample of Khovanov and Kuperberg [57] for the negative exponent property (compare to Theorem 3.2.5 and the Remark 3.2.6) gives rise to another interesting exam-ple, i.e. the corresponding foam will be (up to a scalar factor) a non-trivial idempotent of the web algebra. The preferred flowpfrom Definition 3.12.11 onwpictured below (for both orientations
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ofw) has weightwt(p) = 12. Thus, the corresponding foam hasq-degree0.
wt = 12
One easily checks that the canonical flow con this web also has wt(c) = 12. Thus, F(w) has two linear independent foams inq-degree zero. Note that (b) of Proposition 3.12.4 ensures that the foam obtained from the canonical flow is the identity.
3.13. Open issues. Let us mention some open questions that are hopefully answered in future work. Note that some other open questions were already discussed in Section 3.1. We will focus here on four questions the author is currently working on, namely the ones listed below.
(a) We conjecture that there is an ordering how to resolve webs as explained in Section 3.12 such that the resulting basis is a graded cellular basis. For a lot of constructions involv-ing graded cellular bases one needs a particular basis in an explicit form (compare to the discussion in Section 4.7). Hence, it is a future goal to give such a basis forKS.
(b) A generalisation of the results on Howe duality from Section 3.10. That is a pictorial version similar to the results in Section 3.10, but for arbitrary representations ofUq(sln). In order to do so, one would for example consider clasps and clasped web spaces as explained by Kuperberg in [70]. Note that this is not known at the moment, even for n = 2. This would correspond to the coloured versions of theslnpolynomials instead of the uncoloured case.
(c) Instead of a categorification of the invariant tensors, as we have done, one could also try to give a categorification of the full tensor product. Note that in the n = 2 case a categorification is known, e.g. see Chen and Khovanov [24]. It is worth noting that this is related to the question how to construct the quasi-hereditary cover ofKS. Such a cover for Khovanov’s arc algebra, i.e. thesl2case, was studied by Brundan and Stroppel [19], Chen and Khovanov [24] and Stroppel [107].
(d) The Question 3.1.1 asked by Kamnitzer, i.e. how our work is related to the approach from algebraic geometry by Fontaine, Kamnitzer and Kuperberg [33].
4. TECHNICAL POINTS
4.1. Higher categories. In the present section we recall some definitions and theorems from higher category theory.
Note that we always talk about strict (n, n)-categories if we say n-category. Here we allow n∈ {0,1,2, . . . , ω}. One can think of an-category as an-dimensional category.
We usually drop the “strict”, that is all categories are assumed to be strict unless otherwise mentioned. Informally, the strict refers to the fact that composition of higher morphism is “on the nose” associative and satisfies some identity laws.
Moreover, a(n, r)-category (withr ≤n) is a category with only non-trivialk-cells fork≤nof which allk′-cells forr < k′ ≤nare invertible. For example a strict(1,0)-category is a groupoid, while a strict(1,1)-category is a usual category.
Note that(n,1)-categories arise in a lot of examples motivated from topology and are sometimes calledn-categories. But we do not need them in this thesis, so we only refer to Leinster’s [75] book for a more detailed discussion.
Moreover, to avoid all set-theoretical question, which are interesting for their own sake, but not for our purposes, all categories should be essentially small, i.e. the skeleton should be small. By a slight abuse of notation, we always take about sets of objects, morphisms etc.
We start by recalling some “classical” notions that we need in the following.
Definition 4.1.1. Let C be a category. The category is called (strict) monoidal if it is a monoid in the category Cat1, i.e. the category of categories. That is the category C is equipped with a bifunctor⊗: C × C → C such that the following is satisfied.
(a) There exists a unit1∈Ob(C)such that for allC ∈Ob(C) 1⊗C= 1⊗C =C.
(b) For allC1, C2, C3∈Ob(C)the associativity holds, that is (C1⊗C2)⊗C3 =C1⊗(C2⊗C3).
Note that the functoriality implies
(f′◦g′)⊗(f ◦g) = (f′⊗f)◦(g′⊗g) for suitable morphismsf, f′, g, g′.
A weak monoidal category is the same as above, but (a) and (b) hold only up to natural isomor-phism, denotedrC, ℓC andαC1,C2,C3, such that the following two diagrams commute.
(C1⊗1)⊗C2
αC1,1,C2
//
rC1⊗1
''P
PP PP PP PP
PP C1 ⊗(1⊗C2)
1⊗ℓC2
wwnnnnnnnnnnn
C1⊗C2 164
and the so-called pentagram identity
Example 4.1.2. For a given fieldK the categoryC =VecK together with the usual tensor product is a weak, symmetric monoidal category.
In general, since one is mostly interested in more than equivalence classes of objects, strict monoidal categories are rare. But a well-known fact, also known as Mac Lane’s coherence theo-rem, allows us to “ignore” the difference between strict and weak monoidal categories, i.e. by Mac Lane’s coherence theorem we have the following.