A Self-Linking Invariant of Virtual Knots

In this paper we introduce a new invariant of virtual knots and links that is non-trivial for infinitely many virtuals, but is trivial on classical knots and links. The invariant is initially be expressed in terms of a relative of the bracket polynomial and then extracted from this polynomial in terms of its exponents, particularly for the case of knots. This analog of the bracket polynomial will be denoted {K} (with curly brackets) and called the binary bracket polynomial. The key to the combinatorics of the invariant is an interpretation of the state sum in terms of 2-colorings of the associated diagrams. For virtual knots, the new invariant, J(K), is a restriction of the writhe to the odd crossings of the diagram (A crossing is odd if it links an odd number of crossings in the Gauss code of the knot. The set of odd crossings is empty for a classical knot.) For K a virtual knot, J(K) non-zero implies that K is non-trivial, non-classical and inequivalent to its planar mirror image. The paper also condsiders generalizations of the two-fold coloring of the states of the binary bracket to cases of three and more colors. Relationships with graph coloring and the Four Color Theorem are discussed.Comment: 36 pages, 22 figures, LaTeX documen

Complexity of Discrete Energy Minimization Problems

Discrete energy minimization is widely-used in computer vision and machine learning for problems such as MAP inference in graphical models. The problem, in general, is notoriously intractable, and finding the global optimal solution is known to be NP-hard. However, is it possible to approximate this problem with a reasonable ratio bound on the solution quality in polynomial time? We show in this paper that the answer is no. Specifically, we show that general energy minimization, even in the 2-label pairwise case, and planar energy minimization with three or more labels are exp-APX-complete. This finding rules out the existence of any approximation algorithm with a sub-exponential approximation ratio in the input size for these two problems, including constant factor approximations. Moreover, we collect and review the computational complexity of several subclass problems and arrange them on a complexity scale consisting of three major complexity classes -- PO, APX, and exp-APX, corresponding to problems that are solvable, approximable, and inapproximable in polynomial time. Problems in the first two complexity classes can serve as alternative tractable formulations to the inapproximable ones. This paper can help vision researchers to select an appropriate model for an application or guide them in designing new algorithms.Comment: ECCV'16 accepte

Structure of the flow and Yamada polynomials of cubic graphs

We establish a quadratic identity for the Yamada polynomial of ribbon cubic graphs in 3-space, extending the Tutte golden identity for planar cubic graphs. An application is given to the structure of the flow polynomial of cubic graphs at zero. The golden identity for the flow polynomial is conjectured to characterize planarity of cubic graphs, and we prove this conjecture for a certain infinite family of non-planar graphs. Further, we establish exponential growth of the number of chromatic polynomials of planar triangulations, answering a question of D. Treumann and E. Zaslow. The structure underlying these results is the chromatic algebra, and more generally the SO(3) topological quantum field theory.Comment: 22 page

Quantum Phase Transitions in Anti-ferromagnetic Planar Cubic Lattices

Motivated by its relation to an $\cal{NP}$-hard problem, we analyze the ground state properties of anti-ferromagnetic Ising-spin networks embedded on planar cubic lattices, under the action of homogeneous transverse and longitudinal magnetic fields. This model exhibits a quantum phase transition at critical values of the magnetic field, which can be identified by the entanglement behavior, as well as by a Majorization analysis. The scaling of the entanglement in the critical region is in agreement with the area law, indicating that even simple systems can support large amounts of quantum correlations. We study the scaling behavior of low-lying energy gaps for a restricted set of geometries, and find that even in this simplified case, it is impossible to predict the asymptotic behavior, with the data allowing equally good fits to exponential and power law decays. We can therefore, draw no conclusion as to the algorithmic complexity of a quantum adiabatic ground-state search for the system.Comment: 7 pages, 13 figures, final version (accepted for publication in PRA