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Lectures on SL(3) foams and link homology

T0 review · 2 major / 3 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read For every oriented SL(3) web, the state space of the foam TQFT is a free graded abelian group whose graded rank is the Kuperberg invariant, so the quantum sl3 link invariant lifts to a functorial homology theory.

desk verdict Solid, useful lecture notes on SL(3) foams; the main caveat is that the TQFT assertion in Theorem 3.1 is deferred to the literature rather than proved in the text. read the letter →

arxiv 2507.17119 v1 pith:3E2TLX3I submitted 2025-07-23 math.QA math.GTmath.RT

Mikhail Khovanov , Dmitry Solovyev This is my paper
classification math.QAmath.GTmath.RT MSC 05C1557M1557K1618N25
keywords SL(3)foamslinkhomologyKuperberginvariantplanarwebsTQFTTaitcoloringscategorificationfoamevaluation
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

These lecture notes develop a topological quantum field theory whose inputs are $\mathrm{SL}(3)$ foams, two-dimensional complexes embedded in $\mathbb{R}^3$ with seams where three sheets meet, and whose outputs are graded state spaces for planar webs. Their central claim is that, for every oriented $\mathrm{SL}(3)$ web $\Gamma$, the state space $\langle\Gamma\rangle$ is a free graded abelian group whose graded rank equals the Kuperberg invariant $P(\Gamma)$, and that foam cobordisms between webs induce homogeneous maps, making the assignment a TQFT. Applying this to link diagrams, by resolving crossings into webs and taking cones of the maps induced by elementary foam cobordisms, yields a bigraded link homology whose Euler characteristic is the quantum $\mathfrak{sl}_3$ link invariant that the webs compute. The notes also carry out the same universal construction for unoriented foams in characteristic 2, where state spaces are modules over symmetric polynomials and their ranks count Tait colorings for reducible webs, and they review the generalization to $\mathrm{GL}(N)$ foams. A reader should take away that the quantum $\mathfrak{sl}_3$ invariant is not being treated as a bare polynomial but as the shadow of a functorial homology theory with concrete topological building blocks.

What carries the argument

The load-bearing machinery is the universal construction applied to a multiplicative evaluation of closed foams. A Tait coloring labels every facet by one of three colors so that the three facets along each seam have distinct colors; the colored evaluation is a product over dot decorations and over the bicolored surfaces $F_{ij}(c)$, with exponents $\chi(F_{ij}(c))/2$, and the total evaluation is the sum over all Tait colorings. In the oriented case the evaluation can be computed by surgery along seam circles using two Frobenius algebras, the cohomology rings $H^*(\mathbb{CP}^2)$ and $H^*(\mathrm{Fl}_3)$, and the universal construction then defines the state space $\langle\Gamma\rangle$ as the quotient of the free module spanned by foams with boundary $\Gamma$ by the kernel of the gluing-pairing form. What carries the argument is that oriented foams have no vertices, only seam circles, which makes the surgery evaluation finite and explicit.

What would settle it

Take a single closed oriented foam with two seam circles arranged so that the surgery formula can be applied in either order; if the two orders produce different integers, the evaluation is not well-defined and the TQFT of Theorem 3.1 does not exist. Alternatively, compute $\operatorname{grk}\langle\Gamma\rangle$ for an oriented web $\Gamma$ in two different skein-reduction orders and check that the graded ranks agree with $P(\Gamma)$.

Watch

Extended reading notes

Core claim

The paper's core claim is Theorem 3.1: for every oriented $\mathrm{SL}(3)$ web $\Gamma$, the state space $\langle\Gamma\rangle$ is a free graded abelian group with graded rank $\operatorname{grk}\langle\Gamma\rangle = P(\Gamma)$, where $P(\Gamma)$ is the Kuperberg invariant, and the foam-induced maps define a TQFT from the category of oriented webs and foams to graded abelian groups. For an oriented link $L$, the complex built from the cube of web resolutions has homology $H(L)$ satisfying $P(L)=\sum_{i,j}(-1)^i q^j \operatorname{rk} H^{i,j}(L)$, so the homology categorifies the quantum $\mathfrak{sl}_3$ link invariant. In the unoriented characteristic-2 setting, the same universal construction gives free graded modules over the ring of symmetric polynomials in three variables, with rank equal to the number of Tait colorings for every reducible web, and the general foam evaluation extends the story to $\mathrm{GL}(N)$ webs.

Load-bearing premise

The construction depends on the topological fact that every two-colored slice of a foam embedded in $\mathbb{R}^3$ is a closed orientable surface, which makes its Euler characteristic even and the exponents in the evaluation formula integers.

Editorial extensions

If this is right

  • The Kuperberg invariant of a link becomes the Euler characteristic of a bigraded homology, so the polynomial is accompanied by algebraic structure that is functorial under link cobordisms.
  • The tangle extension yields a 2-functor from tangle cobordisms to complexes of graded bimodules up to homotopy, making the theory a fully functorial invariant rather than a diagram-dependent construction.
  • For unoriented webs, reducible webs have free state spaces whose rank equals the number of Tait colorings, giving a categorified, graded version of the coloring count.
  • The same universal-construction method, applied to the general $\mathrm{GL}(N)$ foam evaluation described in the notes, gives a TQFT for $\mathrm{GL}(N)$ webs and categorifies the quantum $\mathrm{GL}(N)$ invariants colored by exterior powers of the fundamental representation.
  • Conditional on the conjectures stated in the notes, the characteristic-2 state-space ranks would imply the Four-Color Theorem.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A testable extension would be to compute the characteristic-2 state space of the smallest non-reducible web, the dodecahedron graph, and compare its graded dimension with its number of Tait colorings; the notes report this as an open problem, so an affirmative computation would strengthen the conjectural route to the Four-Color Theorem.
  • An editor's inference is that the oriented theory's reliance on exactly two Frobenius algebras suggests a hierarchy: link homology for higher-rank spiders or for non-fundamental colors may require a chain of flag-variety cohomology rings, one per fundamental weight.
  • Because the unoriented evaluation is defined only in characteristic 2, a natural next step is to find an integral lift with vertices present; the paper's comparison with the oriented, vertex-free theory indicates where sign conventions would have to be introduced.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 3 minor

Summary. These lecture notes, based on three talks at Tsinghua University in 2023, provide an introduction to SL(3) foams and their use in link homology. The first part develops unoriented SL(3) foams, their evaluation by Tait colorings, and the universal construction of state spaces, following Khovanov--Robert. The second part treats oriented SL(3) foams and states a categorification of the Kuperberg quantum sl_3 invariant, including a TQFT assertion for oriented webs and foams. Later sections review SL(3)-web algebras, tangle invariants, oriented GL(N) foams and the Robert--Wagner evaluation, and the connection to Kronheimer--Mrowka gauge theory and the Four-Color Theorem. The notes are expository: most substantial theorems are drawn from the existing literature, especially Khovanov's sl(3) link homology paper, Khovanov--Robert, Mackaay--Vaz, and Robert--Wagner, and proofs are often only sketched or cited.

Significance. If its statements are accurate, these notes are a useful service to the community: they collect the unoriented and oriented SL(3) foam evaluations, the universal construction, the categorification of the Kuperberg invariant, the web-algebra approach to tangles, and the Robert--Wagner GL(N) evaluation into a single readable document, with instructive examples such as the sphere, theta-foam, and digon relations. The notes also helpfully situate the Kronheimer--Mrowka conjecture and the open Problem 2.1. However, the paper is a review rather than a new research paper, and its value depends on the precision with which the cited results are presented. The main technical weakness is that Theorem 3.1, the central categorification statement, is asserted without an actual proof or an exact citation for the monoidality part. The orientability of bicolored surfaces in Proposition 1.1 is adequately justified for foams embedded in R^3, so I do not regard that topological point as a serious gap; the load-bearing unsupported step is the TQFT/monoidality assertion in the oriented theory.

major comments (2)
  1. [§3.2, Theorem 3.1] Theorem 3.1 asserts that the oriented state-space functor is a TQFT, which in particular requires the monoidality map ⟨Γ1⟩⊗⟨Γ2⟩→⟨Γ1⊔Γ2⟩ to be an isomorphism for all oriented webs. The text preceding the theorem only says that the direct-sum decompositions categorifying Kuperberg's skein rules are 'straightforward to derive' and cites [46, Section 3.4]. Freeness and the graded-rank formula do not by themselves imply monoidality: in the unoriented theory the analogous map (16) is proved only when at least one web is reducible (Proposition 2.7) and is left open in general (Problem 2.1). Since the TQFT assertion is precisely what upgrades the construction from a lax TQFT to a TQFT, the manuscript should either supply a proof of the monoidality isomorphism or state Theorem 3.1 as a theorem of a specific cited work, with a precise reference. As written, the theorem overstates what is demonstrated in these notes.
  2. [§3.2, evaluation of closed oriented foams] The oriented evaluation is defined by a surgery procedure: one pushes a circle off each seam circle, applies the neck-cutting relation (Figure 3.10), and evaluates the resulting surfaces and theta-foams using the traces (19) and (20). The text says that 'using the rules above, an arbitrary closed foam F in R^3 can be evaluated to an integer', but it does not prove that the result is independent of the choices involved (choice of push-off circles, order of surgeries) or invariant under isotopy of the foam. This well-definedness is a load-bearing ingredient for both Theorem 3.1 and the link invariant in Section 3.3. A precise reference to the original proof in [46] or [78] should be given, or a proof should be sketched; otherwise the evaluation underlying the TQFT is not fully established in these notes.
minor comments (3)
  1. [§2.1, proof of Proposition 2.2] In the paragraph after equations (13) and (14), 'combined with (1314)' should read 'combined with (13)–(14)' or 'with (14)'.
  2. [§2.1, proof of Theorem 1.1] After the phrase 'for some l ∈ 2Z', the symbol ℓ appears in the displayed formulas without being defined; it appears to denote l/2, and this should be stated explicitly.
  3. [§3.2, formula (20)] In the definition of the trace ε on H^*(Fl_3), the third case '0, for (a,b,c) ≠ (2,1,0)' is confusing because (2,1,0) is a cyclic permutation of (0,2,1), which is already handled by the second case. The intended wording is presumably '0 otherwise'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the notes are a review whose central theorems are cited to published independent sources, and the universal-construction setup does not define the Kuperberg ranks into the state spaces by construction.

full rationale

The paper is a lecture-note survey, and its central claims are presented as reviews of established results. Theorem 3.1, asserting grk<Γ> = P(Γ) and TQFT-ness, is justified by direct sum decompositions that the text says are 'straightforward to derive' and cites [46, Section 3.4]; this is a citation to a published proof by one of the present authors, not a redefinition. The universal construction of Section 1.3 defines state spaces from a multiplicative evaluation of closed foams, and the oriented evaluation of Section 3.2 is built from the Frobenius algebras H*(CP^2) and H*(Fl_3) with traces (19) and (20). The Kuperberg invariant P(Γ) is not fed into the evaluation as an input; matching its skein rules requires the nontrivial direct-sum decompositions. The reader's flagged Euclidean-topology point (Proposition 1.1, even Euler characteristic of bicolored surfaces) is proved geometrically in the text and is a hypothesis used to make the exponents in (2) integral; it is not an assumption of the desired conclusion. The only genuinely unsupported point is the monoidality assertion in Theorem 3.1, which the notes do not prove and instead attribute to prior work; that is a proof-gap or rigor concern, not a circular reduction. Under the rules, self-citation with independent published content and external verification (e.g., Lewark's computations, Mackaay--Vaz comparison) does not raise the circularity score.

Assumptions & free parameters 0 free parameters · 2 assumptions · 0 invented entities

This is an expository paper; its content rests on prior published theorems (especially [55], [64], [103]) that are cited and not reproved. There are no new free parameters or invented entities.

assumptions (2)
  • domain assumption The Kuperberg web invariant P(Γ) is well-defined via the skein relations in Figure 3.3 (Kuperberg [64]).
    The notes build the categorification on this invariant and cite it without proof.
  • domain assumption The Robert-Wagner evaluation for GL(N) foams is well-defined and lands in symmetric polynomials, and the state spaces of MOY webs are free graded modules with graded rank equal to the MOY invariant (Propositions 5.2 and 5.3 from [103]).
    These are the main external theorems used in Section 5; the notes cite [103] rather than proving them.

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Pith. "Pith review of Lectures on SL(3) foams and link homology." pith.science (2026). https://pith.science/paper/3E2TLX3I

@misc{pith2026250717119,
  author       = {Pith},
  title        = {Pith review of: Lectures on SL(3) foams and link homology},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3E2TLX3I}},
  note         = {Machine review of arXiv:2507.17119}
}
abstract

These notes are based on the three lectures that one of the authors gave at Tsinghua University in the summer of 2023 as part of the workshop on Geometric Representation Theory and Applications. They contain an introduction to the evaluation of $\mathsf{SL}(3)$ foams and the associated topological theory of trivalent planar graphs and foam cobordisms between them. A categorification of the Kuperberg quantum $\mathfrak{sl}_3$ web and link invariant and the Robert-Wagner $\mathsf{SL}(N)$ foam evaluation are reviewed as well.

Figures

Figures reproduced from arXiv: 2507.17119 by the authors.

Figure 1.1
Figure 1.1. Three types of points of a foam. From left to right: a regular point, a seam point, and a vertex. Remark 1.1. Locally a foam looks like a spine of a 3-manifold. Consider a triangulation T of a 3-manifold M then take the Poincare dual P D(T). The two-skeleton P D(T) 2 of the Poincare dual of T is a spine of M. Singularities of a spine are seams and vertices, as depicted in [PITH_FULL_IMAGE:figures/full_fig_p005_1_1.png] view at source ↗
Figure 1.2
Figure 1.2. The Θ-foam has a singular seam which is a circle. Its singular graph s(Θ) = S 1 consists of a verticeless circle, and the foam has three facets, which are all (open) two-disks. Remark 1.2. According to our definition a facet of F is open in F. Sometimes by a facet we will mean its closure in F. We will either talk about open or closed facets, or it will be clear from the context whether an open or a closed facet is … view at source ↗
Figure 1.3
Figure 1.3. Foam F with three facets and a single singular vertex. Unlike previous examples, this foam has a vertex point. The singular graph of this foam, s(F) = S 1 ∨ S 1 , is a 4-valent graph with a single vertex and two loops. The set of facets f(F) consists of three open disks D2 . Example 1.4. Example 1.3 can be extended by replacing the meridional disk by n parallel disks and the longitudional disk by m parallel disks. T… view at source ↗
Figures from the paper (74 more)
Figure 1.4
Figure 1.4. Figure 1.4: Facets along each seam should be colored by three distinct colors. In Example 1.1, the foam admits 3m different Tait colorings, where m is the number of connected components of the surface S, since there are three color choices for each component. Θ-foam in Example 1…
Figure 1.5
Figure 1.5. Figure 1.5: Surface F12(c); facet colored 3 is omitted. Next we examine the neighbourhood of a vertex in Fij (c). Picking distinct colors for two adjacent facets forces a unique coloring on the remaining four facets at the vertex, see [PITH_FULL_IMAGE:figures/full_fig_p007_1_5.png]
Figure 1.6
Figure 1.6. Figure 1.6: An example of F13(c) and corresponding colored graph. Taking the union of facets colored i or j shows that Fij (c) is a surface in the neighbour￾hood of any vertex v of F, and a suitable neighbourhood of v in Fij (c) is homeomorphic to R 2 . Note that the bicolored s…
Figure 1.7
Figure 1.7. Figure 1.7: Merging several dots on a facet into a single weighted dot. The dots can be thought of as observables of the theory we are about to construct. Next we set up the algebraic side of the story. We work over a field k of characteristic two, char k = 2. The characteristic…
Figure 1.8
Figure 1.8. Figure 1.8: The two-sphere foam. Label 1 depics the coloring of the unique facet by that color. Let us look at some examples before sketching a proof of the theorem. Example 1.5. Consider the two-sphere foam F = S 2 decorated by n dots, see [PITH_FULL_IMAGE:figures/full_fig_p00…
Figure 1.9
Figure 1.9. Figure 1.9: Coloring cσ of a Θ-foam. For any coloring cσ and any pair of colors i, j, i < j the bicolored surface Fij (cσ) is a two-sphere. Consequently, the denominator term does not depend on the coloring, and Fij (cσ) = S 2 , χij (cσ) 2 = 1, [PITH_FULL_IMAGE:figures/full_fig…
Figure 1.10
Figure 1.10. Figure 1.10: Swapping colors 1, 2 on a two-sphere component of F12(c). The two-sphere S results in the denominator xi + xj in the evaluations ⟨F, c⟩ and ⟨F, c′ ⟩. The idea is to compare these two evaluations and show that xi + xj in the denominator will cancel out in their sum ⟨…
Figure 1.11
Figure 1.11. Figure 1.11: R-valued bilinear form on α(Γ) is obtained by gluing two foams F1 and F2 along the common boundary web Γ = ∂F1 = ∂F2 and evaluating the resulting closed foam. Extending R-linearly, one obtains a symmetric bilinear form on Fr(Γ). Foams F1, F2 in this construction may…
Figure 1.12
Figure 1.12. Figure 1.12: Composing F with various foams F0 with boundary Γ0 induces a map of state spaces α(F) : α(Γ0) −→ α(Γ1). In this example Γ0 has two connected components [PITH_FULL_IMAGE:figures/full_fig_p015_1_12.png]
Figure 2.1
Figure 2.1. Figure 2.1: The evaluation ⟨F1F2⟩ is 0 since this foam has no Tait colorings Proof. To understand the state space of Γ we look at all foams F1 such that ∂F1 = Γ, then we pair it up with F2 such that ∂F2 = Γ and compute the evaluation ⟨F1F2⟩. Foam F1F2 is schematically depicted o…
Figure 2.2
Figure 2.2. Figure 2.2: Foams F1 and F2. We claim that these two foams have the same evaluation, ⟨F1⟩ = ⟨F2⟩. Indeed, pick a Tait coloring c of the foam F1, shown in [PITH_FULL_IMAGE:figures/full_fig_p019_2_2.png]
Figure 2
Figure 2. Figure 2: in the center and right and denoted [PITH_FULL_IMAGE:figures/full_fig_p019_2.png]
Figure 2.3
Figure 2.3. Figure 2.3: Coloring c of F1 and its extensions to colorings cj , ck of F2. Let us compare ⟨F1, c⟩ and ⟨F2, c′ ⟩, where c ′ stands for ck or cj . Looking at the respective bicolored surfaces, we see that F1,ij (c) and F2,ij (c ′ ) are homeomorphic and likewise for F1,ik(c) and F…
Figure 2.4
Figure 2.4. Figure 2.4: An equality between foam evaluations, where the first foam on the right-hand side has a dot on the top facet away from the reader, while the second foam on the right-hand side has a dot on the bottom facet facing the reader [PITH_FULL_IMAGE:figures/full_fig_p020_2_4.png]
Figure 2.5
Figure 2.5. Figure 2.5: Identity foam cobordism for the digon facet and its coloring c. There are two colorings c ′ of foams G1 and G2 on the right-hand side which extend the coloring c of the foam F. We denote them cj and ck, with the index referring to the color of the back facet of the u…
Figure 2.6
Figure 2.6. Figure 2.6: Extending coloring c of F to colorings cj , ck of G1, G2. Now let us compare ⟨F, c⟩ and ⟨G1, c′ ⟩ + ⟨G2, c′ ⟩. First consider intermediate foam G3 with induced coloring c ′ , as depicted in [PITH_FULL_IMAGE:figures/full_fig_p021_2_6.png]
Figure 2.7
Figure 2.7. Figure 2.7: Foam G3 and its colorings cj , ck. Direct computation leads to the following relations (13) ⟨G1, cj ⟩ + ⟨G2, cj ⟩ = (xj + xk)⟨G3, cj ⟩, (14) ⟨G1, ck⟩ + ⟨G2, ck⟩ = 2xk⟨G3, ck⟩ = 0. Also note that χ(Fij (c)) = χ(G3,ij (cj )), χ(Fik(c)) = χ(G3,ik(cj )), but χ(Fjk(c)) = …
Figure 2.8
Figure 2.8. Figure 2.8: Foams for the direct sum decomposition of a digon web. Cobordisms in [PITH_FULL_IMAGE:figures/full_fig_p023_2_8.png]
Figure 2
Figure 2. Figure 2 [PITH_FULL_IMAGE:figures/full_fig_p024_2.png]
Figure 2.10
Figure 2.10. Figure 2.10: A neck-cutting relation Proof. Proof is similar to that of the identity in [PITH_FULL_IMAGE:figures/full_fig_p024_2_10.png]
Figure 2.11
Figure 2.11. Figure 2.11: Foam maps for the direct sum decomposition in Lemma 2.2. The remaining relations for the direct sum decomposition: βiαj = δij follow from evaluations of dotted spheres in Example 1.5. □ The following two relations are straightforward to verify. Proposition 2.4. “Tri…
Figure 2.12
Figure 2.12. Figure 2.12: Trivalent bubble relation. Proposition 2.5. “Vertices removal” relation depicted in [PITH_FULL_IMAGE:figures/full_fig_p026_2_12.png]
Figure 2.13
Figure 2.13. Figure 2.13: Two vertices connected by an edge can be cancelled, re￾sulting in a simpler foam with the same evaluation. These, in turn, give rise to the following isomorphism Lemma 2.3. The following webs have isomorphic state spaces ∼ = Proof. The relations in Figures 2.12, 2.1…
Figure 2.14
Figure 2.14. Figure 2.14: Cobordisms that give mutually-inverse isomorphisms of state spaces for a web with a triangle and a contracted web. These identities follow by looking at possible colorings of foams αβ and βα. Colorings of αβ are in a natural bijection with colorings of the identity …
Figure 2.15
Figure 2.15. Figure 2.15: A decomposition of the identity endomorphism of the square web. We refer to [55, Proposition 2.24] for a proof. This relation, together with a relation in Remark 2.1 and a relation to simplify a tube with two parallel disks leads to the following decomposition. Lemm…
Figure 2.16
Figure 2.16. Figure 2.16: This web can be reduced using skein relations 2.1, 2.2 and 2.4. For a web Γ we define a Tait coloring c as a map c : e(Γ) −→ {1, 2, 3} from the set of edges of Γ to the 3-element set of colors so that the colors of the three edges at each vertex of Γ are distinct. I…
Figure 2.17
Figure 2.17. Figure 2.17: Skein relations on t(Γ). These relations mirror the above direct sum decompositions for the state spaces of webs. Induction on the complexity of a web Γ yields the following result [PITH_FULL_IMAGE:figures/full_fig_p028_2_17.png]
Figure 2.18
Figure 2.18. Figure 2.18: Left: foam for the generator x of the state space ⟨O⟩. Center and right: the identity element and monomial x 2 1x2 of ⟨Θ⟩. The state spaces are free graded modules over the ground ring R of graded ranks 1, [3] = q 2 + 1 + q −2 and [3][2] = (q 2 + 1 + q −2 )(q + q −1…
Figure 2.19
Figure 2.19. Figure 2.19: Graph of the dodecahedron is not reducible. Determining state spaces of non-reducible graphs is an open problem, see Boozer [17, 18]. One can propose the following problem. Problem 2.1. Is the state space ⟨Γ⟩ of any planar web Γ a free R-module ⟨Γ⟩ of rank t(Γ)? [P…
Figure 3.1
Figure 3.1. Figure 3.1: In-vertex and an out-vertex of an oriented SL(3) web. Webs were first considered by Kuperberg, who used them to develop a uniform ap￾proach to quantum invariants for Lie algebras of rank two [64]. For the root system A2 Kuperberg webs come from representation theory …
Figure 3.2
Figure 3.2. Figure 3.2: Left: a trivalent out-vertex, respectively in-vertex, repre￾sents a vector in the one-dimensional space of quantum sl3 invariants of V ⊗3 , respectively in InvUq(sl3)((V ∗ ) ⊗3 ). Right: vertical lines denote the identity endomorphism of V and V ∗ , depending on line…
Figure 3.3
Figure 3.3. Figure 3.3: Kuperberg skein relations for the SL(3) web invariant. We use a shortcut writing Γ in place of P(Γ). Relations in [PITH_FULL_IMAGE:figures/full_fig_p033_3_3.png]
Figure 3.4
Figure 3.4. Figure 3.4: Resolving crossings into linear combinations of webs in the Kuperberg invariant. to planar trivalent graphs, proves that such a web always has a region with at most 5 sides. Due to the parity constraint, such a region has 0, 2 or 4 sides, with a 0-sided region being …
Figure 3.5
Figure 3.5. Figure 3.5: Compatible orientations of facets near a seam of an oriented SL(3) foam. Next, consider a vertex of an unoriented foam, as in [PITH_FULL_IMAGE:figures/full_fig_p036_3_5.png]
Figure 3.6
Figure 3.6. Figure 3.6: A closed oriented foam. Singular circles (seams) are high￾lighted in blue. Two of the facets are shaded for convenience of pre￾sentation. Fixing an orientation along a singular circle determines the orientation for all facets and all circles, since this foam is conne…
Figure 3.7
Figure 3.7. Figure 3.7: Cobordism F merging Γ ⊔ Γ into Γ is indicated in grey. This cobordism is defined for any web Γ with a symmetry axis. F [PITH_FULL_IMAGE:figures/full_fig_p038_3_7.png]
Figure 3.8
Figure 3.8. Figure 3.8: Foam that represents the unit element in the ring ⟨Γ⟩, see also [PITH_FULL_IMAGE:figures/full_fig_p038_3_8.png]
Figure 3.9
Figure 3.9. Figure 3.9: Merging a circle into one of the three edges of the Θ-graph [PITH_FULL_IMAGE:figures/full_fig_p038_3_9.png]
Figure 3.10
Figure 3.10. Figure 3.10: The neck-cutting relation [PITH_FULL_IMAGE:figures/full_fig_p039_3_10.png]
Figure 3.11
Figure 3.11. Figure 3.11: Foam on bottom left represents a monomial x a 1x b 2x c 3 ∈ H∗ (Fl3). Foam in on the top left describes the trace map ε : H∗ (Fl3) −→ Z. Closed foam F Fa,b,c on the right evaluates to ε(x a 1x b 2x c 3 ) ∈ Z. A foam of the first type (connected surface, possibly wit…
Figure 3
Figure 3. Figure 3 [PITH_FULL_IMAGE:figures/full_fig_p041_3.png]
Figure 3.13
Figure 3.13. Figure 3.13: Skein relation for the Reshetikhin-Turaev quantum SL(3) invariant, a.k.a. the Kuperberg link invariant, and its normalization. This invariant is also a specialization of the HOMFLYPT 2-variable in￾variant, via (t, q) 7→ (q 3 , q). = = {−2} 0 { 3} 0 {−3} { 2} −1 0 0 …
Figure 3.14
Figure 3.14. Figure 3.14: Converting the crossing into its web resolutions and form￾ing the cone of the map between corresponding complexes induced by the foam F or F. resolutions consists of two arcs, the other is a web with two trivalent vertices. There are 2n possible resolutions of D int…
Figure 3
Figure 3. Figure 3: and its reflection [PITH_FULL_IMAGE:figures/full_fig_p044_3.png]
Figure 3.15
Figure 3.15. Figure 3.15: Foam F between two adjacent resolutions Γ0, Γ1 of D induces the map ⟨F⟩ : ⟨Γ0⟩ −→ ⟨Γ1⟩. Foam F is the reflection of F in a horizontal plane, inducing the map ⟨F⟩ : ⟨Γ1⟩ −→ ⟨Γ0⟩. Summing over all resolutions of the remaining crossings gives corresponding maps between…
Figure 4.1
Figure 4.1. Figure 4.1: An example of Γ′ ∈ Webϵ ′ ϵ and its composition with Γ ∈ Webϵ , which results in Γ′Γ ∈ Webϵ ′ . Here ϵ = (+, +, −, −, −, +) and ϵ ′ = (−, −, +, +). The composition web is simplified using the second skein relation in [PITH_FULL_IMAGE:figures/full_fig_p048_4_1.png]
Figure 4.2
Figure 4.2. Figure 4.2: For ϵ = ϵ ′ = (+, +), Webϵ ′ ϵ is a free Z[q, q−1 ]-module with a two-element basis B ++ ++ = {W1, W2}. Category Web is monoidal. The monoidal structure is given on objects by concate￾nating two sign sequences, ϵ1 ⊗ ϵ2 := ϵ1 ⊔ ϵ2 . The tensor product of webs is given…
Figure 4.3
Figure 4.3. Figure 4.3: An example illustrating the monoidal structure for webs ⊗ : Webϵ ′ 1 ϵ1 ⊗ Webϵ ′ 2 ϵ ′ 2 → Webϵ ′ 1⊔ϵ ′ 2 ϵ1⊔ϵ2 . Category Web is pivotal. This means that each object has a dual, with the dual of a sequence ϵ given by the sequence ϵ ∗ which is ϵ written in the opposi…
Figure 4.4
Figure 4.4. Figure 4.4: Action of duality morphisms given by pivotal structure on Webϵ ′ ϵ for ϵ = (−, −), ϵ ′ = (+, +, −). Here duality morphisms are emphasized in blue, and the resulting web gives an element of Webϵ ′ ϵ ∗ . Using pivotal structure one can rotate webs in any direction. Rem…
Figure 4.5
Figure 4.5. Figure 4.5: An example of a web in the lower-plane R 2 − with boundary ϵ = (+, −, +, +, +, −, +). This is a non-elliptic web with an inner region. Essentially the same web in the disk D2 is bounded by the gray circle. Category Web is braided, with the braiding structure on pairs…
Figure 4.6
Figure 4.6. Figure 4.6: An example of an oriented (4, 4)-tangle T ∈ Tan with ∂0T = (−, +, −, +) and ∂1T = (+, −, −, +). There is a functor Ku : Tan −→ Web which is the identity on objects. To define it on a morphism (tangle) T, project T generically from R 2 × [0, 1] onto the strip R × [0, …
Figure 4.7
Figure 4.7. Figure 4.7: An example of a graph ΓjΓi , which is a gluing of two objects in B ϵ , here ϵ = (+, +, −, −). As an intermediate object, for a balanced sequence ϵ introduce the category Hϵ . Objects of Hϵ are non-elliptic webs Γ with boundary conditions ∂1Γ = ϵ and ∂0Γ = ∅. We denot…
Figure 4.8
Figure 4.8. Figure 4.8: Elements Γ1, Γ2 of B ϵ , where ϵ = (+, +, −, −). It is straightforward to convert webs in the lower half-plane to webs in a disk, see Remark 4.1 and [PITH_FULL_IMAGE:figures/full_fig_p053_4_8.png]
Figure 4.9
Figure 4.9. Figure 4.9: Foam F ∈ HomHϵ (Γ1, Γ2), where Γ1, Γ2 are highlighted in blue, singular saddle is highlighted in gray. This foam is homeomorphic rel boundary to the foam considered in [PITH_FULL_IMAGE:figures/full_fig_p054_4_9.png]
Figure 4
Figure 4. Figure 4: on the left. In general, for a foam [PITH_FULL_IMAGE:figures/full_fig_p054_4.png]
Figure 4.11
Figure 4.11. Figure 4.11: State spaces assigned to pairs of generating objects Γ1, Γ2 for ϵ = (+, +, −, −), all with the shift {4} from (25). Γ2Γ1 ∂F Γ2 Γ1 [PITH_FULL_IMAGE:figures/full_fig_p055_4_11.png]
Figure 4.10
Figure 4.10. Figure 4.10: Identification of ∂F with Γ2Γ1. For this sequence the category has only two objects Γ1, Γ2, see [PITH_FULL_IMAGE:figures/full_fig_p055_4_10.png]
Figure 4.12
Figure 4.12. Figure 4.12: Multiplication of elements ⟨Γ1Γ1⟩ and ⟨Γ1Γ2⟩. Category Hϵ has finitely many objects, and it is a pre-additive category. The sum of its hom spaces, over all possible pairs of objects, is an associative ring. Namely, define the graded ring Hϵ by (27) Hϵ := M Γi,Γj∈Bϵ …
Figure 4.13
Figure 4.13. Figure 4.13: Γ1 Γ1 ∅ [PITH_FULL_IMAGE:figures/full_fig_p057_4_13.png]
Figure 4.14
Figure 4.14. Figure 4.14: An example of H(U) for ϵ = (−, −, −) and ϵ ′ = (−, +). For these sequences diagrams Γi , Γj are unique, and H(U) is the state space of the single web shown on the right and called the Θ-web, with the grading shifted up by 3. In other words, we complete U to a closed…
Figure 4.15
Figure 4.15. Figure 4.15: Example of a cobordism S from U0 to U1, where U0, U1 ∈ Wϵ ′ ϵ for ϵ = ϵ ′ = (+, −). To S we assign a homomorphism H(S) of (Hϵ ′ , Hϵ )-bimodules. To define it, recall that a web with boundary U is converted into closed webs ΓjUΓi to get the corresponding bimodule, s…
Figure 4.16
Figure 4.16. Figure 4.16: Left-hand side represents manifold M as a cobordism in HomZ(K)(N1, N2), right-hand side represents manifold M as an element of the k-vector space Z(N2N1). In the category Z(K), it seems natural to further restrict to objects N with ∂N ∼= K such that each connected c…
Figure 5.1
Figure 5.1. Figure 5.1: Reshetikhin-Turaev invariant f(T). This is the Reshetikhin-Turaev construction [100], which is a (braided monoidal) functor from the category OFTang of oriented framed tangles, with components labelled by irreducible representations Vλ of g, to the category Rep(Uq(g)…
Figure 5
Figure 5. Figure 5: on the left, but not in a [PITH_FULL_IMAGE:figures/full_fig_p069_5.png]
Figure 5.2
Figure 5.2. Figure 5.2: Vertices of MOY graphs, where thick lines split and merge. representations of quantum groups: Λ a+b q (V ) −→ Λ a q (V ) ⊗ Λ b q (V ), Λ a q (V ) ⊗ Λ b q (V ) −→ Λ a+b q (V ) [PITH_FULL_IMAGE:figures/full_fig_p069_5_2.png]
Figure 5.3
Figure 5.3. Figure 5.3: Two types of vertices in SL(N) MOY webs, with all orien￾tations either in or out. The case N = 3, restricted to webs with all edges of weight 1, reproduces webs for the Kuperberg invariant [64] and foam evaluation for its categorification in Section 3. A crossing dec…
Figure 5.4
Figure 5.4. Figure 5.4: Decomposition of an (a, b)-crossing into a q-linear combi￾nation of planar MOY webs. Normalization of the unknot is chosen as in [PITH_FULL_IMAGE:figures/full_fig_p070_5_4.png]
Figure 5.5
Figure 5.5. Figure 5.5: A circle of thickness a evaluates to the quantum binomial coefficient, also see (17) for the definition of quantum integer [n] [PITH_FULL_IMAGE:figures/full_fig_p070_5_5.png]
Figure 5.6
Figure 5.6. Figure 5.6: Three types of points of a GL(N) foam, left to right: regular points, points along a seam, and vertices. ure 5.8 and the middle picture in [PITH_FULL_IMAGE:figures/full_fig_p071_5_6.png]
Figure 5.7
Figure 5.7. Figure 5.7: Associativity diagram for merging three strands of thick￾ness a, b, c into a single strand of thickness a + b + c [PITH_FULL_IMAGE:figures/full_fig_p071_5_7.png]
Figure 5.8
Figure 5.8. Figure 5.8: Orientations of three facets along a seam of a GL(N) foam. Similar to SL(3) foams, dots on facets of a GL(N) foam are allowed. A dot d on a facet f of thickness a carries a homogeneous symmetric functions in a variables (42) Pd ∈ Z[y1, . . . , ya] Sa . Homogeneity co…
Figure 5.9
Figure 5.9. Figure 5.9: Left: the flow rule along the seam. Right: an example of a flow along the seam, where a = 2, c(f1) = {2, 5} and b = 1, c(f2) = {4}. oriented closed surface. It is embedded in R 3 since F ⊂ R 3 . Being a closed surface in F ⊂ R 3 , it has even Euler characteristic χ(F…
Figure 5.10
Figure 5.10. Figure 5.10: The second diagram in Figure 5.10 shows the same cross-section viewed [PITH_FULL_IMAGE:figures/full_fig_p074_5_10.png]
Figure 5.10
Figure 5.10. Figure 5.10: Positive and negative types of seam circles on the 2- colored subfoam Fi(c) ∪ Fj (c) ⊂ F, where i < j. Central dot denotes orientation of the singular circle pointing in the direction outside of the plane of the figure, while a cross denotes the opposite orientation…

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