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András Juhász

Publications and source records attributed to András Juhász.

At least 19 recordsLinked to original sources
arXiv2026

Computations of the slice genus and the unknotting number of links via machine learning

Links are disjoint unions of circles smoothly embedded in $S^3$. We use reinforcement learning and Bayesian optimisation to obtain new upper bounds on several link invariants that are not known to be algorithmically computable: the slice genus and the unknotting number for links, and the strong slice genus for algebraically split links. We also compute lower bounds using known invariants. Combining the upper and lower bounds, we obtain new exact values in many cases. Our unknotting agents can reproduce the non-additivity of the unknotting number for several counterexamples due to Brittenham and Hermiller, in some cases finding new unknotting trajectories.

math.GT↗
arXiv2026

Quantum braid Floer homology

Given a braid whose closure is a knot, we associate to it a doubly pointed Heegaard diagram with additional marked points that we call a quantum Heegaard diagram. This is obtained by surgery from the unified intersection model for the Alexander and Jones polynomials due to the first author. By translating the Alexander and quantum gradings from the disc model to the Heegaard diagram, we obtain a bifiltration on the hat knot Floer complex. The degree-$(0,0)$ component of the differential with respect to this bifiltration defines a triply graded invariant of the braid that we call quantum braid Floer homology. It is generally not invariant under Markov conjugation or stabilisation. It contains a distinguished non-zero element, and admits a spectral sequence to knot Floer homology. Furthermore, its graded Euler characteristic is a two-variable polynomial that specialises to the Alexander and Jones polynomials.

math.GT↗
arXiv2025

Data for Mathematical Copilots: Better Ways of Presenting Proofs for Machine Learning

The datasets and benchmarks commonly used to train and evaluate the mathematical capabilities of AI-based mathematical copilots (primarily large language models) exhibit several shortcomings and misdirections. These range from a restricted scope of mathematical complexity to limited fidelity in capturing aspects beyond the final, written proof (e.g. motivating the proof, or representing the thought processes leading to a proof). These issues are compounded by a dynamic reminiscent of Goodhart's law: as benchmark performance becomes the primary target for model development, the benchmarks themselves become less reliable indicators of genuine mathematical capability. We systematically explore these limitations and contend that enhancing the capabilities of large language models, or any forthcoming advancements in AI-based mathematical assistants (copilots or ``thought partners''), necessitates a course correction both in the design of mathematical datasets and the evaluation criteria of the models' mathematical ability. In particular, it is necessary for benchmarks to move beyond the existing result-based datasets that map theorem statements directly to proofs, and instead focus on datasets that translate the richer facets of mathematical research practice into data that LLMs can learn from. This includes benchmarks that supervise the proving process and the proof discovery process itself, and we advocate for mathematical dataset developers to consider the concept of "motivated proof", introduced by G. Pólya in 1949, which can serve as a blueprint for datasets that offer a better proof learning signal, alleviating some of the mentioned limitations.

cs.LG↗
arXiv2025

The unknotting number, hard unknot diagrams, and reinforcement learning

We have developed a reinforcement learning agent that often finds a minimal sequence of unknotting crossing changes for a knot diagram with up to 200 crossings, hence giving an upper bound on the unknotting number. We have used this to determine the unknotting number of 57k knots. We took diagrams of connected sums of such knots with oppositely signed signatures, where the summands were overlaid. The agent has found examples where several of the crossing changes in an unknotting collection of crossings result in hyperbolic knots. Based on this, we have shown that, given knots $K$ and $K'$ that satisfy some mild assumptions, there is a diagram of their connected sum and $u(K) + u(K')$ unknotting crossings such that changing any one of them results in a prime knot. As a by-product, we have obtained a dataset of 2.6 million distinct hard unknot diagrams; most of them under 35 crossings. Assuming the additivity of the unknotting number, we have determined the unknotting number of 43 at most 12-crossing knots for which the unknotting number is unknown.

math.GT↗
arXiv2024

On Fox's trapezoidal conjecture

We investigate Fox's trapezoidal conjecture for alternating links. We show that it holds for diagrammatic Murasugi sums of special alternating links, where all sums involved have length less than three (which includes diagrammatic plumbing). It also holds for links containing a large twist region, which we call twist-concentrated. Furthermore, we show some weaker inequalities between consecutive coefficients of the Alexander polynomial of an alternating 3-braid closure, and extend this to arbitrary alternating links. We then study an extension of the trapezoidal conjecture due to Hirasawa and Murasugi, which states that the stable length of the Alexander polynomial of an alternating link can be bounded from above using the signature. We estabilish this and determine when equality holds for diagrammatic Murasugi sums of special alternating knots where each sum has length less than three, and also for twist-concentrated 3-braids. Finally, we study the behavior of the Hirasawa-Murasugi inequality under concordance.

math.GT↗
arXiv2024

Examples of topologically unknotted tori

We show that certain smooth tori with group $\mathbb{Z}$ in $S^4$ have exteriors with standard equivariant intersection forms, and so are topologically unknotted. These include the turned 1-twist-spun tori in the 4-sphere constructed by Boyle, the union of the genus one Seifert surface of Cochran and Davis that has no slice derivative with a ribbon disc, and tori with precisely four critical points whose middle level set is a 2-component link with vanishing Alexander polynomial. This gives evidence towards the conjecture that all $\mathbb{Z}$-surfaces in $S^4$ are topologically unknotted, which is open for genus one and two. It is unclear whether these tori are smoothly unknotted, except for tori with four critical points whose middle level set is a split link. The double cover of $S^4$ branched along any of these surfaces is a potentially exotic copy of $S^2 \times S^2$, and, in the case of turned twisted tori, we show they cannot be distinguished from $S^2 \times S^2$ using Seiberg-Witten invariants.

math.GT↗
arXiv2024

New invariants for virtual knots via spanning surfaces

We define three different types of spanning surfaces for knots in thickened surfaces. We use these to introduce new Seifert matrices, Alexander-type polynomials, genera, and a signature invariant. One of these Alexander polynomials extends to virtual knots and can obstruct a virtual knot from being classical. Furthermore, it can distinguish a knot in a thickened surface from its mirror up to isotopy. We also propose several constructions of Heegaard Floer homology for knots in thickened surfaces, and give examples why they are not stabilization invariant. However, we can define Floer homology for virtual knots by taking a minimal genus representative. Finally, we use the Behrens-Golla $δ$-invariant to obstruct a knot from being a stabilization of another.

math.GT↗
arXiv2024

Stabilization distance bounds from link Floer homology

We consider the set of connected surfaces in the 4-ball with boundary a fixed knot in the 3-sphere. We define the stabilization distance between two surfaces as the minimal $g$ such that we can get from one to the other using stabilizations and destabilizations through surfaces of genus at most $g$. Similarly, we consider a double point distance between two surfaces of the same genus, which is the minimum over all regular homotopies connecting the two surfaces of the maximal number of double points appearing in the homotopy. To many of the concordance invariants defined using Heegaard Floer homology, we construct an analogous invariant for a pair of surfaces. We show that these give lower bounds on the stabilization distance and the double point distance. We compute our invariants for some pairs of deform-spun slice disks by proving a trace formula on the full infinity knot Floer complex, and by determining the action on knot Floer homology of an automorphism of the connected sum of a knot with itself that swaps the two summands. We use our invariants to find pairs of slice disks with arbitrarily large distance with respect to many of the metrics we consider in this paper. We also answer a slice disk analogue of Problem 1.105 (B) from Kirby's problem list by showing the existence of non-0-cobordant slice disks.

math.GT↗
arXiv2022

The signature and cusp geometry of hyperbolic knots

We introduce a new real-valued invariant called the natural slope of a hyperbolic knot in the 3-sphere, which is defined in terms of its cusp geometry. We show that twice the knot signature and the natural slope differ by at most a constant times the hyperbolic volume divided by the cube of the injectivity radius. This inequality was discovered using machine learning to detect relationships between various knot invariants. It has applications to Dehn surgery and to 4-ball genus. We also show a refined version of the inequality where the upper bound is a linear function of the volume, and the slope is corrected by terms corresponding to short geodesics that link the knot an odd number of times.

math.GT↗
arXiv2020

Transverse invariants and exotic surfaces in the 4-ball

Using 1-twist rim surgery, we construct infinitely many smoothly embedded, orientable surfaces in the 4-ball bounding a knot in the 3-sphere that are pairwise topologically isotopic, but not ambient diffeomorphic. We distinguish the surfaces using the maps they induce on perturbed sutured Floer homology. Along the way, we show that the cobordism map induced by an ascending surface in a Weinstein cobordism preserves the transverse invariant in knot Floer homology.

math.GT↗
arXiv2020

New Heegaard Floer slice genus and clasp number bounds

We define several concordance invariants using knot Floer homology which give improvements over known slice genus and clasp number bounds from Heegaard Floer homology. We also prove that the involutive correction terms of Hendricks and Manolescu give both a slice genus and a clasp number bound.

math.GT↗
arXiv2020

Knot cobordisms, bridge index, and torsion in Floer homology

Given a connected cobordism between two knots in the 3-sphere, our main result is an inequality involving torsion orders of the knot Floer homology of the knots, and the number of local maxima and the genus of the cobordism. This has several topological applications: The torsion order gives lower bounds on the bridge index and the band-unlinking number of a knot, the fusion number of a ribbon knot, and the number of minima appearing in a slice disk of a knot. It also gives a lower bound on the number of bands appearing in a ribbon concordance between two knots. Our bounds on the bridge index and fusion number are sharp for $T_{p,q}$ and $T_{p,q}\# \overline{T}_{p,q}$, respectively. We also show that the bridge index of $T_{p,q}$ is minimal within its concordance class. The torsion order bounds a refinement of the cobordism distance on knots, which is a metric. As a special case, we can bound the number of band moves required to get from one knot to the other. We show knot Floer homology also gives a lower bound on Sarkar's ribbon distance, and exhibit examples of ribbon knots with arbitrarily large ribbon distance from the unknot.

math.GT↗
arXiv2020

Concordance surgery and the Ozsváth-Szabó 4-manifold invariant

We compute the effect of concordance surgery, a generalization of knot surgery defined using a self-concordance of a knot, on the Ozsváth-Szabó 4-manifold invariant. The formula involves the graded Lefschetz number of the concordance map on knot Floer homology. The proof uses the sutured Floer TQFT, and a version of sutured Floer homology perturbed by a 2-form.

math.GT↗
arXiv2019

Distinguishing slice disks using knot Floer homology

We study the classification of slice disks of knots up to isotopy and diffeomorphism using an invariant in knot Floer homology. We compute the invariant of a slice disk obtained by deform-spinning, and show that it can be effectively used to distinguish non-isotopic slice disks with diffeomorphic complements. Given a slice disk of a composite knot, we define a numerical stable diffeomorphism invariant called the rank. This can be used to show that a slice disk is not a boundary connected sum, and to give lower bounds on the complexity of certain hyperplane sections of the slice disk.

math.GT↗
arXiv2019

Contact handles, duality, and sutured Floer homology

We give an explicit construction of the Honda--Kazez--Matić gluing maps in terms of contact handles. We use this to prove a duality result for turning a sutured manifold cobordism around, and to compute the trace in the sutured Floer TQFT. We also show that the decorated link cobordism maps on the hat version of link Floer homology defined by the first author via sutured manifold cobordisms and by the second author via elementary cobordisms agree.

math.GT↗
arXiv2018

Naturality and mapping class groups in Heegaard Floer homology

We show that all versions of Heegaard Floer homology, link Floer homology, and sutured Floer homology are natural. That is, they assign concrete groups to each based 3-manifold, based link, and balanced sutured manifold, respectively. Furthermore, we functorially assign isomorphisms to (based) diffeomorphisms, and show that this assignment is isotopy invariant. The proof relies on finding a simple generating set for the fundamental group of the "space of Heegaard diagrams," and then showing that Heegaard Floer homology has no monodromy around these generators. In fact, this allows us to give sufficient conditions for an arbitrary invariant of multi-pointed Heegaard diagrams to descend to a natural invariant of 3-manifolds, links, or sutured manifolds.

math.GT↗
arXiv2018

Spectral order for contact manifolds with convex boundary

We extend the Heegaard Floer homological definition of spectral order for closed contact 3-manifolds due to Kutluhan, Matić, Van Horn-Morris, and Wand to contact 3-manifolds with convex boundary. We show that the order of a codimension zero contact submanifold bounds the order of the ambient manifold from above. As the neighborhood of an overtwisted disk has order zero, we obtain that overtwisted contact structures have order zero. We also prove that the order of a small perturbation of a $2π$ Giroux torsion domain has order at most two, hence any contact structure with positive Giroux torsion has order at most two (and, in particular, a vanishing contact invariant).

math.GT↗