Let Cm be the closure of the Hecke algebra with m strings Hm in the oriented framed Homfly skein C of the annulus [11,5,9,2], which provides the natural parameter space for the Homfly satellite invariants of a knot. The submodule C+ ⊂ C spanned by the union ∪ m≥0 Cm is an algebra, isomorphic to the algebra of the symmetric functions. Turaev's geometrical basis for C+ consists of monomials in closed m-braids Am, the closure of the braid σm−1 · · · σ2σ1.We collect and expand formulae relating elements expressed in terms of symmetric functions to Turaev's basis. We reformulate the formulae of Rosso and Jones for quantum sl(N ) invariants of cables [14] in terms of plethysms of symmetric functions, and use the connection between quantum sl(N ) invariants and the skein C+ to give a formula for the satellite of a cable as an element of the Homfly skein C+. We can then analyse the case where a cable is decorated by the pattern P d which corresponds to a power sum in the symmetric function interpretation of C+ to get direct relations between the Homfly invariants of some diagrams decorated by power sums.

We prove that the hypothetical extreme Khovanov cohomology of a link is the cohomology of the independence simplicial complex of its Lando graph. We also provide a family of knots having as many non-trivial extreme Khovanov cohomology modules as desired, that is, examples of H-thick knots which are as far of being H-thin as desired.

We find families of prime knot diagrams with arbitrary extreme coefficients in their Jones polynomials. Some graph theory is presented in connection with this problem, generalizing ideas by Yongju Bae and Morton [3] and giving a positive answer to a question in their paper.

Homogeneous links were introduced by Peter Cromwell, who proved that the projection surface of these links, that given by the Seifert algorithm, has minimal genus. Here we provide a different proof, with a geometric rather than combinatorial flavor. To do this, we first show a direct relation between the Seifert matrix and the decomposition into blocks of the Seifert graph. Precisely, we prove that the Seifert matrix can be arranged in a block triangular form, with small boxes in the diagonal corresponding to the blocks of the Seifert graph. Then we prove that the boxes in the diagonal has non-zero determinant, by looking at an explicit matrix of degrees given by the planar structure of the Seifert graph. The paper contains also a complete classification of the homogeneous knots of genus one.

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