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

Hom Quandles

If $A$ is an abelian quandle and $Q$ is a quandle, the hom set $\mathrm{Hom}(Q,A)$ of quandle homomorphisms from $Q$ to $A$ 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

The forbidden number of a knot

Every classical or virtual knot is equivalent to the unknot via a sequence of extended Reidemeister moves and the so-called forbidden moves. The minimum number of forbidden moves necessary to unknot a given knot is an invariant we call the {\it forbidden number}. We relate the forbidden number to several known invariants, and calculate bounds for some classes of virtual knots.Comment: 14 pages, many figures; v2 improves the upper bounds from the crossing number, and adds more detail to the data presented in the conclusio

The Hadamard Core of the Totally Nonnegative Matrices

An m-by- n matrix A is called totally nonnegative if every minor of A is nonnegative. The Hadamard product of two matrices is simply their entry-wise product. This paper introduces the subclass of totally nonnegative matrices whose Hadamard product with any totally nonnegative matrix is again totally nonnegative. Many properties concerning this class are discussed including: a complete characterization for min{m,n}\u3c4 \u3e; a characterization of the zero–nonzero patterns for which all totally nonnegative matrices lie in this class; and connections to Oppenheim\u27s inequality

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

What to Do on Your Summer Vacation

Summer vacation brings back fond memories: playing frisbee in the park, bike-riding till dusk, sipping lemonade on the porch with Grandpa, collecting shells at the beach, solving that difficult math problem you\u27ve been working on for the past several weeks. Wait a minute, you say? Many mathematics majors don\u27t realize that numerous summer opportunities exist (many of them paid, so you don\u27t need to get that job bagging groceries, too). A math program provides intellectual stimulation during those hot summer months, bringing your level of concentration up from swatting flies and applying sunscreen. The following descriptions are from actual participants in a few of the many available programs. You might even know students at your own school who would love to share their experiences with you. Your professors and academic advisors also serve as invaluable resources, and the Internet has a wealth of in formation. Start looking now to plan your summer of 2001