A Method of Riemann–Hilbert Problem for Zhang’s Conjecture 1 in a Ferromagnetic 3D Ising Model: Trivialization of Topological Structure

A method of the Riemann–Hilbert problem is applied for Zhang’s conjecture 1 proposed in Philo. Mag. 87 (2007) 5309 for a ferromagnetic three-dimensional (3D) Ising model in the zero external field and the solution to the Zhang’s conjecture 1 is constructed by use of the monoidal transform. At first, the knot structure of the ferromagnetic 3D Ising model in the zero external field is determined and the non-local behavior of the ferromagnetic 3D Ising model can be described by the non-trivial knot structure. A representation from the knot space to the Clifford algebra of exponential type is constructed, and the partition function of the ferromagnetic 3D Ising model in the zero external field can be obtained by this representation (Theorem I). After a realization of the knots on a Riemann surface of hyperelliptic type, the monodromy representation is realized from the representation. The Riemann–Hilbert problem is formulated and the solution is obtained (Theorem II). Finally, the monoidal transformation is introduced for the solution and the trivialization of the representation is constructed (Theorem III). By this, we can obtain the desired solution to the Zhang’s conjecture 1 (Main Theorem). The present work not only proves the Zhang’s conjecture 1, but also shows that the 3D Ising model is a good platform for studying in deep the mathematical structure of a physical many-body interacting spin system and the connections between algebra, topology, and geometry.

INCOMPRESSIBLE ONE-SIDED SURFACES IN EVEN FILLINGS OF FIGURE 8 KNOT SPACE

In the closed, non-Haken, hyperbolic class of examples generated by (2p, q) Dehn fillings of Figure 8 knot space, the geometrically incompressible one-sided surfaces are identified by the filling ratio [Formula: see text] and determined to be unique in all cases. When applied to one-sided Heegaard splitting, this can be used to classify all geometrically incompressible splittings in this class of closed, hyperbolic examples; no analogous classification exists for two-sided Heegaard splitting.

POLYGONAL KNOT SPACE NEAR ROPELENGTH-MINIMIZED KNOTS

For a polygonal knot K, it is shown that a tube of radius R(K), the polygonal thickness radius, is an embedded torus. Given a thick configuration K, perturbations of size r < R(K) define satellite structures, or local knotting. We explore knotting within these tubes both theoretically and numerically. We provide bounds on perturbation radii for which one can obtain small trefoil and figure-eight summands and use Monte Carlo simulations to estimate the relative probabilities of these structures as a function of the number of edges.

Natural classification of knots

The principal objective of the knot theory is to provide a simple way of classifying and ordering all the knot types. Here, we propose a natural classification of knots based on their intrinsic position in the knot space that is defined by the set of knots to which a given knot can be converted by individual intersegmental passages. In addition, we characterize various knots using a set of simple quantum numbers that can be determined upon inspection of minimal crossing diagram of a knot. These numbers include: crossing number; average three-dimensional writhe; number of topological domains; and the average relaxation value.

COMBINATORICS AND FOUR BRIDGED KNOTS

The ℝ P 3-Conjecture states a non-trivial knot in S 3 cannot yield ℝ P 3 by a Dehn surgery. Generically, in the knot-space S3-N(K), the intersection of a projective plane ℝP2 in ℝ P 3, and any 2-sphere S2 in S3 pierced by K, is a 1-complex which can be viewed as a graph in either the projective plane or the 2-sphere. Gordon and Luecke have used similar graphs arising as the intersection of two 2-spheres, to prove that a knot in S3 is determined by its complement. A part of this paper concerns some new combinatorial results on these graphs. They are considered as an unavoidable step towards showing that the ℝ P 3-Conjecture is true. Moreover, we use these results to prove that any non-trivial knot that could yield ℝ P 3 has at least five bridges.

GEOMETRIC KNOT SPACES AND POLYGONAL ISOTOPY

The space of n-sided polygons embedded in three-space consists of a smooth manifold in which points correspond to piecewise linear or "geometric" knots, while paths correspond to isotopies which preserve the geometric structure of these knots. The topology of these spaces for the case n=6 and n=7 is described. In both of these cases, each knot space consists of five components,but contains only three (when n=6) or four (when n=7) topological knot types. Therefore "geometric knot equivalence" is strictly stronger than topological equivalence. This point is demonstrated by the hexagonal trefoils and heptagonal figure-eight knots, which, unlike their topological counterparts, are not reversible. Extending these results to the cases n≥8 will also be discussed.