16,794 research outputs found
On possible existence of HOMFLY polynomials for virtual knots
Virtual knots are associated with knot diagrams, which are not obligatory
planar. The recently suggested generalization from N=2 to arbitrary N of the
Kauffman-Khovanov calculus of cycles in resolved diagrams can be
straightforwardly applied to non-planar case. In simple examples we demonstrate
that this construction preserves topological invariance -- thus implying the
existence of HOMFLY extension of cabled Jones polynomials for virtual knots and
links.Comment: 12 page
Evolution method and "differential hierarchy" of colored knot polynomials
We consider braids with repeating patterns inside arbitrary knots which
provides a multi-parametric family of knots, depending on the "evolution"
parameter, which controls the number of repetitions. The dependence of knot
(super)polynomials on such evolution parameters is very easy to find. We apply
this evolution method to study of the families of knots and links which include
the cases with just two parallel and anti-parallel strands in the braid, like
the ordinary twist and 2-strand torus knots/links and counter-oriented 2-strand
links. When the answers were available before, they are immediately reproduced,
and an essentially new example is added of the "double braid", which is a
combination of parallel and anti-parallel 2-strand braids. This study helps us
to reveal with the full clarity and partly investigate a mysterious
hierarchical structure of the colored HOMFLY polynomials, at least, in
(anti)symmetric representations, which extends the original observation for the
figure-eight knot to many (presumably all) knots. We demonstrate that this
structure is typically respected by the t-deformation to the superpolynomials.Comment: 31 page
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