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Topological entanglement in polymers and biopolymers is a topic that involves different fields of science such as chemistry, biology, physics, and mathematics. One of the main issues in this topic is to understand how the entanglement complexity can depend on factors such as the degree of polymerization, the quality of the solvent, and the temperature or the degree of confinement of the macromolecule. In this respect a statistical approach to the problem is natural and in the last few years there has been a lot of work on the study of the entanglement complexity of polymers within the statistical mechanics framework. A review on this topic is given here stressing the main results obtained and describing the tools most used with this approach

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We present the exact solutions of various directed walk models of polymers confined to a slit and interacting with the walls of the slit via an attractive potential. We consider three geometric constraints on the ends of the polymer and concentrate on the long chain limit. Apart from the general interest in the effect of geometrical confinement this can be viewed as a two-dimensional model of steric stabilization and sensitized flocculation of colloidal dispersions. We demonstrate that the large width limit admits a phase diagram that is markedly different from the one found in a halfplane geometry, even when the polymer is constrained to be fixed at both ends on one wall. We are not able to find a closed form solution for the free energy for finite width, at all values of the interaction parameters, but we can calculate the asymptotic behaviour for large widths everywhere in the phase plane. This allows us to find the force between the walls induced by the polymer and hence the regions of the plane where either steric stabilization or sensitized flocculation would occur.PACS numbers: 05.50.+q, 05.70.fh, 61.41.+e Short title: A directed walk model of a long chain polymer in a slit with attractive walls A directed walk model of a long chain polymer in a slit with attractive walls 2

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We consider several different directed walk models of a homopolymer adsorbing at a surface when the polymer is subject to an elongational force which hinders the adsorption. We use combinatorial methods for analyzing how the critical temperature for adsorption depends on the magnitude of the applied force and show that the crossover exponent φ changes when a force is applied. We discuss the characteristics of the model needed to obtain a re-entrant phase diagram.