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SU(3) instanton homology for webs and foams

Peter B. Kronheimer, Tomasz S. Mrowka

TL;DR

This work extends KM-Tait’s SU(2)-based framework to SU(3), building a bifold SU(3) instanton homology L^# for webs and foams that is functorial under cobordisms and equipped with skein exact triangles and an octahedral diagram. A key achievement is showing that for planar webs K in R^3, the dimension of L^#(S^3,K) equals the number of Tait colorings, while the Euler characteristic χ(L^#_*(K)) equals the Yamada polynomial evaluated at 1, i.e., χ(L^#_*(K)) = 𝒀(K) = R[K](1), up to a sign depending on the number of vertices. The theory introduces an edge-decomposition, absolute Z/2 gradings (via framings), and a trifold atom to avoid reducibles, enabling computations for classic webs (unknots, the theta web, the tetrahedron) and nonplanar graphs (K_{3,3}, Kinoshita theta). Connections to Khovanov–Rozansky homology are discussed, including potential spectral sequences and deformations, and the framework generalizes to TH-based atoms with altered cubic relations. Overall, the paper provides a robust SU(3) gauge-theoretic paradigm for planar vs nonplanar graphs, linking topology, representation varieties, and combinatorial colorings in a unified Floer-theoretic setting.

Abstract

An instanton homology is constructed for webs and foams, using gauge theory with structure group SU(3), adapting previous work of the authors for the SO(3) case. Skein exact triangles are established, and using an eigenspace decomposition arising from operators associated to the edges, it is shown that the dimension of the SU(3) homology counts Tait colorings when the web is planar. Unlike the SO(3) case, the SU(3) homology is mod-2 graded. Its Euler characteristic can be interpreted as a signed count of Tait colorings, or equivalently as the value at 1 of the Yamada polynomial invariant. Some examples and variants of the construction are also discussed.

SU(3) instanton homology for webs and foams

TL;DR

This work extends KM-Tait’s SU(2)-based framework to SU(3), building a bifold SU(3) instanton homology L^# for webs and foams that is functorial under cobordisms and equipped with skein exact triangles and an octahedral diagram. A key achievement is showing that for planar webs K in R^3, the dimension of L^#(S^3,K) equals the number of Tait colorings, while the Euler characteristic χ(L^#_*(K)) equals the Yamada polynomial evaluated at 1, i.e., χ(L^#_*(K)) = 𝒀(K) = R[K](1), up to a sign depending on the number of vertices. The theory introduces an edge-decomposition, absolute Z/2 gradings (via framings), and a trifold atom to avoid reducibles, enabling computations for classic webs (unknots, the theta web, the tetrahedron) and nonplanar graphs (K_{3,3}, Kinoshita theta). Connections to Khovanov–Rozansky homology are discussed, including potential spectral sequences and deformations, and the framework generalizes to TH-based atoms with altered cubic relations. Overall, the paper provides a robust SU(3) gauge-theoretic paradigm for planar vs nonplanar graphs, linking topology, representation varieties, and combinatorial colorings in a unified Floer-theoretic setting.

Abstract

An instanton homology is constructed for webs and foams, using gauge theory with structure group SU(3), adapting previous work of the authors for the SO(3) case. Skein exact triangles are established, and using an eigenspace decomposition arising from operators associated to the edges, it is shown that the dimension of the SU(3) homology counts Tait colorings when the web is planar. Unlike the SO(3) case, the SU(3) homology is mod-2 graded. Its Euler characteristic can be interpreted as a signed count of Tait colorings, or equivalently as the value at 1 of the Yamada polynomial invariant. Some examples and variants of the construction are also discussed.
Paper Structure (65 sections, 70 theorems, 214 equations, 23 figures)

This paper contains 65 sections, 70 theorems, 214 equations, 23 figures.

Table of Contents

  1. Introduction
  2. Statement of a result
  3. Outline
  4. Acknowledgements.
  5. Bifolds and SU(3) instantons
  6. Orbifolds and bifolds
  7. Bifold bundles and connections
  8. The configuration space
  9. Classification of bifold bundles
  10. The inclusion of SO(3)
  11. Anti-self-dual bifold connections
  12. Uhlenbeck compactness
  13. Instanton homology for bifold connections
  14. Atoms and a trifold
  15. Defining instanton homology
  16. ...and 50 more sections

Key Result

Theorem 1.1

If $K\subset \mathbb{R}^{2} \subset S^{3}$ is a planar web, then the dimension of the $\mathop{\mathit{SU}}\nolimits(3)$ instanton homology $L^{\sharp}(S^{3},K)$, as a vector space over the field $\mathbb{F}$ of two elements, is equal to the number of Tait colorings of $K$.

Figures (23)

  • Figure 1: The tetrahedron.
  • Figure 2: The octahedral diagram.
  • Figure 3: The non-trivial part of the composite cobordism $b \cup t \cup q$ (portrayed schematically because it is not embedded in $\mathbb{R}^{3}$). The gray dots are vertices of the web $L_{0}$. The shaded disk $\Delta_{0}$ is part of the foam. The hatched disk $\Delta_{1}$ is not. The picture continues periodically with period $3$ in both directions.
  • Figure 4: Two isomorphic foams $W'_{2}$ (left) and $W"_{2}$ (right), each with boundary the unlink $U_{2}$, resulting from the two decompositions of $W$.
  • Figure 5: The web $J$ arising as the boundary of $\check B_{21}$, for the chain homotopy $k_{0}$.
  • ...and 18 more figures

Theorems & Definitions (135)

  • Theorem 1.1
  • Remark
  • Remark
  • Proposition 2.1
  • proof
  • Corollary 2.2
  • Remark
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • ...and 125 more