Computing a Knot Invariant as a Constraint Satisfaction Problem

There seems to be no generic method to find its solution, i.e., a colored configuration with satisfied manner, called a coloring class, and to count the total number of them. In this note, we present a statistical mechanical formulation of the p-colorability problem of knots, which provides an algorithm to find the solution. The method also allows one to get some deeper insight into the complexity of the problem from the viewpoint of the constraint satisfaction problem.

Formulation of the problem. The knot topology can be analyzed by using the knot trajectory projected onto the plane with conserving over-and under-crossing conditions of local components. It is called the knot diagram. As symbolically exemplified in Fig. 1(a), it consists of n arcs and n crossing points. In p-colorability problem, we attempt to color each arcs using p colors under a particular type of the constraint (see Fig. 1

be the color of i-th arc. Each crossing point consists of three arcs, at which a constraint is defined locally. Suppose that two arcs i and i + 1 are separated by an overcrossing of an arc k (ii) The number m(i) of edges connected to i-th variable node is random under the constraint

Hence, graphs corresponding to knots have equal number n of clause and variable nodes, and are allowed to have random connecting property patterns (by (ii)) with the condition (i)

imposed. The satisfiability function h i (c i , c i+1 , c k )is defined as,

The degree of the net satisfiability is quantified by the function

where U(β) represents the internal energy.

First we generated 3-d random sample trajectories with figure-8 knot type by performing a standard Langevin-dynamics (LD) simulation using a closed beads-spring model polymers (Fig. 2). Random graphs were obtained by projecting the trajectories onto the plane. We then carried out MC simulation explained above for each graph and obtained entropy as a function of inverse temperature (Fig. 3). To obtain smooth profiles in Fig. 3 Discussion and perspectives The present method is, in principle, applicable to much more complex knots than considered here with larger minimal crossing numbers. The results from several different coloring number p, and if necessary combined with other kind of the invariants, would have a good classification ability of knots.

So far we have only focused on the ground states of the model Hamiltonian eq.( 3). Ground states are obviously most important in the context of knot theory with a definite meaning corresponding to the invariance. It does not, however, necessarily exclude the possible implications of the excited states. For instance, we expect that the internal energy of our model at finite temperature may contain some useful information on the knot complexity. The coloring problem on random graphs often show characteristic phase transition behavior involving both statics 8) and dynamics 9) from simple structureless phase to glassy phase with many metastable states. 4,7) It has been intensively studied from the spin glass perspective 5) and often discussed in relation to computational complexity. 4,6) A unique feature in the present system comes from the fact that the ground states of our model are connected to the topological invariance. Therefore, we can control the apparent conformational complexity in arbitrary ways, i.e., the crossing number n in the knot diagram, while keeping the ground states invariant via Reidemeister moves 1) (See the caption in Fig. 2). The model surely exhibits extensively separated ground states, but its total number does not grow exponentially with the system size n. This fact means that the model does not exhibit clustering and condensation transitions on solution space structure 7) in literal terms. Detailed investigations on such a model system may be interesting towards better understanding of the glassy properties with rugged landscapes 10) and empirical hardness of searching problems. [11][12][13]