17,163 research outputs found
Mutant knots with symmetry
Mutant knots, in the sense of Conway, are known to share the same Homfly
polynomial. Their 2-string satellites also share the same Homfly polynomial,
but in general their m-string satellites can have different Homfly polynomials
for m>2. We show that, under conditions of extra symmetry on the constituent
2-tangles, the directed m-string satellites of mutants share the same Homfly
polynomial for m<6 in general, and for all choices of m when the satellite is
based on a cable knot pattern.
We give examples of mutants with extra symmetry whose Homfly polynomials of
some 6-string satellites are different, by comparing their quantum sl(3)
invariants.Comment: 15 page
The Burau matrix and Fiedler's invariant for a closed braid
It is shown how Fiedler's `small state-sum' invariant for a braid can be
calculated from the 2-variable Alexander polynomial of the link which consists
of the closed braid together with the braid axis.Comment: 7 pages LaTeX2e. To appear in Topology and its Application
Mutual braiding and the band presentation of braid groups
This paper is concerned with detecting when a closed braid and its axis are
'mutually braided' in the sense of Rudolph. It deals with closed braids which
are fibred links, the simplest case being closed braids which present the
unknot. The geometric condition for mutual braiding refers to the existence of
a close control on the way in which the whole family of fibre surfaces meet the
family of discs spanning the braid axis. We show how such a braid can be
presented naturally as a word in the `band generators' of the braid group
discussed by Birman, Ko and Lee in their recent account of the band
presentation of the braid groups. In this context we are able to convert the
conditions for mutual braiding into the existence of a suitable sequence of
band relations and other moves on the braid word, and derive a combinatorial
method for deciding whether a braid is mutually braided.Comment: 13 pages, 10 figures. To appear in the proceedings of 'Knots in
Hellas 98
Complete Separability and Fourier representations of n-qubit states
Necessary conditions for separability are most easily expressed in the
computational basis, while sufficient conditions are most conveniently
expressed in the spin basis. We use the Hadamard matrix to define the
relationship between these two bases and to emphasize its interpretation as a
Fourier transform. We then prove a general sufficient condition for complete
separability in terms of the spin coefficients and give necessary and
sufficient conditions for the complete separability of a class of generalized
Werner densities. As a further application of the theory, we give necessary and
sufficient conditions for full separability for a particular set of -qubit
states whose densities all satisfy the Peres condition
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