Evaluating the with tensor networks

Konstantinos Meichanetzidis1 and Stefanos Kourtis2 1School of Physics and Astronomy, University of Leeds, Leeds LS2 9JT, United Kingdom 2Physics Department, Boston University, Boston, Massachusetts 02215, USA (Dated: September 16, 2019) We introduce tensor network contraction for the evaluation of the Jones polynomial of arbitrary . The value of the Jones polynomial of a is reduces to the partition function of a q-state anisotropic Potts model with complex interactions, which is defined on a planar signed graph that corresponds to the knot. For any integer q, we cast this partition function into tensor network form, which inherits the interaction graph structure of the Potts model instance, and employ fast tensor network contraction protocols to obtain the exact tensor trace, and thus the value of the Jones polynomial. By sampling random knots via a grid-walk procedure and computing the full tensor trace exactly, we demonstrate numerically that the Jones polynomial can be evaluated in time that scales subexponentially with the number of crossings in the typical case. This allows us to evaluate the Jones polynomial of knots that are too complex to be treated with other available methods. Our results establish tensor network methods as a practical tool for the study of knots.

I. INTRODUCTION tistical mechanical models [37]. The Jones polynomial, in particular, is related to a q-state classical Potts model is immensely interdisciplinary, with re- with Anisotropic Complex Interactions (PACI). [38, 39]. sults and open questions spanning many fields of science, Remarkably, the partition function Z(q) of PACI, which such as physics [1–8], quantum computation [9–12], quan- is defined on an irregular planar graph whose structure tum cryptography [13, 14], chemistry and biology [15– is defined by the topology of a knot K is essentially the 18], study of every day life knotting of strands [19], and Jones polynomial VK (t(q)) evaluated at complexity theory [20–23]. A key notion in knot the- 1 ory is that of a — a quantity extracted t(q) = (q + √q q 4 2) , (1) 2 from a knot K which changes only under topology non- − − p preserving knot operations, such as passing the knot up to a normalization. through itself or cutting and recombining its strand. The Partition functions of classical models are of great in- Jones polynomial VK (t)[24] — a Laurent polynomial in terest in condensed matter physics and powerful algo- t C — is one such invariant that pervades knot the- rithms have been developed to compute them, albeit ∈ ory. Knots K,K0 are distinct if VK (t) = VK0 (t). The mostly on graphs with periodic structure. Tensor net- Jones polynomial is thus pertinent to questions6 related work methods are an especially successful class of such to knottedness, such as the unknotting problem, a deci- techniques, which typically employ the renormalization- sion problem which is known to be in NP but unknown group procedure to efficiently approximate partition whether it lies in P [25]. Hence, in addition to being cen- functions of classical lattice models very accurately [40– tral to the aforementioned applications of knot theory, 46]. Recently, it was demonstrated that tensor network evaluating the Jones polynomial is also a fundamental contraction schemes can also be very fast in obtaining the computational problem. partition function of classical models exactly, even on un- Exact evaluation of the Jones polynomial is generally structured graphs with bounded degree and even when a #P-hard problem; computing VK (t) takes time that is the underlying computation is a #P-hard problem [47]. expected to scale exponentially with the number of cross- In this work, we exploit the connection with statistical ings in a knot. Exceptions to this occur for t restricted mechanics and the efficiency of tensor network methods to certain roots of unity, where VK (t) corresponds to to evaluate the Jones polynomial in the general #P-hard quantum amplitudes of a quantum field theory [2], un- case. Specifically, we introduce tensor network contrac- derstood as braiding of anyons [26]. In particular, for tion algorithms that can evaluate VK (t) at values of t 2πi/3 2πi/3 2 t = 1, i, e , (e ) , VK (t) can be evaluated away from the “easy” ones, yet achieve demonstrably efficiently± ± [27±]. Moreover,± quantum algorithms can ef- advantageous computation times that indicate subexpo- arXiv:1807.02119v2 [cond-mat.stat-mech] 13 Sep 2019 ficiently approximate the Jones polynomial at principal nential scaling as a function of the number of crossings roots of unity in both the conventional quantum circuit in K for the typical case. This affords us access to the model [28, 29] and the setting of topological quantum value of the Jones polynomial of knots with 6 to 10 times computation [30]. On the other hand, exponential classi- as many crossings as what has been previously achieved cal algorithms that yield the full expression for the Jones in the literature with other methods. Our work thus fur- polynomial [31–36] in the general case have been imple- nishes a useful numerical tool for the evaluation of an mented and are readily usable, but have a relatively small essential knot invariant. reach (up to 20 crossings). The rest of the paper is organised as follows. In sec- Many knot∼ invariants are intimately connected to sta- tionII we review the mathematical connection between 2

� � the Jones polynomial and the partition function of the − + PACI model. We follow with sectionIII where we present + − tensor network contraction methods for evaluating it along with convincing numerical evidence of favourable resource scaling. We conclude with sectionIV. + −

II. JONES POLYNOMIAL EVALUATION AS PARTITION FUNCTION OF PACI

We begin with a preliminary review of the relation be- tween the PACI partition function and the Jones polyno- mial, starting with the relevant knot theory terminology. A knot K consists of an embedding of the circle S1 in R3. A knot diagram is the projection of the knot to R2, where the information about which strand is over which at ev- ery crossing c is preserved. Intuitively, a knot diagram is what one produces when one attempts to draw a knot in two dimensions. Discarding the information about over and under crossing we obtain the knot shadow. A knot can be oriented by choosing a direction along the strand. There are two ways to do this but they are equivalent. Each crossing obtains a twist sign c accord- ing to the direction of the strands exiting the crossing; if the strands cross in a clockwise (counterclockwise) fash- ion then the crossing obtains a positive (negative) twist sign (see Fig.1(top-left)). The sum of all the twist signs is called , wK = c c, and characterises the knot chirality. The Jones polynomial is sensitive to the knot chirality as it can distinguishP mirrored knots. For any knot diagram, a planar graph G called the Tait graph is defined as follows. The two-dimensional regions defined by the knot diagram can be bicoloured with, say, black and white, so that no two adjacent re- gions share a colour. There are two ways to do this, and so we choose the convention that the unique unbounded region (background) is white. In Fig.1 we show an ex- ample of a knot K (middle) along with its bicolouring Figure 1. (Top) Twist (left) and Tait (right) signs for (bottom). Then, vertices v V , where V is the ver- crossings of an oriented grid walk. (Middle) Oriented ran- tex set of G, correspond to∈ the black regions. Edges dom knot diagram generated by a random grid walk for grid size L = 12. (Bottom) Bicoloured knot diagram, its unsim- c = (v, v0) E, where (v, v0) V V and E is the edge set of G∈, are such that they∈ connect× black regions plified Tait graph G composed of vertices (blue dots) asso- ciated with black regions, and edges (green) decorated with through the knot diagram crossings. The graph G thus ε = +/− (up/down red triangles) associated with crossings. obtained from K is shown in Fig.1 (bottom), where ver- c The corresponding tensor network GT comprises variable ten- tices are represented by blue dots and edges by green sors (blue dots) connected through q-dimensional green lines lines. The vertex degree is denoted dv and counts the with clause tensors (red triangles), pointing upwards (down- number of incident edges to that vertex. We have used wards) when representing a J+(J−) Potts interaction. This the same symbol, c, for crossings and the corresponding knot is the right-handed√ Trefoil and for q = 3√, 5 the Jones polynomial is V (t(3)) = i 3, V (t(5)) = 1 (3 + 5) + ( 1 (3 + edges. The edges are decorated by Tait signs εc which √ √ 2 2 3 1 4 are determined by the following rule. If the region to the 5)) − ( 2 (3 + 5)) , as confirmed by our . left (right) is black when exiting a crossing on the over strand, then the crossing obtains a positive (negative) Tait sign (see Fig.1(top-right)). The sum of all tait signs is called Tait number, τK = c εc. In Fig.1 (bottom), of a knot K [48, 49]. A Potts model is placed on the Tait the Tait signs are represented by red triangles decorating graph G by defining spins with q available states σv = P the edges, pointing up(down) for positive(negative). 0, . . . , q 1 to reside on the vertices v = 1, . . . , nv. The We now restate the relation between the q-state PACI Tait signs− ε = that decorate the edges of G determine c ± partition function Z(q) and the Jones polynomial VK (t) the interaction strength between spins, which take two 3

3 corresponding values J C. This rule which assigns 10 ± ∈ nc interactions between the q-state spins renders the Potts h i nv model anisotropic. Their relation with the Jones variable h i J± is e = t∓ and the Jones variable is determined by

− edges 2

fixing q N via Eq. (1). The Potts partition function / 10 over all spin-states∈ σ is { }

Z(q) = Tσv σv0 , (2)

σ (v,v0) #vertices 101 X{ } Y ε T = 1 (1 + t− c )δ . (3) σv σv0 − σv σv0 Multiplying the partition function Z(q) with the appro- 101 102 priate prefactor (q), which accounts for twists and en- A L sures that the returns V = 1, t C, we write 2.0 ∀ ∈ VK (t(q)) = (q)Z(q) , (4) A 1.8 1 1 ( nv 1) 3 wK 1 7.4 2 2 − − 4 4 τK (q) = ( t(q) t(q)− ) ( t(q) ) t(q) . i 7.0 v

A − − − 1.6 i n G h 6.6 / i h III. EXACT PACI PARTITION FUNCTION c 6.2

n 1.4 FROM TENSOR NETWORK CONTRACTION h 5.8 1.2 101 102 Our goal is the evaluation of Z(q) of PACI at some L q N of our choice in order to use Eq.(4) and obtain the 1.0 ∈ value of the Jones polynomial at t(q) C as defined in 0 20 40 60 80 100 Eq.(1). Note that computing the prefactor∈ (q) in Eq.(4) L A is in P as all t(q), wK , τK are efficiently computable. Thus one would focus on computing Z(q) as efficiently Figure 2. (Top) Scaling of average number of vertices hnvi as possible. (triangles) and edges hnci (dots) (logscale) of simplified Tait However, the evaluation of Z(q) on arbitrary graphs is graphs with the the grid size L (logscale) of the random grid a #P-hard problem [50]. Regardless of this complexity, walk. Dashed and solid red lines are linear fits on the last in this work we will use tensor network methods [47] to 15 data points with slopes 2.0132 and 2.0742, respectively. (Bottom) Ratio hnci over hnvi versus L converging to ∼ 2, obtain Z(q) for Tait graphs G exactly. From a graph 13 compatible with the lower bound 7 (dashed line) for random G we construct a tensor network GT encoding PACI as planar graphs. (Inset) Scaling versus L (logscale) of average follows. Each vertex v is endowed with a spin tensor (also maximal degree h∆Gi of simplified Tait graphs G, for L > 22. known as a COPY tensor) of the form For each L ∈ [6, 100] we sampled 200 knots and error bars are standard mean error. T˜ dv = δ , (5) σ σv 1σv 2...σv dv { v }i=1 which is a generalized qdv -dimensional Kronecker tensor. Each edge obtains a vertex on which we place the in- In general, finding the optimal contraction sequence teraction tensor T of Eq. (3), which is always a q q so that ∆ grows favourably slowly, is an NP-complete matrix. Reinterpreting Fig.1 (bottom), we see the tensor× H ˜ graph-theoretic problem [51–54]. Practical contraction network GT comprising T and T tensors (blue dots and schemes for tensor networks is an active field of re- red triangles respectively). search [55]. We employ contraction methods introduced Full contraction of GT yields Z(q). This amounts to in Ref. [47], where we refer the reader for explicit de- performing a sequence of tensor contractions, each being tails of the methods, here dubbed greedy and METIS, a dot product over the common index of two adjacent which were developed for fast evaluation of partition tensors in GT (green lines in Fig.1 (bottom). The parti- functions similar to Z(q). In the greedy method, the tion function is then equivalently expressed as “cheapest” edge contraction in terms of the resulting ∆H is performed. On the other hand, METIS heuristically ˜ d Z(q) = Tσv ,σv0 T σ v . (6) { v }i=1 constructs a separator hierarchy using the METIS algo- σ (σ ,σ 0 ) v X{ } vYv Y rithm [56], attempting to minimize the cut length while Every contraction step yields a graph minor H of the splitting the graph into comparably large components. initial graph. Thus, when contracting a tensor net- The contraction is performed following the separator hi- work GT, there occurs at least one tensor of dimen- erarchy in a coarse graining fashion. For details we point sion equal to the maximal vertex degree over all minors to Ref. [47]. We provide an example script demonstrating ∆H = maxH maxv dv. the evaluation of VK (t) in an online repository [57]. 4

A. Subexponential memory and runtime scaling 50 60 40 40 To investigate the performance of our numerical scheme, we require a procedure for generating random METIS H 20 30 knots, whose Tait graphs we can use to evaluate the 0 Jones polynomial. Since Tait graphs are planar and con- 0 20 40 60 G greedy h i 20 greedy nected by construction, one may be tempted to just gen- H h H i erate random connected planar graphs. However, not any max degree METIS h H i planar connected graph corresponds to a knot shadow. 10 A generic planar connected graph corresponds to a shadow, where a link is viewed as the embedding of multi- 0 ple nonintersecting S1 components. Instead, we employ 0 20 40 60 80 100 the random grid walk method [58, 59] to sample ran- L dom knot diagrams, ensuring by construction that the 40 ∆ number of components is always one. A grid walk con- h Gi ∆greedy sists of horizontal segments and vertical segments, where 30 h H i ∆METIS vertical segments always pass over horizontal ones. The h H i walk is encoded by a random permutation of coordinates x, y S , where L is the linear grid size, and steps of the 20 ∈ L form (xi, yi) (xi, yi+1) (xi+1, yi+1). Since all knots

→ → max degree have a grid walk representation, any knot is accessible 10 via this procedure. An example grid walk is shown in Fig.1 (middle). For a given orientation of the grid diagram, each cross- 0 5 10 15 20 25 30 ing has a twist sign. All possible configurations are shown in Fig.1 (top-left), and summing over them we obtain the nc h i writhe. The bicoloured knot along with its G are shown p Figure 3. Scaling with grid walk size L (top) and with cross- Fig.1 (bottom). Keeping in mind that vertical segments √ pass over horizontal ones, the colour pattern around a ing number nc (bottom) of average maximal degree h∆H i encountered under (red squares) and (green crossing determines the Tait signs εc, as shown in Fig.1 greedy METIS (top-right), and so the Tait number is readily available. triangles) contraction of G. Solid lines are linear fits (on the last 10 data points) showing the asymptotic behaviour. With the random grid walk construction that allows Black dots represent average maximal degrees h∆ i of G. us to randomly sample knots, we now investigate prop- G (Inset) Instance-by-instance comparison of ∆H for the two erties of the corresponding Tait graphs G. First, we per- contraction methods. The same data is used as for Fig.2. form Reidemeister moves that leave the knot topology Data points hnci are obtained by binning the interval between invariant but simplify the graph. In particular, a Rei- max nc for min L and min nc for max L and placing symbols at demeister I introduces or removes a twist in the knot. the mean of each bin. This is due to the fact that by sampling We employ it to remove loops, i.e. edges of the form graphs for each L we obtain a finite-variance distribution over (v, v), as well as spikes, i.e,. degree-1 vertices, from the nc. Error bars are standard mean error. Tait graph. Note that a Reidemeister I move changes the writhe by 1. A Reidemeister II move amounts to overlay- ing a strand over or under another, or inversely, combing with the graph size [61]. This is also confirmed for the the strands so they do not cross. These moves are usually simplified Tait graphs sampled by the random grid walk, referred to as poke and unpoke, respectively. In terms of as shown in Fig.2 (inset). Note that the data presented G, we perform unpokes in order to remove double edges in Fig.2 are also a manifestation of the fact that the c, c0 = (v, v0) with εc = εc0 . We then study the scaling tensor networks, or equivalently, the interaction graphs of graph invariants of the− resulting simplified graphs. of the PACI model relevant to the problem at hand, are In Fig.2 (top) we provide evidence for quadratic scal- irregular. This means that they are not amenable to 2 efficient methods that yield the Potts model’s partition ing of average number of vertices, nv L , and average 2 h i ∼ function on regular graphs, such as the Corner Transfer number of edges, nc L , for simplified Tait graphs G obtained by the randomh i ∼ grid walk. In Fig.2 (bottom) it Matrix Renormalization Group in the case of the square is shown that the ratio of the average number of edges lattice. over the average number of vertices converges to 2 as the size of the Tait graphs increases. This conver-∼ 13 B. Numerical results gent behavior is compatible with the lower bound 7 of this ratio for random planar graphs [60]. Furthermore, for random planar graphs the average maximal degree The central quantity of interest for the purposes of ∆ , defined as ∆ = max d , scales logarithmically tensor network contraction is the maximal degree ∆ h Gi G v v H 5 encountered in the sequence of minors H occurring dur- 103 ing contracting G. This quantity characterizes the com- q = 5 q = 4 plexity of the algorithm, in the sense that runtime and

) q = 3 ∆H memory requirements scale as O(q ). Z 1

r 10

In Fig.3 we show the scaling of the average ∆ H with the grid-walk size L. We find an asymptotically linear √n 1 scaling with L, which implies runtime scaling O(q c ). median ( 10− Both contraction methods perform similarly, with METIS exhibiting marginally better scaling, yet only outper- forming greedy for larger graphs (nc & 900). We there- fore use greedy to explicitly time the computation of 50 100 150 200 Z(q) for realistically accessible graph sizes. Runtime re- nc sults for the cases of q = 3, 4, 5 are shown in Fig.4.

Figure 4. Scaling of median runtime rZ(q) for computing Z(q) The favorable typical-case scaling allows us to evaluate at q = 3, 4, 5 via contracting G with as a function √ T greedy the Jones polynomial for knots with nc = 200 for q = 3 of nc for typical instances: For nc we sampled 1000 knots and with nc = 135 for q = 5, using moderate computa- by inverting the quadratic fit hnci(L) of Fig.2 (Top) to obtain tional resources. For comparison, the largest calculations L(hnci) and sampling for the appropriate L until 1000 knots of the full expression for the Jones polynomial reported were obtained. The runtimes shown were computed for the in the literature are for nc = 22 [31–35]. Furthermore, for graphs with ∆H = median(∆H ) for each nc. Dashed line of the exact evaluation of Z(q), we compare our algorithm’s slope 0.93 indicates the scaling of TDTM [62] and the slopes for q-dependent performance with that of the q-independent greedy are 0.68, 0.92, and 1.11, showing superiority for q = 3 and matching performance for q = 4. Computations were tree-decomposed transfer matrix algorithm (TDTM)[62], performed on a single processor (Intel Xeon CPU E5-2667 0 which in the literature is presented for random planar 2.90GHz) processor with ∼80 GB RAM. graphs of size up to nc = 100. In Fig.4 we show that for q = 3 the greedy tensor network algorithm outperforms TDTM for typical instances. IV. CONCLUSIONS AND OUTLOOK The main bottleneck in these benchmarks is memory usage. For each nc there are exceptional knots that yield In conclusion, we have developed a concrete methodol- atypically large ∆H and thus evaluation of VK (t) re- ogy, based on tensor networks, for the evaluation of the quires contraction of large tensors. With larger q, these Jones polynomial of arbitrary knots and demonstrated fa- exceptional cases may overflow the available memory, vorable performance of actual implementations. Due to even though typical cases with the same nc are easily the broad relevance of knot invariants, our methods have amenable. On the other hand, for any particular knot of wide applicability: classification of knotted polymers [3], interest, one can test various graph contraction schemes quantification of turbulence in classical and quantum flu- to find the most favorable ∆H and thus gauge the re- ids [5], and study of the Jones conjecture [63] are just a sources required a priori. Then one can study typical few examples of problems that require computation of cases alone, as we have demonstrated in Fig.4, where for knot polynomials. Furthermore, since Tait graphs are every nc we have obtained the runtimes for the graphs defined for links, our algorithm trivially can be extended with ∆H = median(∆H ). to the study of links, as well. Recall that a link is the embedding of disjoint circles in R3 with the knot as a Importantly, the asymptotic performance of our ten- special case. We therefore believe that the techniques sor network method does not depend on the content of introduced here can have multifaceted impact. the tensors, and so it is expected to perform as favor- They also admit several extensions. For example, it ably for random planar instances of the Potts model, i.e. is possible to obtain the coefficients of VK (t) via poly- including those not corresponding to the Jones polyno- nomial interpolation between evaluations at a number of mial. Recall that Fig.2 provides evidence that the graphs values of t equal to the degree of VK (t), bounds to which on which we have benchmarked our tensor network algo- are easily obtainable from the knot’s bicolouring(which rithm can be considered as random planar graphs. There- is efficient) and scale polynomialy with the number of fore, and especially for the case of q = 3, this is a nontriv- crossings [1]. Moreover, in analogy with condensed mat- ial result, as even incremental speedups in solving #P- ter applications of tensor networks, where truncation of hard problems are rare. We note that the slope change in singular values along edges of the network lead to accu- the scaling of the median runtimes in Fig.4 is likely due rate approximations of a desired physical quantity, ap- to the absence of CPU cache misses when tensor sizes propriate truncation procedures may allow one to obtain remain small throughout the contraction of the network. controlled approximations of the Jones polynomial, and We therefore disregard small systems below this slope potentially other knot invariants. It is also interesting to change when we obtain runtime scalings. consider whether our algorithms can be extended to cases 6 of q R [64, 65]. Indeed, recall that the evaluation of ∈i2π/n VK (e ) corresponding to q 4 is a BQP-complete problem, except when t = 1≤, i, e2πi/3, (e2πi/3)2 corresponding to q = 1, 2, 3, 4 for± which± ± the problem± is in P. This means that the quantum computation for the cases q = 1, 2, 3, 4 can be efficiently evaluated. Nev- ertheless, our algorithms are agnostic to the contents of the tensors and thus our results can extend to and be impactful for the study of the Potts model itself, as well as related graph theoretic problems such as k- colouring, where the interactions Jij decorating an inter- action graph’s edge (i, j) need not be compatible with the topology of a link diagram.

ACKNOWLEDGMENTS

We thank D. Aasen, G. Brennen, C. Chamon, P. Mar- tin, A. Michailidis, J. Pachos, and Z. Papic for comments on the manuscript and inspiring discussions. S.K. was partially supported through the Boston University Cen- ter for Non-Equilibrium Systems and Computation. Pre- liminary numerical benchmarks were performed on the Boston University Shared Computing Cluster, which is administered by Boston University Research Computing Services. K.M. acknowledges the EPSRC Doctoral Prize Fellowship for financial support, J. K. Pachos for provid- ing a workstation at the School of Physics & Astronomy, University of Leeds, and B. Coecke for providing access to the Duvel server at the Department of Computer Sci- ence, University of Oxford, where runtime results where obtained. 7

[1] Louis H. Kauffman, Knots and Physics, 4th ed. (World http://www.pnas.org/content/104/42/16432.full.pdf. Scientific, 2013). [20] Joan S. Birman and Michael D. Hirsch, “A new [2] Edward Witten, “Quantum field theory and the Jones algorithm for recognizing the unknot,” (1998), polynomial,” Comm. Math. Phys. 121, 351–399 (1989). 10.2140/gt.1998.2.175, arXiv:math/9801126. [3] Jian Qin and Scott T. Milner, “Counting polymer knots [21] Joel Hass and Jeffrey C. Lagarias, “The number of to find the entanglement length,” Soft Matter 7, 10676 reidemeister moves needed for unknotting,” (1998), (2011). arXiv:math/9807012. [4] Alberto Enciso and Daniel Peralta-Salas, “Knots and [22] Greg Kuperberg, “Knottedness is in NP, modulo GRH,” links in fluid mechanics,” Procedia IUTAM 7, 13 – 20 (2011), arXiv:1112.0845. (2013), iUTAM Symposium on Topological Fluid Dy- [23] , “A polynomial upper bound on reide- namics: Theory and Applications. meister moves,” (2013), arXiv:1302.0180. [5] Renzo L. Ricca, “Impulse of vortex knots from diagram [24] Vaughan F. R. Jones, “A polynomial invariant for knots projections,” Procedia IUTAM 7, 21 – 28 (2013), iUTAM via von Neumann algebras,” Bull. Amer. Math. Soc. Symposium on Topological Fluid Dynamics: Theory and (N.S.) 12, 103–111 (1985). Applications. [25] Joel Hass, Jeffrey C. Lagarias, and Nicholas Pippenger, [6] Francesca Maggioni, Sultan Z. Alamri, Carlo F. “The computational complexity of knot and link prob- Barenghi, and Renzo L. Ricca, “Vortex knots dynamics lems,” J. ACM 46, 185–211 (1999). in euler fluids,” Procedia IUTAM 7, 29 – 38 (2013), iU- [26] A.Yu. Kitaev, “Fault-tolerant quantum computation by TAM Symposium on Topological Fluid Dynamics: The- anyons,” Annals of Physics 303, 2 – 30 (2003). ory and Applications. [27] F. Jaeger, D. L. Vertigan, and D. J. A. Welsh, “On the [7] Xin Liu and Renzo L. Ricca, “Tackling fluid structures computational complexity of the Jones and tutte poly- complexity by the Jones polynomial,” Procedia IUTAM nomials,” Mathematical Proceedings of the Cambridge 7, 175 – 182 (2013), iUTAM Symposium on Topological Philosophical Society 108, 3553 (1990). Fluid Dynamics: Theory and Applications. [28] Dorit Aharonov, Vaughan Jones, and Zeph Landau, [8] Y. M. Cho, Seung Hun Oh, and Pengming Zhang, “A polynomial quantum algorithm for approximating the “Knots in physics,” International Journal of Modern Jones polynomial,” (2005), arXiv:quant-ph/0511096. Physics A 33, 1830006 (2018). [29] S. Iblisdir, M. Cirio, O. Boada, and G.K. Brennen, [9] Pawel Wocjan and Jon Yard, “The Jones polyno- “Low depth quantum circuits for ising models,” Annals mial: quantum algorithms and applications in quantum of Physics 340, 205 – 251 (2014). complexity theory,” arXiv:quant-ph/0603069 (2006), [30] Eric C. Rowell and Zhenghan Wang, “Mathemat- arXiv:0603069 [quant-ph]. ics of topological quantum computing,” (2017), [10] Jiannis K. Pachos, Introduction to Topological Quantum arXiv:1705.06206. Computation (Cambridge University Press, 2012). [31] Mustafa Hajij and Jesse Levitt, “An efficient algorithm [11] Ryan L. Mann and Michael J. Bremner, “On the com- to compute the colored Jones polynomial,” (2018), plexity of random quantum computations and the jones arXiv:1804.07910. polynomial,” (2017), arXiv:1711.00686. [32] http://katlas.org/wiki/The Mathematica Package KnotTheory. [12] Leslie Ann Goldberg and Heng Guo, “The complexity [33] https://www.math.uic.edu/t3m/SnapPy/. of approximating complex-valued ising and tutte parti- [34] J Hoste and M Thistlethwaite, “Knotscape, tion functions,” computational complexity 26, 765–833 a calculation program.” (2017). https://www.math.utk.edu/ morwen/knotscape.html. [13] Edward Farhi, David Gosset, Avinatan Hassidim, An- [35] Tetsuo Deguchi and Kyoichi Tsurusaki, “A new al- drew Lutomirski, and Peter Shor, “Quantum money gorithm for numerical calculation of link invariants,” from knots,” (2010), arXiv:1004.5127. Physics Letters A 174, 29 – 37 (1993). [14] Annalisa Marzuoli and Giandomenico Palumbo, “Post [36] A.E.M. El-Misiery and El-Sayed M. El-Horbaty, “An quantum cryptography from mutant prime knots,” In- algorithm for calculating jones polynomials,” Applied ternational Journal of Geometric Methods in Modern Mathematics and Computation 74, 249 – 259 (1996). Physics 08, 1571–1581 (2011). [37] F. Y. Wu, “Knot theory and statistical mechanics,” Rev. [15] Kate E. Horner, Mark A. Miller, Jonathan W. Steed, and Mod. Phys. 64, 1099–1131 (1992). Paul M. Sutcliffe, “Knot theory in modern chemistry,” [38] F. Y. Wu, “The Potts model,” Rev. Mod. Phys. 54, 235– Chem. Soc. Rev. 45, 6432–6448 (2016). 268 (1982). [16] Alexander R. Klotz, Beatrice W. Soh, and Patrick S. [39] Fran ¸coisGraner and James A. Glazier, “Simulation of Doyle, “Motion of knots in dna stretched by elongational biological cell sorting using a two-dimensional extended fields,” Phys. Rev. Lett. 120, 188003 (2018). Potts model,” Phys. Rev. Lett. 69, 2013–2016 (1992). [17] Nicole C H Lim and Sophie E Jackson, “Molecular knots [40] M. Levin and C. P. Nave, “Tensor Renormalization in biology and chemistry,” Journal of Physics: Condensed Group Approach to Two-Dimensional Classical Lattice Matter 27, 354101 (2015). Models,” Phys. Rev. Lett. 99, 120601 (2007). [18] Marc L. Mansfield, “Are there knots in proteins?” Nature [41] H. C. Jiang, Z. Y. Weng, and T. Xiang, “Accurate De- Structural Biology 1, 213 EP – (1994). termination of Tensor Network State of Quantum Lat- [19] Dorian M. Raymer and Douglas E. Smith, “Spontaneous tice Models in Two Dimensions,” Phys. Rev. Lett. 101, knotting of an agitated string,” Proceedings of the Na- 090603 (2008). tional Academy of Sciences 104, 16432–16437 (2007), [42] Z.-C. Gu and X.-G. Wen, “Tensor-entanglement-filtering 8

renormalization approach and symmetry-protected topo- [54] R´emy Belmonte, Petr A. Golovach, Pim van ’t Hof, and logical order,” Phys. Rev. B 80, 155131 (2009). Dani¨el Paulusma, “Parameterized Complexity of Two [43] Z. Y. Xie, J. Chen, M. P. Qin, J. W. Zhu, L. P. Yang, and Edge Contraction Problems with Degree Constraints,” T. Xiang, “Coarse-graining renormalization by higher- in Parameterized Exact Comput., edited by G. Gutin and order singular value decomposition,” Phys. Rev. B 86, S. Szeider (Springer, 2013) pp. 16–27. 045139 (2012). [55] Shi-Ju Ran, Emanuele Tirrito, Cheng Peng, Xi Chen, [44] G. Evenbly and G. Vidal, “Tensor Network Renor- Gang Su, and Maciej Lewenstein, “Review of tensor net- malization,” Phys. Rev. Lett. 115, 180405 (2015), work contraction approaches,” (2017), arXiv:1708.09213. arXiv:1412.0732. [56] George Karypis and Vipin Kumar, “A Fast and High [45] Hui-Hai Zhao, Z. Y. Xie, T. Xiang, and Masatoshi Quality Multilevel Scheme for Partitioning Irregular Imada, “Tensor network algorithm by coarse-graining Graphs,” SIAM J. Sci. Comput. 20, 359–392 (1998). tensor renormalization on finite periodic lattices,” Phys. [57] https://gitlab.com/kourtis/tensorCSP. Rev. B 93, 125115 (2016). [58] Peter R. Cromwell, “Embedding knots and links in an [46] G. Evenbly, “Algorithms for tensor network renormaliza- open book I: Basic properties,” Topology and its Appli- tion,” Phys. Rev. B 95, 045117 (2017). cations 64, 37–58 (1995). [47] Stefanos Kourtis, Claudio Chamon, Eduardo R. Mucci- [59] Chaim Even-Zohar, “Models of random knots,” Jour- olo, and Andrei E. Ruckenstein, “Fast counting with nal of Applied and 1, 263–296 tensor networks,” (2018), arXiv:1805.00475. (2017). [48] S.-C. Chang and R. Shrock, “Zeros of Jones polynomials [60] Gerke Stefanie, Schlatter Dirk, Steger Angelika, for families of knots and links,” Physica A: Statistical and Taraz Anusch, “The random planar graph pro- Mechanics and its Applications 301, 196 – 218 (2001). cess,” Random Structures & Algorithms 32, 236–261, [49] F.Y Wu and J Wang, “Zeroes of the Jones polynomial,” https://onlinelibrary.wiley.com/doi/pdf/10.1002/rsa.20186. Physica A: Statistical Mechanics and its Applications [61] Colin McDiarmid and Bruce Reed, “On the maximum 296, 483 – 494 (2001). degree of a random planar graph,” Combinatorics, Prob- [50] Joseph Geraci and Daniel A. Lidar, “On the exact evalu- ability and Computing 17, 591601 (2008). ation of certain instances of the Potts partition function [62] Andrea Bedini and Jesper Lykke Jacobsen, “A tree- by quantum computers,” Communications in Mathemat- decomposed transfer matrix for computing exact potts ical Physics 279, 735–768 (2008). model partition functions for arbitrary graphs, with ap- [51] Toshimasa Watanabe, Tadashi Ae, and Akira Nakamura, plications to planar graph colourings,” Journal of Physics “On the NP-hardness of edge-deletion and -contraction A: Mathematical and Theoretical 43, 385001 (2010). problems,” Discrete Applied Mathematics 6, 63 – 78 [63] Robert E. Tuzun and Adam S. Sikora, “Verification of (1983). the Jones unknot conjecture up to 22 crossings,” (2016), [52] Takao Asano and Tomio Hirata, “Edge-contraction prob- arXiv:1606.06671. lems,” Journal of Computer and System Sciences 26, 197 [64] Ferdinando Gliozzi, “Simulation of Potts models with – 208 (1983). real q and no critical slowing down,” Phys. Rev. E 66, [53] John M. Lewis and Mihalis Yannakakis, “The node- 016115 (2002). deletion problem for hereditary properties is NP- [65] A. K. Hartmann, “Calculation of partition functions by complete,” Journal of Computer and System Sciences 20, measuring component distributions,” Phys. Rev. Lett. 219 – 230 (1980). 94, 050601 (2005).