arXiv subjects

Hugh R. Morton

Publications and source records attributed to Hugh R. Morton.

6 recordsLinked to original sources

arXiv2009

A basis for the full Homfly skein of the annulus

The meridian maps of the full Homfly skein of the annulus are linear endomorphisms induced by the insertion of a meridian loop, with either orientation, around a diagram in the annulus. The eigenvalues of the meridian maps are known to be distinct, and are indexed by pairs of partitions of integers p and n into k and k* parts respectively. We give here an explicit formula for a corresponding eigenvector as the determinant of a (k*+k)x(k*+k) matrix whose entries are skein elements corresponding to partitions with a single part. This extends the results of Kawagoe and Lukac, for the case p=0, giving a basis for the subspace of the skein spanned by closed braids all oriented in the same direction. Their formula uses the Jacobi-Trudy determinants for Schur functions in terms of complete symmetric functions. Our matrices have a similar pattern of entries in k rows, and a modified pattern in k* rows, resulting in a combination of closed braids with up to n strings oriented in one direction and p in the reverse direction. We discuss the 2-variable knot invariants resulting from decoration of a knot by these skein elements, and their relation to unitary quantum invariants.

math.GT↗

arXiv2003

Conjugacy for positive permutation braids

Positive permutation braids on n strings, which are defined to be positive n-braids where each pair of strings crosses at most once, form the elementary but non-trivial building blocks in many studies of conjugacy in the braid groups. We consider conjugacy among these elementary braids which close to knots, and show that those which close to the trivial knot or to the trefoil are all conjugate. All such n-braids with the maximum possible crossing number are also shown to be conjugate. We note that conjugacy of these braids for n<6 depends only on the crossing number. In contrast, we exhibit two such braids on 6 strings with 9 crossings which are not conjugate but whose closures are each isotopic to the (2,5) torus knot.

math.GT↗

arXiv2002

Power sums and Homfly skein theory

The Murphy operators in the Hecke algebra H_n of type A are explicit commuting elements, whose symmetric functions are central in H_n. In [Skein theory and the Murphy operators, J. Knot Theory Ramif. 11 (2002), 475-492] I defined geometrically a homomorphism from the Homfly skein C of the annulus to the centre of each algebra H_n, and found an element P_m in C, independent of n, whose image, up to an explicit linear combination with the identity of H_n, is the m-th power sum of the Murphy operators. The aim of this paper is to give simple geometric representatives for the elements P_m, and to discuss their role in a similar construction for central elements of an extended family of algebras H_{n,p}.

math.GT↗

arXiv2002

Homfly Polynomials of Generalized Hopf Links

Following the recent work by T.-H. Chan in [HOMFLY polynomial of some generalized Hopf links, J. Knot Theory Ramif. 9 (2000) 865--883] on reverse string parallels of the Hopf link we give an alternative approach to finding the Homfly polynomials of these links, based on the Homfly skein of the annulus. We establish that two natural skein maps have distinct eigenvalues, answering a question raised by Chan, and use this result to calculate the Homfly polynomial of some more general reverse string satellites of the Hopf link.

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arXiv2001

The Homfly polynomial of the decorated Hopf link

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 to find the 2-variable Homfly invariant <λ,μ> of the Hopf link arising from decorations Q_λand Q_μin terms of the Schur symmetric function s_μof 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_λand V_μ, which is a 1-variable specialisation of <λ,μ>, can be expressed in terms of an N x N minor of the Vandermonde matrix (q^{ij}).

math.GT↗

arXiv2001

Skein theory and the Murphy operators

The Murphy operators in the Hecke algebra H_n of type A are explicit commuting elements whose sum generates the centre. They can be represented by simple tangles in the Homfly skein theory version of H_n. In this paper I present a single tangle which represents their sum, and which is obviously central. As a consequence it is possible to identify a natural basis for the Homfly skein of the annulus, C. Symmetric functions of the Murphy operators are also central in H_n. I define geometrically a homomorphism from C to the centre of each algebra H_n, and find an element in C, independent of n, whose image is the m-th power sum of the Murphy operators. Generating function techniques are used to describe images of other elements of C in terms of the Murphy operators, and to demonstrate relations among other natural skein elements.

math.GT↗