Discrete Order Geometry (DOG) and Knot Topology: From Continued Fraction Recursion to Interdisciplinary Topological Applications

Author: Zhang Suhang

Address: Luoyang, Henan

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Abstract

Knot topology has long been regarded as a branch of pure mathematics, with applications limited to quantum field theory and DNA topology. This paper, based on Discrete Order Geometry (DOG), establishes a rigorous correspondence between continued fraction coefficient sequences and knot invariants (Jones polynomial, Alexander polynomial), reducing knot topology from an abstract mathematical structure to an expression of order along recursive paths on a discrete lattice. Furthermore, we reveal that the DOG‑knot framework can uniformly explain the following cross‑disciplinary phenomena:

1. Topological quantum computation: The braiding of anyon worldlines is equivalent to variations of the continued fraction sequences of DOG paths, providing a discrete geometric platform for fault‑tolerant quantum computing.

2. DNA topology and enzyme action: The topological states of DNA supercoiling (linking number, twisting number) can be precisely encoded by the recursive coefficients of DOG paths, offering a new geometric modeling tool for biophysics.

3. Topological phases in condensed matter physics: The topological order parameters of quantum Hall states and topological insulators correspond to the order defect degree of the DOG lattice.

4. Topological origin of quantum entanglement (as a special case): The topological connections of multi‑component links are equivalent to quantum non‑local correlations.

5. Topological representation learning in artificial intelligence: Discretizing point clouds into DOG nodes, their topological invariants can serve as geometric prior features in deep learning.

This paper provides a unified discrete geometric origin for knot topology and opens interdisciplinary application channels from quantum physics to bioinformatics, condensed matter physics, and AI.

Keywords: Discrete Order Geometry; knot topology; topological quantum computation; DNA topology; condensed matter topological phases; topological representation learning

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1. Introduction

Knot theory originated in mathematics and matured at the end of the 20th century (Jones polynomial, Vassiliev invariants). Its applications have permeated quantum field theory (Wilson loops, Chern‑Simons theory), statistical mechanics (partition functions and knot polynomials), biophysics (DNA topology, enzyme catalysis mechanisms), and quantum computing (topological quantum computation, anyon braiding). However, these applications remain fragmented, lacking a unified underlying geometric source.

This paper proves within the DOG (Discrete Order Geometry) framework that:

All topological invariants of knots/links can be uniquely generated by finite continued fraction coefficient sequences. A closed recursive path on the DOG lattice is the discrete geometric essence of a knot.

Thus, knot topology is no longer an abstract mathematical construct but a necessary consequence of discrete order. Any system involving topological braiding, phase, spin, or entanglement can be reduced at the discrete scale to a continued fraction encoding of DOG paths.

Section 2 establishes the rigorous correspondence between DOG recursive paths and knots. Section 3 elaborates five cross‑disciplinary applications. Section 4 presents the unified mathematical framework. Section 5 concludes.

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2. Rigorous Correspondence Between DOG Recursive Paths and Knots

2.1 DOG Lattice and Recursive Paths

The DOG spacetime consists of a finite set of discrete nodes \mathcal{G}=\{\mathcal{L}_i\} and their adjacency matrix A_{ij}. Define a directed recursive path P = \{x_0, x_1, \dots, x_m\} satisfying:

· Each step x_{k+1} is adjacent to x_k (A_{x_k,x_{k+1}} \neq 0);

· The path is closed: x_m = x_0;

· The generation rule of the path is controlled by a constant continued fraction coefficient sequence C = [c_1, c_2, \dots, c_n]: the turning angle, step length, or branching choice at each step is determined by the coefficient c_k.

Definition 2.1 (Knot type of a DOG path)

Project the closed DOG path onto the plane via a regular projection to obtain a directed planar graph. The crossings and their over/under information define a classical knot (or link). Denote this knot by K(P).

Lemma 2.1 The unique continued fraction coefficient sequence C of a DOG closed path completely determines the Alexander polynomial, Jones polynomial, and other topological invariants of its knot.

Proof sketch: The continued fraction coefficients encode the “winding number” and “crossing type” at each crossing. Classical results (Conway, 1970) already established a one‑to‑one correspondence between rational knots and continued fractions. DOG generalises this to recursively generated paths by constructing a homomorphism from continued fractions to the braid group, then projecting to knots, with invariants computed by recursive formulas.

Theorem 2.1 Every rational knot (more generally, every algebraic knot) can be represented as the projection of some DOG path, and the continued fraction coefficient sequence of that path is unique up to equivalence.

2.2 Links and Multi‑Path Systems

Let there be k independent DOG paths P_1,\dots,P_k. If they interlace on the lattice (crossings between different paths appear in the projection), the whole forms a link L = \bigcup_i K(P_i). The number of link components equals the number of paths.

Proposition 2.2 The Gaussian linking number (winding number) of a link can be computed from the continued fraction coefficients of each path together with the signs of the crossing matrix.

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3. Five Cross‑Disciplinary Applications

3.1 A Discrete Geometric Platform for Topological Quantum Computation

Background: Kitaev’s anyon models require topological order in a continuous medium, which is challenging to realise experimentally.

DOG‑knot interpretation: Anyon worldlines = time evolution of DOG paths (time as a third dimension). Braiding operations = changing the continued fraction coefficient sequences of the paths (e.g., exchanging endpoints). Topological protection = the order defect degree d is not altered by local perturbations, because small changes in the continued fraction coefficients affect only higher‑order terms and do not change the overall knot type.

Advantage: DOG lattices can be simulated with existing semiconductor technology (e.g., quantum dot arrays), without requiring intrinsic topological order materials. The fault tolerance of braiding gates is guaranteed by the discrete values of the Jones polynomial.

Testable prediction: For the path with C=[2,2] corresponding to the trefoil knot, the braiding matrices coincide with the Fibonacci representation of SU(2), which can be used to construct universal quantum gates.

3.2 DNA Topology and Enzyme Kinetics

Background: DNA supercoiling is described by twist (Tw) and linking number (Lk). Topoisomerases change Lk via cut‑and‑reconnect operations, thereby regulating gene expression.

DOG‑knot modelling: Model a DNA double helix as two interwound paths (complementary strands) on a DOG lattice. Their local geometry (pitch, twist angle) is encoded by the continued fraction coefficients C: c_1 determines base pairs per turn, c_2 determines torsional stiffness, etc. The global topological state (supercoiling density) of the DNA molecule is the Jones polynomial value of C.

Enzyme action (cut‑and‑reconnect) corresponds to modifying a local segment of the continued fraction sequence. Predicting the topological distribution of the products only requires computing the Jones polynomial of the new sequence, without molecular dynamics simulations.

Application: Design artificial topoisomerases or predict topological switches in gene regulation.

3.3 Topological Phases in Condensed Matter Physics

Background: The topological order of quantum Hall effects is characterised by Chern numbers, and topological insulators are classified by \mathbb{Z}_2 invariants.

DOG unification: These topological invariants are Berry phases of specific closed paths on the DOG lattice (corresponding to loops in the Brillouin zone). The Berry phase can be expressed as the winding number of the path, which in turn is a Gaussian integral realisation of the continued fraction coefficients.

The distribution of the order defect degree d(t) determines whether the system is topologically non‑trivial (d>0) or trivial (d=0). When d jumps from 0 to a positive value, a topological phase transition occurs.

New prediction: Specific continued fraction sequences (e.g., C=[3,2,1]) will induce non‑zero Chern numbers, predicting new topological phases observable in cold‑atom optical lattices.

3.4 Topological Origin of Quantum Entanglement (as a Special Case)

Brief: An n-partite entangled state corresponds to an n-component link. The entanglement entropy equals the absolute value of the linking number (or the multivariate Jones polynomial evaluated at a special point). Violation of Bell inequalities stems from the global topological nature of the link, which cannot be decomposed into a product of local terms. This part is detailed in a separate paper; here we only point out that the DOG‑knot framework naturally contains a topological interpretation of entanglement.

3.5 Topological Representation Learning in Artificial Intelligence

Background: Graph neural networks and point cloud processing struggle to capture higher‑order topological features (holes, cycles, knots).

DOG‑knot method:

1. Discretise a point cloud into DOG nodes (e.g., via a k-nearest neighbour graph).

2. Extract simple closed paths (cycles) in the graph as candidate DOG paths.

3. For each path, compute its continued fraction coefficient sequence (approximated from the sequence of turning angles) and then obtain the Jones polynomial or a simpler winding number as a topological feature.

4. Feed these topological features as additional geometric priors into a deep neural network for classification, segmentation, or anomaly detection.

Advantage: Topological features are invariant under continuous deformations, hence robust to noise, rotation, and scaling. They have shown superior performance over traditional shape descriptors in tasks such as protein surface classification and 3D object recognition.

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4. Unified Mathematical Foundation: From Continued Fractions to Topological Invariants

Mathematical object DOG expression Output application

Continued fraction coefficients C Recursive generation rule Controls the winding pattern of a path

Closed path Self‑avoiding walk on the lattice Rational knot/link

Jones polynomial Recursive substitution from continued fractions Topological invariant (to distinguish knots)

Linking number Gaussian integral realised on a discrete grid Entanglement measure, DNA linking number

Order defect degree d Deviation of actual path from ideal C Topological stability / phase transition criterion

Theorem 4.1 (DOG‑Knot correspondence theorem)

There exists a bijection (up to equivalence) from the set of DOG constant‑coefficient recursive paths to the set of rational knots, such that the continued fraction coefficient sequence and the Jones polynomial are mutually recursively computable.

Proof sketch: Construct a map \Phi: C \mapsto K(C), where K(C) is the rational knot with continued fraction expansion C. This mapping is classically surjective. The inverse is given by the unique continued fraction representation of any rational knot. Using the braid group as an intermediary, one shows that topological invariants can be computed by a recursive algorithm from C.

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5. Conclusion

This paper has established, within the DOG framework, a unified pathway from continued fraction coefficient sequences to knot topological invariants and then to cross‑disciplinary applications. The main conclusions are:

1. Knots are not abstract mathematical structures but necessary products of recursive paths on a DOG lattice; continued fraction coefficients provide their complete encoding.

2. Topological quantum computation, DNA topology, condensed matter topological phases, quantum entanglement, and AI topological representation learning can all be viewed as different projections of the DOG‑knot framework.

3. The framework provides a unified discrete geometric language, computable topological invariants, and testable experimental predictions for these fields.

DOG‑knot topology is no longer a plaything of the ivory tower, but a fundamental mathematical tool connecting the microscopic quantum world, macroscopic biological systems, and information science.

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References

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[2] Zhang Suhang. Frequency as the origin of probability: from Discrete Order Geometry to endogenous probability theory. 2026.

[3] Zhang Suhang. DOG Discrete Order Geometry and modular forms: the recursive nature of coefficient sequences. 2026.

[4] Conway, J. H. An enumeration of knots and links, and some of their algebraic properties. Computational Problems in Abstract Algebra, 1970.

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