Narrow Results

The main goal is to find the Homfly polynomial of a link formed by decorating each component of the Hopf link with the closure of a directly oriented tangle. Such decorations are spanned in the Homfly skein of the annulus by elements Q_λ, depending on partitions λ. We show how the 2-variable Homfly invariant <λ, μ> of the Hopf link arising from decorations Q_λ and Q_μ can be found from the Schur symmetric function s_μ of an explicit power series depending on λ. We show also that the quantum invariant of the Hopf link coloured by irreducible sl(N)_q modules V_λ and V_μ, which is a 1-variable specialisation of <λ, μ>, can be expressed in terms of an N × N minor of the Vandermonde matrix (q^ij).

Pairs of genus 2 mutant knots can have different Homfly polynomials, for example some 3-string satellites of Conway mutant pairs. We give examples which have different Kauffman 2-variable polynomials, answering a question raised by Dunfield et al. in their study of genus 2 mutants. While pairs of genus 2 mutant knots have the same Jones polynomial, given from the Homfly polynomial by setting v = s², we give examples whose Homfly polynomials differ when v = s³. We also give examples which differ in a Vassiliev invariant of degree 7, in contrast to satellites of Conway mutant knots.

In this paper, an expression for the Homfly polynomial of a classical pretzel link with a fixed orientation was obtained. As an application, we proved that a (p, q, r)-pretzel knot is achiral if and only if {p, q, r} = {2, 1, 1}, {-2, -1, -1}, {2, -1, 3}, {-2, 1, -3}, {1, -1, k}, {1, k, -2 -k} or {-1, -k, 2 + k}, where k is an odd integer.

We study the properties of a knot K to be 2-adjacent to another knot W by analyzing their Conway polynomials, Jones polynomials and Homfly polynomials and give some very useful conditions. We discuss whether each pair of knots can be 2-adjacent to each other, i.e. whether 2-adjacency is a symmetric relation. We discuss also whether the trivial knot, the trefoil knot and the figure-eight knot can be 2-adjacent to any knot in Rolfsen's table and the opposite cases, except for 9₃₄ it is not decided whether it is 2-adjacent to 4₁. Finally, we give some examples to answer I. Torisu's problem partly and etc.

In this study, we introduce new criteria for a given link to detect the non-$p$-periodicity for a prime $p\ge 5$. For this we use a congruence of the quantum ${𝔰𝔩}_N$-invariant of periodic links. Our result is naturally consistent with Traczyk’s criterion and Przytycki’s criterion, and can be regarded as a generalization of these classical criteria. We shall give computational results of all the (alternating and non-alternating) knots of crossing number $\le 16$ for their non-$p$-periodicity ($p\ge 5$). The computational results also show that our criteria has somewhat different nature with the classical criterion of Murasugi.

It is shown that the knot invariant arising from an irreducible representation of a quantum group is, under certain conditions, an evaluation of the Homfly or Dubrovnik polynomial of the knot.

Besides the known cases of the fundamental representation for each of the quantum groups in the series A_n, B_n, C_n and D_n, the results cover the special cases of the 3-dimensional representation of SU(2) and the 6-dimensional representation of SU(4), which can be viewed as the fundamental representations of SO(3) and SO(6) respectively. The second of these cases leads to a new relation between an evaluation of the Dubrovnik polynomial of a knot and an evaluation of the Homfly polynomials of two 2-cables about the knot.

We consider the problem of distinguishing mutant knots using invariants of their satellites. We show, by explicit calculation, that the Homfly polynomial of the 3-parallel (and hence the related quantum invariants) will distinguish some mutant pairs.

Having established a condition on the colouring module which forces a quantum invariant to agree on mutants, we explain several features of the difference between the Homfly polynomials of satellites constructed from mutants using more general patterns. We illustrate this by our calculations; from these we isolate some simple quantum invariants, and a framed Vassiliev invariant of type 11, which distinguish certain mutants, including the Conway and Kinoshita-Teresaka pair.

We prove that the skein polynomial and the Jones polynomial of oriented links are not changed be the rotation operation of Anstee, Przytycki and Rolfsen [1], provided the rotor part of the diagram is of a certain special type.