Classifying the topology of closed curves is a central problem in low dimensional topology with applications beyond mathematics spanning protein folding, polymer physics and even magnetohydrodynamics. The central problem is how to determine whether two embeddings of a closed arc are equivalent under ambient isotopy. Given the striking ability of neural networks to solve complex classification tasks, it is therefore natural to ask if the knot classification problem can be tackled using Machine Learning (ML). In this paper, we investigate generic shortcut methods employed by ML to solve the knot classification challenge and specifically discover hidden non-topological features in training data generated through Molecular Dynamics simulations of polygonal knots that are used by ML to arrive to positive classifications results. We then provide a rigorous foundation for future attempts to tackle the knot classification challenge using ML by developing a publicly-available (i) dataset, that aims to remove the potential of non-topological feature classification and (ii) code, that can generate knot embeddings that faithfully explore chosen geometric state space with fixed knot topology. We expect that our work will accelerate the development of ML models that can solve complex geometric knot classification challenges.
Knot theory, a topic within low-dimensional topology, concerns the study of embeddings of the form S 1 → R 3 , knots. Knots are said to be equivalent if one embedding can be transformed into another under ambient isotopy, i.e. a smooth deformation of the embedding without breaking the curve or passing through oneself. Ambient isotopies can be realized diagrammatically through Reidemeister moves, such that two embeddings are said to be topologically equivalent if they can be transformed into each other through a series of Reidemeister moves [1].
The challenge of classifying knots up to topological equivalence dates back to Peter Guthrie Tait who was inspired by Lord Kelvin’s idea of knotted vortex atoms [2]. In current tabulations, different knots are labelled as P Q , where P denotes the minimal number of crossings in any projection of a knot, and Q denotes a (somewhat arbitrary) order within knots with same crossing number. Beyond 10 crossings, an additional labelling is conventionally added to denote properties such as alternating, a, or non-alternating, n, giving P nQ (Fig. 1A).
Knots are classified by topological invariants, i.e. quantities that can be computed on 3D embeddings or their 2D projections, and that are invariant under ambient isotopy, such as Reidemeister moves. However, finding a “complete invariant”, i.e. one that can uniquely classify any two knots, remains an open problem in mathematics [1]. Indeed, mathematicians have developed several invariants spanning both geometric and topological construction, for example the Jones polynomial [3], HOMFLY-PT polynomial [4], hyperbolic volume [5], and Vassiliev invariants [6]. However, each invariant developed thus far is not known to be able to uniquely identify every knot, i.e. it is so far always possible to find two topologically distinct knots that share the same topological invariant. An enlightening example is the comparison of the unknot (0 1 ) with the Conway knot (11 n34 ), drawn in Figure 1: the unknot cannot be smoothly deformed into the 11 crossing Conway knot, however, they both have the same (trivial) Alexander polynomial.
In the last ∼10 years, machine learning (ML) has become an ideal tool to learn complex patterns and solve classification tasks. To do so, an ML model learns a function f from labeled data X (j) = {x 1 , …, x i }, y (j) , such that f : X (j) → y (j) . ML techniques therefore lend themselves to be applied to problems in knot theory and have shown some preliminary success in both classification and conjecture generation [7][8][9][10][11].
In addition to the mathematical challenge of classifying knots based on their embedding, knots in biophysical systems such as DNA, proteins, polymers, or fields display additional physical constraints that are manifested through the specific geometric embedding of the knotted object [12,13]. Importantly, some geometric features such as protein folds, are important for the function of such biophysical molecule [14]. It is thus intriguing to conjecture that physical embeddings have properties which are reflected by the underlying topology of the object. The hope is that we can discover quantities that connect functionally important geometric features to structurally informative topological motifs in knotted biological and physical matter [14].
Motivated by this idea, several papers have proposed ML models to analyse embeddings of polygonal knots using geometric features, such as coordinates of the discretized segments and other geometrically-inferred measurements FIG. 1: Shortcut learning of knot topology. A.) The unknot (0 1 ) and the Conway knot (11 34 ) have the same (trivial) Alexander polynomial. However one cannot be deformed into the other. B.) A sketch of a neural network taking a knot embedding coordinates as input and outputting a binary unknot versus trefoil (0 1 vs. 3 1 ) classification. C.) An example of a configurational landscape of a knot embedding. The two main directions here are “size” and “writhe”, i.e. the amount of self-crossing of the curve. Molecular Dynamics simulations intrinsically bias sampling towards low free energy embeddings and therefore generate knot conformations with narrow distributions of geometric properties. D.) Example of a training dataset in which the ML takes an obvious shortcut learning based on the size of embeddings. All the trivial knots (0 1 ) are smaller than the trefoils (3 1 ) in the dataset. When the ML is challenged with 0 1 that are large, and 3 1 that are small it leads to incorrect classification. Figure layout inspired from [20].
such as curvature and local writhe [15][16][17][18][19]. Despite the fact that these features are not topologically invariant, supervised ML models appeared to successfully solve knot classification tasks (with > 99% accuracy), in turn sparking the question of whether ML models are “learning” topological invariants from algebraic patterns within these nontopological, geometric features. The idea
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