American Mathematical Monthly - January 2017

Welcome to the New Year! In the January issue of the Monthly you can resolve to: study the complexity of knots using bridge numbers, bridge distances, and the Gordian graph; learn how a Slinky slinks as it falls; review the history of the Putnam Competition; ponder the moduli space of triangles (which is a another triangle)—and much more.

In addition, there are problems to keep you warm. And Jason Rosenhouse, the Monthly’s new Reviews Editor, introduces himself and his plans for the section via a discussion of mathematics books he has hated and loved.

As the new Editor of the Monthly, I open the issue with a letter of greeting and gratitude.

— Susan Jane Colley, Editor

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A Letter from the Editor

p. 3.

Susan Jane Colley

DOI: http://dx.doi.org/10.4169/amer.math.monthly.124.1.3

Neighbors of Knots in the Gordian Graph

p. 4.

Ryan Blair, Marion Campisi, Jesse Johnson, Scott A. Taylor, and Maggy Tomova

The Gordian graph is the graph with vertex set the set of knot types and edge set consisting of pairs of knots that have a diagram wherein they differ at a single crossing. The bridge number is a classical knot invariant that is a measure of the complexity of a knot. It can be refined by another, recently discovered, knot invariant known as “bridge distance.” We show, using arguments that are almost entirely elementary, that each vertex of the Gordian graph is adjacent to a vertex having arbitrarily high bridge number and bridge distance.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.124.1.4

The Falling Slinky

p. 24.

Robert J. Vanderbei

It is an interesting and counterintuitive fact that a Slinky released from a hanging position does not begin to fall all at once but rather each part of the Slinky only starts to fall when the collapsed part above it reaches its level. The analyses published so far have given physical arguments to explain this property of Slinkies. In particular, they have relied on the fact that a perturbation to a Slinky travels through the Slinky as a wave and therefore has a certain propagation speed. Releasing a Slinky that was being held at the top is a perturbation at the top, and it takes time for that perturbation to propagate downward. This “high-level” analysis is, of course, correct. But, it is also interesting to analyze the dynamics from a purely mathematical perspective. We present such a careful mathematical analysis. It turns out that we can derive an explicit formula for the solution to the differential equation, and from that solution, we see that the effect of gravity exactly counteracts the tension in the Slinky. The mathematical analysis turns out to be as interesting as the physics.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.124.1.24

Irreducibility Crieria for Reciprocal Polynomials and Applications

p. 37.

Antonio Cafure and Eda Cesaratto

We present criteria for determining irreducibility of reciprocal polynomials over the field of rational numbers. We also obtain some combinatorial results concerning the irreducibility of reciprocal polynomials. As a consequence of our approach, we are able to deal with other problems such as factorization properties of Chebyshev polynomials of the first and second kind and with the classical problems of computing minimal polynomials of algebraic values of trigonometric functions.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.124.1.37

Seventy-Five Years of the Putnam Mathematical Competition

p. 54.

Joseph A. Gallian

We review the 75-year history of the world’s foremost college-level mathematics competition. Evidence is provided to support the assertion that exceptional performance in premier secondary-school-level and college-level mathematics competitions correlates well with an exceptional research career in mathematics.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.124.1.54

On a 2015 Putnam Problem Related to a Double Recursion

p. 60.

Nicolae Anghel

We generalize a 2015 Putnam competition problem via a binomial-type double recursion.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.124.1.60

Notes

Why Do All Triangles Form a Triangle?

p. 70.

Ian Stewart

We provide a geometric explanation, based on symmetry, for why the moduli space of all triangles up to similarity is itself a triangle. Symmetries occur because the lengths of the sides define triples in $\mathbb {R}$ ³ so are acted on by the symmetric group $\mathbb {S}$ ₃, which is isomorphic to the symmetry group $\mathbb {D}$ ₃ of an equilateral triangle. The moduli space for triangles is a fundamental domain for the action of $\mathbb {D}$ ₃ on an equilateral triangle in $\mathbb {R}$ ³ determined by all triangles with unit perimeter and is chosen from a subdivision into six congruent triangles. Isosceles and equilateral triangles occupy special locations determined by their symmetries. The sides of a right triangle lie on one of three double cones in $\mathbb {R}$ ³, and those of unit perimeter lie on a segment of a hyperbola in the moduli space.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.124.1.70

Chebyshev Polynomials and the Minimal Polynomial of cos(2π/n)

p. 74.

Yusuf Z. Gürtaş

The minimal polynomial of cos(2π/n) allows one to realize the value of cos(2π/n) as the root of a polynomial with rational coefficients. These polynomials prove to be instrumental in expressing some relations satisfied by Chebyshev polynomials as a product. In this article a few relations satisfied by Chebyshev polynomials of the first and second kind and the minimal polynomial of cos(2π/n) are presented. The proof of the main theorem shows how cyclotomic polynomials can be used to link these two kinds of polynomials.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.124.1.74

Two-Sided, Unbiased Version of Hall's Marriage Theorem

p. 79.

Eli Shamir and Benny Sudakov

The standard conditions in Hall’s perfect matching theorem for a bipartite graph G require that all subsets from one side of G are expanding. The unbiased extension identifies mixtures of subsets from both sides such that their expansions imply the standard conditions—hence a perfect matching.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.124.1.79

Hausdorffization and Homotopy

p. 81.

Bart van Munster

As discussed in an earlier article in this Monthly, there is a universal construction in topology that turns any space into a Hausdorff space, which is known as the Hausdorffization, Hausdorffication, Hausdorffification or Hausdorff quotient. In this article, I will discuss the surprising compatibility of this Hausdorff quotient with homotopy.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.124.1.81