44 research outputs found
Hom Quandles
If is an abelian quandle and is a quandle, the hom set
of quandle homomorphisms from to has a natural
quandle structure. We exploit this fact to enhance the quandle counting
invariant, providing an example of links with the same counting invariant
values but distinguished by the hom quandle structure. We generalize the result
to the case of biquandles, collect observations and results about abelian
quandles and the hom quandle, and show that the category of abelian quandles is
symmetric monoidal closed.Comment: 15 pages; revision 1 removes an incorrect remark; revision 2 corrects
some small typos. To appear in J. Knot Theory Ramification
Crossed modules of racks
We generalize the notion of a crossed module of groups to that of a crossed
module of racks. We investigate the relation to categorified racks, namely
strict 2-racks, and trunk-like objects in the category of racks, generalizing
the relation between crossed modules of groups and strict 2-groups. Then we
explore topological applications. We show that by applying the rack-space
functor, a crossed module of racks gives rise to a covering. Our main result
shows how the fundamental racks associated to links upstairs and downstairs in
a covering fit together to form a crossed module of racks.Comment: 25 pages, 1 figure, accepted in Homology, Homotopy and Application
Musical Actions of Dihedral Groups
The sequence of pitches which form a musical melody can be transposed or
inverted. Since the 1970s, music theorists have modeled musical transposition
and inversion in terms of an action of the dihedral group of order 24. More
recently music theorists have found an intriguing second way that the dihedral
group of order 24 acts on the set of major and minor chords. We illustrate both
geometrically and algebraically how these two actions are {\it dual}. Both
actions and their duality have been used to analyze works of music as diverse
as Hindemith and the Beatles.Comment: 27 pages, 11 figures. To appear in the American Mathematical Monthly
Exotic Statistics for Strings in 4d BF Theory
After a review of exotic statistics for point particles in 3d BF theory, and
especially 3d quantum gravity, we show that string-like defects in 4d BF theory
obey exotic statistics governed by the 'loop braid group'. This group has a set
of generators that switch two strings just as one would normally switch point
particles, but also a set of generators that switch two strings by passing one
through the other. The first set generates a copy of the symmetric group, while
the second generates a copy of the braid group. Thanks to recent work of
Xiao-Song Lin, we can give a presentation of the whole loop braid group, which
turns out to be isomorphic to the 'braid permutation group' of Fenn, Rimanyi
and Rourke. In the context 4d BF theory this group naturally acts on the moduli
space of flat G-bundles on the complement of a collection of unlinked unknotted
circles in R^3. When G is unimodular, this gives a unitary representation of
the loop braid group. We also discuss 'quandle field theory', in which the
gauge group G is replaced by a quandle.Comment: 41 pages, many figures. New version has minor corrections and
clarifications, and some added reference
Torsion in one-term distributive homology
The one-term distributive homology was introduced by J.H.Przytycki as an
atomic replacement of rack and quandle homology, which was first introduced and
developed by R.Fenn, C.Rourke and B.Sanderson, and J.S.Carter, S.Kamada and
M.Saito. This homology was initially suspected to be torsion-free, but we show
in this paper that the one-term homology of a finite spindle can have torsion.
We carefully analyze spindles of block decomposition of type (n,1) and
introduce various techniques to compute their homology precisely. In addition,
we show that any finite group can appear as the torsion subgroup of the first
homology of some finite spindle. Finally, we show that if a shelf satisfies a
certain, rather general, condition then the one-term homology is trivial.Comment: 17 pages, 2 PS-Tricks figure