A weighted graph polynomial from chromatic invariants of knots

Polynomial graph invariants from homomorphism numbers

The number of homomorphisms hom(G,Kk) from a graph G to the complete graph Kk is the value of the chromatic polynomial of G at a positive integer k. This motivates the following (cf. [3]): Definition 1 A sequence of graphs (Hk), k = (k1, . . . , kh) ∈ N , is strongly polynomial if for every graph G there is a polynomial p(G; x1, . . . , xh) such that hom(G,Hk) = p(G; k1, . . . , kh) for every k...

On Legendrian Knots and Polynomial Invariants

It is proved in this note that the analogues of the Bennequin inequality which provide an upper bound for the Bennequin invariant of a Legendrian knot in the standard contact three dimensional space in terms of the lower degree in the framing variable of the HOMFLY and the Kauffman polynomials are not sharp. Furthermore, the relationships between these restrictions on the range of the Bennequin...

The New Polynomial Invariants of Knots and Links

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Polynomial invariants of graphs on surfaces and virtual knots

Alexander Polynomial, Finite Type Invariants and Volume of Hyperbolic Knots

We show that given n > 0, there exists a hyperbolic knot K with trivial Alexander polynomial, trivial finite type invariants of orders ≤ n and such that the volume of the complement of K is larger than n. We compare our result with known results about the relation of hyperbolic volume and Alexander polynomials of alternating knots. We also discuss how our result compares with relations between ...