Gluons and Knots

arXiv:2610.10673 · hep-th, math-ph, math.GT, math.MP · Submitted 2026-10-07 · Read on arXiv

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Introduction to the show: ident: High Energy Physics: Theory. The day in high-energy theory: gravity, strings, fields and the questions left open.

Kai: We're looking at the paper: "Gluons and Knots".

Mira: Here's the gist of the paper. Gluons and Knots argues that gluon scattering in four dimensions can be rephrased as a state count involving knots to provide a connection to topological field theory and string theory.

Kai: The amplitude is constant in chambers of kinematic space, its resonances are located at walls, and its factorization is determined by wall-crossing.

Mira: Topological Interpretation of the Amplitude The scattering amplitude of a single minus helicity gluon with arbitrarily many positive helicity gluons is topological, constant in chambers of kinematic space, its resonances are located at walls and its factorization is determined by wall-crossing.

Kai: Yangian invariance dictates the amplitude to be determined by a 2d null Wilson loop that can be lifted into a positive and transverse 3d link.

Paper summary: Kai: So, we’re looking at this paper called "Gluons and Knots." It’s about how gluon scattering in four dimensions can be rephrased using knots. Mira, can you give us the core idea?

Mira: Well, the central thesis is that the scattering amplitude for a single minus helicity gluon with any number of positive helicity gluons is topological. It claims this amplitude stays constant within chambers of kinematic space, and its resonances happen at walls.

Lev: So it’s not a smooth function across all space, but it jumps discontinuously at these walls, right? That sounds like something that would be really hard to simulate on real hardware if you don't have the right structure.

Kai: Exactly. And what makes this connection interesting is how they factor the amplitude through wall-crossing. It suggests that instead of a complicated calculation, we might be looking at some kind of state counting involving knots to connect it to topological field theory and string theory <ref:2610.10673#pg1>.

Mira: They do this by saying the amplitude is determined by a two-dimensional null Wilson loop that gets lifted into a positive and transverse three-dimensional link. They then use wall-crossing and the Weinberg soft theorem to think about this as skein relations and Reidemeister moves of subknots <ref:2610.10673#pg2>.

Lev: That lift into three dee links, that’s where I get a bit nervous for hardware implementation <ref:2610.10673#pg1>. Translating a 2D polygon into actual physical links with positive and transverse orientations sounds like it introduces a lot of complexity in terms of orientation tracking and self-linking numbers <ref:2610.10673#pg1>.

Kai: Right, Lev. And they actually construct a state-counting description by adding the corners of their HOMFLY-PT polynomials, which is essentially what they call the corners for this construction <ref:2610.10673#pg1>. It’s a new way to count things that goes beyond the classical limit and connects to those string theory ideas.

Mira: So, it’s not just a math trick; it’s proposing that these gluon interactions have a structure based on knot topology, which is what makes this paper so significant for the theoretical side <ref:2610.10673#pg2>.

Lev: But how does this state counting actually translate into something we could measure? I mean, if we’re trying to build error correction for real hardware, we need concrete numbers on what’s being counted versus what's the overhead of the actual computation <ref:2610.10673#pg3>.

Kai: The paper does try to give some concrete results. They talk about studying a null Wilson polygon, which factorizes at cusps and self-intersections as imprints of unitarity <ref:2610.10673#pg2>. They then use this 2D polygon to lift to three dee links using its orientation, which allows them to use the Morton-Franks-Williams inequality to design a novel corner quantity that satisfies wall-crossing laws <ref:2610.10673#pg2>.

Paper summary: Mira: And these wall-crossing laws are treated like an equation of motion, which leads to three fundamental rules—W1, W2, and W3—that describe how the amplitude changes when you cross certain walls <ref:2610.10673#pg2>.

Lev: Those rules sound like constraints on the system's dynamics. W1 says a vertex crosses a nonadjacent edge, taking P- to P+ with a jump given by A(P+) minus A(P-) which equals chi A(QL)A(QR) <ref:2610.10673#pg2>. That jump term is what’s going to be tricky for error correction if it depends on those other terms <ref:2610.10673#pg2>.

Kai: And W2 says the corner at a vertex straightens, with the jump being A(Pj:right) minus A(Pj:left) which equals A(P j) <ref:2610.10673#pg2>. It’s showing how the structure of the polygon dictates local behavior <ref:2610.10673#pg3>.

Mira: And W3 at a generic triple wall means three pairwise nonadjacent edges become concurrent and the amplitude stays unchanged because A(P+) equals A(P-) <ref:2610.10673#pg2>. That invariance is what makes the topological interpretation so strong.

Lev: So, to put this in terms of running a computer, you’re essentially dealing with these constraints that dictate how the amplitude behaves as you move through different regions of parameter space <ref:2610.10673#pg2>. It sounds like a very rigid mathematical structure underneath the scattering process.

Kai: The paper then moves into the state sum formulation where they define an amplitude C(P) over uniform decompositions of the Wilson loop, using a corner coefficient c(Q) from the HOMFLY-PT polynomial <ref:2610.10673#pg1>. The final state sum is defined as C(P) equals negative one times X PS uniform negative one times S Y Q in PS c(Q) <ref:2610.10673#pg3>.

Mira: They find that for Catalan stars, the amplitude A(Kr) is found to be equal to negative one times r Catr <ref:2610.10673#pg3>. That’s a very specific result they derived from this state sum approach.

Lev: If we look at the numerical verification part, they match this with a closed form of the Berends-Giele recursion for Catalan stars <ref:2610.10673#pg2>. They show that A(Kr) equals negative one times r Catr, which means the amplitude can grow quickly with n equals two r plus one, saturating a bound A(Pn) less than or equal to Catr <ref:2610.10673#pg2>.

Kai: And they mention that this matches the knot formula where only the empty smoothing S equals zero survives for r greater than or equal to two <ref:2610.10673#pg2>. They also find that decay loops yield C(P)R1 equals negative one times l P plus k, uniform, zero otherwise <ref:2610.10673#pg2>.

Mira: The implication here is that this state sum C(P) is interpreted as an observable in topological field theory and a candidate for the topological string <ref:2610.10673#pg3>. They suggest that CˆP (a, z) is a motivic amplitude that might be obtained by promoting Feynman rules to a non-commutative background <ref:2610.10673#pg3>.

Paper summary: Lev: If it’s a candidate for topological string theory, we need to consider the assumptions they made about the underlying field theory structure, especially since they are connecting it beyond the classical limit <ref:2610.10673#pg1>. I wonder if that connection holds up when you try to map this onto a lattice or a quantum computer architecture.

Kai: That’s where Lev’s expertise comes in, though. The paper suggests that W3 topological invariance might be related to the Yang-Baxter condition, which would describe braiding <ref:2610.10673#pg3>. This braiding structure is what allows them to connect the wall-crossing calculus to those fundamental moves of subknots <ref:2610.10673#pg2>.

Mira: If W1 is loop splitting and W3 is topological invariance, then it implies a chiral resolution of that singular null Wilson loop <ref:2610.10673#pg3>. That would be the mechanism for how they handle the singularities in the diagrammatic formula <ref:2610.10673#pg2>.

Lev: From an engineering standpoint, if we take that chiral resolution idea and try to build a real system, we have to figure out how to implement those braiding operations or whatever structure is required by the Yang-Baxter condition <ref:2610.10673#pg3>. It seems like the complexity of that structure would land heavily on the error correction overhead <ref:2610.10673#pg2>.

Kai: The paper does have a limitation they state—that reproducing the gluon amplitude from knots may be counterintuitive because those topological tools are insensitive to cusps and don't self-intersect <ref:2610.10673#pg2>. That’s a hurdle for direct intuition.

Mira: They also mention that the construction relies on an assumption about the existence of a Lean-formalized dependency graph, detailing precise assumptions including Reidemeister III invariance <ref:2610.10673#pg1>. If that specific invariance doesn't hold in the physical theory, then their entire derivation based on those rules breaks down.

Lev: So it’s a rigorous mathematical mapping, but we have to trust that the underlying topological moves translate exactly into the physical dynamics they are modeling <ref:2610.10673#pg2>. It's a strong claim for anything involving non-perturbative physics <ref:2610.10673#pg3>.

Kai: To wrap up on this, "Gluons and Knots" is proposing a precise connection between gluon scattering in four dimensions and knot theory through state counting via HOMFLY-PT polynomials <ref:2610.10673#pg1>. It suggests that the amplitude behaves topologically with resonances at walls determined by wall-crossing <ref:2610.10673#pg2>.

Mira: It’s a conjecture for now, but it provides a way to approach topological field theory and string theory from the language of knots <ref:2610.10673#pg3>.

Lev: For someone just listening, what this means is that we have a new mathematical structure suggesting that complex gauge interactions might be governed by fundamental topological objects <ref:2610.10673#pg2>. It’s a deep connection between particle physics and topology.

Conclusion: Kai: So, to wrap up this discussion on "Gluons and Knots," we’re looking at how they connect gluon scattering to knot theory through state counting <ref:2610.10673#pg1>.

Mira: It boils down to the idea that these gluon interactions have a topological structure, constant in certain spaces, with resonances showing up at walls determined by wall-crossing phenomena <ref:2610.10673#pg2>.

Lev: From an engineering standpoint, we’re looking at how much overhead it takes for our real hardware to run these kinds of calculations involving those topological constraints <ref:2610.10673#pg2>.

Kai: The paper proposes that this state counting—using things like HOMFLY-PT polynomials—is a way to get a description that links particle physics directly to topological field theory and string theory <ref:2610.10673#pg3>.

Mira: They’re suggesting this state sum is an observable for topological field theory, maybe even the candidate for the topological string itself <ref:2610.10673#pg3>.

Lev: If that connection holds up, it means we might be able to see how those complex gauge interactions are governed by fundamental topological objects rather than just standard field equations <ref:2610.10673#pg2>.

Kai: The authors found some specific results for Catalan stars, matching the amplitude with a closed form from the Berends-Giele recursion <ref:2610.10673#pg2>.

Mira: That means for those specific cases, you get a very concrete formula for the amplitude that matches what they derived from counting knots <ref:2610.10673#pg3>.

Lev: It shows how the structure of the knot theory directly dictates the behavior of the gluon scattering process <ref:2610.10673#pg2>.

Kai: The authors are suggesting a path forward by looking at holomorphic Chern-Simons theory in complex twistor space <ref:2610.10673#pg3>.

Mira: In that picture, the wall-crossing rules translate into loop splitting and topological invariance, which might resolve those singularities in the null Wilson loop <ref:2610.10673#pg3>.

Lev: That sounds like we’d be looking at a very constrained system, where the dynamics are dictated by fundamental moves of subknots <ref:2610.10673#pg2>.

Kai: This work opens up a way to look at gluon scattering using the language of knots, suggesting deep links to string theory beyond what we see in the classical limit <ref:2610.10673#pg1>.

School of Natural Sciences, Institute for Advanced Study, Princeton

hep-th, math-ph, math.GT, math.MP

Submitted: 2026-10-07

Updated: 2026-10-07

Comments: 11 pages

Code: https://github.com/aguevara22/black-box-arena

Project page: https://aguevara22.github.io/GKSM.pdf

License: http://creativecommons.org/licenses/by/4.0/

Importance score: 91/100

The gist: Gluons and Knots argues that gluon scattering in four dimensions can be rephrased as a state count involving knots to provide a connection to topological field theory and string theory.

Key concepts

Topological Interpretation of the Amplitude: The scattering amplitude is treated as a topological quantity that is constant within certain kinematic regions called chambers. Its behavior at boundaries (walls) determines how it factorizes, which is governed by wall-crossing laws derived from knot theory.
Wall-Crossing Laws: These are three fundamental rules (W1, W2, W3) that describe how the amplitude changes when kinematic configurations cross specific boundaries. These rules are treated as equations of motion and dictate the relationship between different parts of the scattering process.
State Sum Formulation: The authors define a state sum description for the amplitude using uniform decompositions of Wilson loops. This involves calculating coefficients from HOMFLY-PT polynomials to construct a final amplitude formula that relates gluon scattering to knot invariants.

Terminology

Chern-Simons Theory — 8× in this paper · explained in 2 episodes
Configuration Space — 4× in this paper · explained in 2 episodes
Integrability — 4× in this paper · explained in 2 episodes
Equation of Motion — 3× in this paper
Convex Hull — 2× in this paper · explained in 2 episodes
Harness — 2× in this paper · explained in 2 episodes
Trace — 2× in this paper · explained in 2 episodes
Verification — 2× in this paper · explained in 2 episodes
Core — 1× in this paper · explained in 2 episodes
Decoupling — 1× in this paper · explained in 2 episodes

Summary

Gluons and Knots argues that gluon scattering in four dimensions can be rephrased as a state count involving knots to provide a connection to topological field theory and string theory. The amplitude is constant in chambers of kinematic space, its resonances are located at walls, and its factorization is determined by wall-crossing.

Topological Interpretation of the Amplitude

The scattering amplitude of a single minus helicity gluon with arbitrarily many positive helicity gluons is topological, constant in chambers of kinematic space, its resonances are located at walls and its factorization is determined by wall-crossing. Yangian invariance dictates the amplitude to be determined by a 2d null Wilson loop that can be lifted into a positive and transverse 3d link. By reinterpreting wall-crossing and the Weinberg soft theorem as skein relations and Reidemeister moves of the subknots, the authors construct a state-counting description consisting of adding the ‘corners’ of their HOMFLY-PT polynomials. This provides a conjectured but precise connection to topological field and string theory beyond the classical limit.

Derivation via Knots and Polygons

The amplitude is an integer function A of the 2d momentum spinors, constant in domains called chambers. The authors rely on unitarity, which leads to the relation An = ±Θ+ [λeL λeR] + continuous. They study a null Wilson polygon, which factorizes both at cusps and self-intersections as imprints of unitarity. The 2d polygon is lifted to 3d links using its orientation to construct positive and transverse knot diagrams. This lift provides a self-linking number and allows the use of the Morton-Franks-Williams (MFW) inequality to design a novel ‘corner’ quantity that satisfies wall-crossing laws.

Wall-Crossing Laws and Integrability

The wall-crossing calculus is treated as an Equation of Motion (EOM) satisfied by the amplitude, which leads to three fundamental rules. These laws are:

W1 (vertex–edge): A vertex crosses a nonadjacent oriented edge, taking P− to P+ with the jump given by A(P+) − A(P−) = χA(QL)A(QR).
W2 (flat): The corner at a vertex straightens, with the jump being A(Pj:right) − A(Pj:left) = A(P j).
W3 (triple): At a generic triple wall, three pairwise nonadjacent edges become concurrent and the amplitude remains unchanged, as illustrated by A(P+) = A(P−).

State Sum Formulation and Results

The paper introduces a state sum description where the amplitude C(P) is defined over uniform decompositions of the Wilson loop. The corner coefficient c(Q) is defined using the HOMFLY-PT polynomial H+Q, which satisfies mindegz H+Q ≥ 0. The final state sum is C(P) = (−1)l X PS uniform (−1)S Y Q∈PS c(Q). For Catalan stars, the amplitude is found to be A(Kr) = (−1)r Catr.

Connection to Topological String Theory

The state sum C(P) is interpreted as an observable of topological field theory and a candidate for the topological string. The contribution from each decomposition is related to expectation values of Wilson loops in 3d Chern-Simons theory, where the fundamental U(N) Wilson loop is given by the HOMFLY polynomial times the unknot expectation. The construction suggests that CˆP (a, z) is a ‘motivic amplitude’ and may be obtained by promoting Feynman rules to a non-commutative background. This links the gluon scattering problem to the topological string [2610.10673>.

Numerical Verification in Special Kinematics

Explicit results are derived for Catalan stars, where the amplitude is matched by a closed form of the Berends-Giele (BG) recursion. The result A(Kr) = (−1)r Catr shows that the amplitude can grow quickly with n = 2r + 1, saturating a bound A(Pn) ≤ Catr. This matches the knot formula (28), where only the empty smoothing S = 0 survives for r ≥ 2. The decay loops yield C(P)R1 = (−1)l(P)+k, uniform, 0, otherwise.

Future Directions and Conjectures

The work suggests that a sharper description could be holomorphic Chern-Simons theory in complex twistor space. In this picture, W1 would be loop splitting, W3 topological invariance (likely related to braiding/Yang-Baxter condition), and the positive lift a chiral resolution of the singular null Wilson loop. The correspondence is expected to be nonperturbative due to supersymmetry protecting the single-minus correspondence.

Acknowledgments

The authors thank N. Arkani-Hamed, F. Coronado, A. Lupsasca, J. Maldacena, G. Salvatori, D. Skinner and A. Strominger for many enlightening discussions. The structural form of equation (28) was originally devised by Black-Box-Arena, an automated AI research harness used to obtain new formulas in physics and mathematics. The author is supported in part by the Simons foundation. The paper includes references to numerous works in gauge theory, gravity, and knot theory such as AdS/CFT [1], celestial holography [2], and the integrability of N = 4 Super Yang-Mills [5]. The paper also cites work on M-theory and the BFSS model in relation to single-minus graviton amplitudes [12]. The extensive list of references supports the deep connections made between these fields. The paper is supported by funding from the Simons foundation and AI assistance from Anthropic, OpenAI and Google Deepmind. The paper is also supported by the Simons foundation. The paper includes references to work on knot invariants such as the HOMFLY–PT polynomial [28] and Lickorish–Millett factorization [35]. The paper is also supported by funding from the Simons foundation. The paper includes references to work on the spectral network construction of GMN for BPS states counting [47]. The paper is also supported by funding from the Simons foundation. The paper includes references to work on knot invariants such as the HOMFLY–PT polynomial [28] and Lickorish–Millett factorization [35]. The paper is also supported by funding from the Simons foundation. The paper includes references to work on knot invariants such as the HOMFLY–PT polynomial [28] and Lickorish–Millett factorization [35]. The paper is also supported by funding from the Simons foundation.

Improvements for AI systems

Improved AI can perform topological field theory calculations by directly applying knot invariants to gluon scattering amplitudes through a state sum description, as seen in equation (28). This allows for a reformulation that simplifies gauge dependent Feynman diagrams into fundamental constituent blocks and satisfies wall-crossing laws via the corner coefficient c(Q).
Improved AI systems will be able to calculate precise scattering amplitudes for single-minus gluons by determining the integer A(P) associated with a 2d null Wilson polygon P, which is constant in kinematic chambers. This capability relies on interpreting the amplitude as a state count over positive 3d links of knots.
Improved AI can solve complex combinatorial problems related to chamber structure by using the wall-crossing calculus described in section IV, where the rules (15), (16), and (17) govern how amplitudes jump across kinematic walls. This allows for the calculation of A(P) in any generic polygon P using a unique function C(P).
Improved AI can generate explicit numerical results for specific kinematics, such as Catalan stars, by solving the Berends-Giele recursion (equation 40). This provides an exact match between the knot theory formulation and the original gluon amplitude formula for certain configurations.
Improved AI will be able to interpret gauge theory observables in terms of topological string theory by identifying a motivic amplitude C(P) which is conjectured to be related to expectation values of Wilson loops in Chern-Simons theory. This bridges the gap between quantum field theory and topological string duality at large N limits.

Abstract

We argue that the scattering amplitude of a single minus helicity gluon with arbitrarily many positive helicity gluons is topological. The amplitude is constant in chambers of kinematic space, its resonances are located at walls and its factorization is determined by wall-crossing. Yangian invariance dictates the amplitude to be determined by a 2d null Wilson loop that can be lifted into a positive and transverse 3d link. By reinterpreting wall-crossing and the Weinberg soft theorem as skein relations and Reidemeister moves of the subknots, we construct a state-counting description consisting of adding the `corners' of their HOMFLY-PT polynomials. Beyond the classical limit, we argue that this provides a conjectured but precise connection to topological field and string theory. The new formula is rigorously derived via a Lean-formalized dependency graph (DAG) as detailed in a companion paper, laying out the precise assumptions including Reidemeister III invariance. It is further tested exhaustively, including novel `star' kinematics where the amplitude and state count are shown to become Catalan numbers, in agreement with knot theory literature.