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Deep Learning the Hyperbolic Volume of a Knot Physics Letters B 799 (2019) 135033 Contents lists available at ScienceDirect Physics Letters B www.elsevier.com/locate/physletb Deep learning the hyperbolic volume of a knot ∗ Vishnu Jejjala a,b, Arjun Kar b, , Onkar Parrikar b,c a Mandelstam Institute for Theoretical Physics, School of Physics, NITheP, and CoE-MaSS, University of the Witwatersrand, Johannesburg, WITS 2050, South Africa b David Rittenhouse Laboratory, University of Pennsylvania, 209 S 33rd Street, Philadelphia, PA 19104, USA c Stanford Institute for Theoretical Physics, Stanford University, Stanford, CA 94305, USA a r t i c l e i n f o a b s t r a c t Article history: An important conjecture in knot theory relates the large-N, double scaling limit of the colored Jones Received 8 October 2019 polynomial J K ,N (q) of a knot K to the hyperbolic volume of the knot complement, Vol(K ). A less studied Accepted 14 October 2019 question is whether Vol(K ) can be recovered directly from the original Jones polynomial (N = 2). In this Available online 28 October 2019 report we use a deep neural network to approximate Vol(K ) from the Jones polynomial. Our network Editor: M. Cveticˇ is robust and correctly predicts the volume with 97.6% accuracy when training on 10% of the data. Keywords: This points to the existence of a more direct connection between the hyperbolic volume and the Jones Machine learning polynomial. Neural network © 2019 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license 3 Topological field theory (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP . Knot theory 1. Introduction The special value N = 2, corresponding to a Wilson loop in the fundamental representation, recovers the Jones polynomial. While Identifying patterns in data enables us to formulate questions the Jones polynomial is defined for any knot, some invariants ex- that can lead to exact results. Since many of these patterns are ist only for subsets of knots. An example of such an invariant is subtle, machine learning has emerged as a useful tool in discover- the hyperbolic volume of the knot’s complement, denoted Vol(K ). ing these relationships. In this work, we apply this idea to invari- It is defined only if the manifold obtained by drilling out the knot ants in knot theory. 3 from its 3-dimensional ambient space admits a complete hyper- A knot is an embedding of a circle in the 3-sphere S . These bolic structure. The vast majority of knots are hyperbolic [32], and objects play important roles in a wide range of fields including par- we will restrict our attention to this case. An important open prob- ticle physics, statistical mechanics, molecular biology, chemistry, lem in knot theory is to establish a conjecture that relates J K ,N (q) sailing, and art [34,21,31,17,3,19]. Fig. 1 depicts several well known to Vol(K ). The volume conjecture [22,28,15] asserts that knots. Knot invariants, which distinguish knots from each other, are independent of how a knot is drawn on the plane (the knot 2π i 2π log | J K ,N (e N )| diagram). Determining relationships between these invariant quan- lim = Vol(K ). (1) N→∞ N tities is a central theme of knot theory. See Appendix A for a brief overview of the invariants discussed in this work. Perhaps the most The main idea of the volume conjecture is that the colored Jones famous invariant of a knot K is the Jones polynomial J K (q), which polynomial in the large color limit contains information about the is a Laurent polynomial with integer coefficients. The original def- volume of K . One might wonder if this property of the colored Jones poly- inition of the Jones polynomial was combinatorial [20], but an in- nomials extends to the original Jones polynomial. Evaluating the trinsically geometric definition and generalization was discovered Jones polynomial at q =−1, there is a surprising approximately soon thereafter [34]. The generalizations found in [34]are known linear relationship between log | J K (−1)| and Vol(K ), but this cor- as “colored” Jones polynomials and represent a family of knot in- relation seems to apply only to a particular class of knots [13]. variants J K ,N (q) labeled by a positive integer N called the color. Additionally, the so-called “volume-ish” theorem [12]gives upper and lower bounds on Vol(K ) in terms of certain coefficients ap- pearing in J q . An improved relationship is achieved [25]byre- Corresponding author. K ( ) * − E-mail addresses: [email protected] (V. Jejjala), [email protected] placing J K ( 1) with the reduced rank of the Khovanov homology, (A. Kar), [email protected] (O. Parrikar). a homology theory HK whose graded Euler characteristic is J K (q). https://doi.org/10.1016/j.physletb.2019.135033 0370-2693/© 2019 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3. 2 V. Jejjala et al. / Physics Letters B 799 (2019) 135033 Fig. 1. From left to right: the unknot, trefoil knot, figure-eight knot, and cinquefoil knot. When strands of the knot cross, the diagram keeps track of which strand is on top and which strand is on the bottom, so the diagram captures all information about the 3-dimensional embedding. For the figure-eight knot, the Jones polynomial in our −2 −1 2 conventions is J figure-eight(q) = q − q + 1 − q + q , and the hyperbolic volume is approximately 2.02988. Image taken from Wikimedia Commons. j j The cost of this improvement is that the Khovanov homology is a where Wθ and bθ are the weight matrices and bias vectors, re- much more refined invariant of K than the Jones polynomial, and spectively, of the jth hidden layer and the summation simply adds one needs to work much harder to compute it [24]. The most op- up the components of the output vector. The input layer is padded timistic interpretation of these results is that there is a nonlinear with zeroes so that the vectors are of uniform length. In our case, relation A mapping J K (q) to Vol(K ) along the lines of Eq. (1), but the inputs are vectors of length 18. The hidden layers have 100 perhaps not quite as simple to write. In this report, we provide ev- neurons each, and the final output layer is a summation over the 1 idence for this idea by directly estimating A using a simple two output of the second hidden layer. In the language of Eq. (3), Wθ is hidden layer fully connected neural network. × 1 2 × a 100 18 matrix, bθ is a length 100 vector, Wθ is a 100 100 ma- 2 trix, and bθ is a length 100 vector, all with variable entries that are 2. Setup and result determined by training the network on data. For data, we use a ta- ble of Jones polynomials and hyperbolic volumes for 313,209 knots A neural network is a function which is constructed by train- obtained from the online databases Knot Atlas [1] and SnapPy [10]. = ing on several examples. Suppose that we have a dataset D This includes all hyperbolic knots in Knot Atlas up to 15 crossings. { } J1, J2, ..., Jm , and to every element of D, there is an associated We implement and train fθ in Mathematica 11 [2]using built in element in another set S: functions that are completely unoptimized for the problem under consideration. The loss function is proportional to the squared er- :{ }→{ }⊂ A J1, J2,..., Jm v1, v2,...,vm S . (2) ror in volume, and parameters are adjusted by stochastic gradient descent. We follow the usual protocol for neural network train- In our case, the J i are the Jones polynomials of knots, and the 1 ing [27]: the network is shown a set of training data, and loss vi are the volumes of those knots. Aneural network fθ is a function gradients on this set are used to adjust the network pa- function (with an a priori chosen architecture) which is designed rameters θ via backpropagation. to approximate the associations A efficiently; the subscript θ de- With this architecture, our network performs significantly bet- notes the internal parameters, called weights and biases, on which ter than any previous method of estimating Vol(K ) from J K (q). Its the neural network depends. In order for the network to learn simplicity and robustness suggest the existence of an almost exact A, we divide the dataset D into two parts: a training set, T = nonlinear relationship between the two invariants which is more { J1, J2, ..., Jn} chosen at random from D, and its complement, c ={ } complicated than Eq. (1), but not by much. T J1, J2, ..., Jm−n . The neural network is taught the associ- In Fig. 2, we show a plot of the accuracy of our trained model ations on the training set by tuning the internal parameters θ to compared with the volume-ish bounds and Khovanov homology approximate A as closely as possible on T . In general, fθ ( J i) = vi without overfitting the data. We must instead minimize a suitably rank methods. Random selection within the volume-ish bounds chosen loss function that captures the difference between the two. leads to an enormous error (Fig. 2a). This is because the volume- Finally, we assess the performance of the trained network by ap- ish bounds are fairly loose, and the range of allowed volumes is ∈ c wide enough that large errors become unavoidable due to random plying it to the unseen inputs J i T and comparing fθ ( J i) to the = true answers vi A( J i).
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