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In 1997, Jean-Claude Hausmann and Allen Knutson introduced a natural and beautiful correspondence between planar n-gons and the Grassmann manifold of 2-planes in real n-space. This construction leads to a natural probability distribution and a natural metric on polygons which has been used in shape classification and computer vision. In this paper, we provide an accessible introduction to this circle of ideas by explaining the Grassmannian geometry of triangles. We use this to find the probability that a random triangle is obtuse, which was a question raised by Lewis Carroll. We then explore the Grassmannian geometry of planar quadrilaterals, providing an answer to Sylvester's four-point problem, and describing explicitly the moduli space of unordered quadrilaterals. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new type of relation between knots called the descendant relation. One knot is a descendant of another knot if can be obtained from a minimal crossing diagram of by some number of crossing changes. We explore properties of the descendant relation and study how certain knots are related, paying particular attention to those knots, called fertile knots, that have a large number of descendants. Furthermore, we provide computational data related to various notions of knot fertility and propose several open questions for future exploration. [ABSTRACT FROM AUTHOR]

We consider the expected value for the total curvature of a random closed polygon. Numerical experiments have suggested that as the number of edges becomes large, the difference between the expected total curvature of a random closed polygon and a random open polygon with the same number of turning angles approaches a positive constant. We show that this is true for a natural class of probability measures on polygons, and give a formula for the constant in terms of the moments of the edge-length distribution. We then consider the symmetric measure on closed polygons of fixed total length constructed by Cantarella, Deguchi, and Shonkwiler. For this measure, we are able to prove that the expected value of total curvature for a closed n-gon is exactly π/2 n + π/4 2n/2n-3 . As a consequence, we show that at least 1/3 of fixed-length hexagons and 1/11 of fixed-length heptagons in R3 are unknotted. [ABSTRACT FROM AUTHOR]

We prove a version of symmetric criticality for ropelength-critical knots. Our theorem implies that a knot or link with a symmetric representative has a ropelength-critical configuration with the same symmetry. We use this to construct new examples of ropelength-critical configurations for knots and links which are different from the ropelength minima for these knot and link types. [ABSTRACT FROM AUTHOR]

We present new computations of tight shapes obtained using the constrained gradient descent code ridgerunner for 544 composite knots with 12 and fewer crossings, expanding our dataset to 943 knots and links. We use the new data set to analyze two outstanding conjectures about tight knots, namely that the ropelengths of composite knots are at least 4π - 4 less than the sums of the prime factors and that the writhes of composite knots are the sums of the writhes of the prime factors. Our numerics support the connect sum conjecture and argue against the additivity of writhe conjecture. We also present data on the number of configurations having straight segments and highly curved kinked regions. [ABSTRACT FROM AUTHOR]

We present an elementary derivation of the "intrinsic" symmetry groups for links of 8 or fewer crossings. We show that standard invariants are enough to rule out all potential symmetries outside the symmetry group of the group of the link for all but one of these links and present explicit isotopies generating the symmetry group for every link. [ABSTRACT FROM AUTHOR]

We consider the "intrinsic" symmetry group of a two-component link L, defined to be the image Σ(L) of the natural homomorphism from the standard symmetry group MCG(S³; L) to the product MCG(S³) x MCG(L). This group, first defined by Whitten in 1969, records directly whether L is isotopic to a link L' obtained from L by permuting components or reversing orientations; it is a subgroup of Γ2, the group of all such operations. For two-component links, we catalog the 27 possible intrinsic symmetry groups, which represent the subgroups of Γ2 up to conjugacy. We are able to provide prime, nonsplit examples for 21 of these groups; some are classically known, some are new. We catalog the frequency at which each group appears among all 77,036 of the hyperbolic two-component links of 14 or fewer crossings in Thistlethwaite's table. We also provide some new information about symmetry groups of the 293 non-hyperbolic two-component links of 14 or fewer crossings in the table. [ABSTRACT FROM AUTHOR]

Abstract: The helicity of a vector field is a measure of the average linking of pairs of integral curves of the field. Computed by a six-dimensional integral, it is widely useful in the physics of fluids. For a divergence-free field tangent to the boundary of a domain in 3-space, helicity is known to be invariant under volume-preserving diffeomorphisms of the domain that are homotopic to the identity. We give a new construction of helicity for closed -forms on a domain in -space that vanish when pulled back to the boundary of the domain. Our construction expresses helicity in terms of a cohomology class represented by the form when pulled back to the compactified configuration space of pairs of points in the domain. We show that our definition is equivalent to the standard one. We use our construction to give a new formula for computing helicity by a four-dimensional integral. We provide a Biot–Savart operator that computes a primitive for such forms; utilizing it, we obtain another formula for helicity. As a main result, we find a general formula for how much the value of helicity changes when the form is pushed forward by a diffeomorphism of the domain; it relies upon understanding the effect of the diffeomorphism on the homology of the domain and the de Rham cohomology class represented by the form. Our formula allows us to classify the helicity-preserving diffeomorphisms on a given domain, finding new helicity-preserving diffeomorphisms on the two-holed solid torus and proving that there are no new helicity-preserving diffeomorphisms on the standard solid torus. We conclude by defining helicities for forms on submanifolds of Euclidean space. In addition, we provide a detailed exposition of some standard ‘folk’ theorems about the cohomology of the boundary of domains in . [Copyright &y& Elsevier]

We find bounds on the difference between the writhing numbers of a smooth curve and a polygonal curve inscribed within. The proof is based on an extension of Fuller's difference of writhe formula to the case of polygonal curves. The results establish error bounds useful in the numerical computation of writhe in terms of bounds on the edge lengths of the polygon and the derivatives of the curve. The bounds are "adaptive" in the sense that they improve when regions of the smooth curve with larger derivatives are approximated by shorter edges of the polygon. [ABSTRACT FROM AUTHOR]

This paper provides bounds for the ropelength of a link in terms of the crossing numbers of its prime components. As in earlier papers, the bounds grow with the square of the crossing number; however, the constant involved is a substantial improvement on previous results. The proof depends essentially on writing links in terms of their arc-presentations, and has as a key ingredient Bae and Park''s theorem that an n-crossing link has an arc-presentation with less than or equal to n+2 arcs. [Copyright &y& Elsevier]

The convex hull of a set K in space consists of points which are, in a certain sense, "surrounded" by K. When K is a closed curve, we define its higher hulls, consisting of points which are "multiply surrounded" by the curve. Our main theorem shows that if a curve is knotted then it has a nonempty second hull. This provides a new proof of the Fáry/Milnor theorem that every knotted curve has total curvature at least 4π. [ABSTRACT FROM AUTHOR]

We define a new class of knot energies (known as renormalization energies) and prove that a broad class of these energies are uniquely minimized by the round circle. Most of O'Hara's knot energies belong to this class. This proves two conjectures of O'Hara and of Freedman, He, and Wang. We also find energies not minimized by a round circle. The proof is based on a theorem of Lu¨ko˝ on average chord lengths of closed curves. [Copyright &y& Elsevier]

Examines the relationship between the calculus of vector fields and the topology of their domains of definition. Statement of the problem; Organization of the proof; Examples of vector fields; Proof of the Hodge decomposition theorem.

We build a new probability measure on closed space and plane polygons. The key construction is a map, given by Hausmann and Knutson, using the Hopf map on quaternions from the complex Stiefel manifold of 2-frames in n-space to the space of closed n-gons in 3-space of total length 2. Our probability measure on polygon space is defined by pushing forward Haar measure on the Stiefel manifold by this map. A similar construction yields a probability measure on plane polygons that comes from a real Stiefel manifold. The edgelengths of polygons sampled according to our measures obey beta distributions. This makes our polygon measures different from those usually studied, which have Gaussian or fixed edgelengths. One advantage of our measures is that we can explicitly compute expectations and moments for chord lengths and radii of gyration. Another is that direct sampling according to our measures is fast (linear in the number of edges) and easy to code. Some of our methods will be of independent interest in studying other probability measures on polygon spaces. We define an edge set ensemble (ESE) to be the set of polygons created by rearranging a given set of n edges. A key theorem gives a formula for the average over an ESE of the squared lengths of chords skipping k vertices in terms of k, n, and the edgelengths of the ensemble. This allows one to easily compute expected values of squared chord lengths and radii of gyration for any probability measure on polygon space invariant under rearrangements of edges. © 2014 Wiley Periodicals, Inc. [ABSTRACT FROM AUTHOR]

The writhing number of a curve in 3-space is the standard measure of the extent to which the curve wraps and coils around itself; it has proved its importance for molecular biologists in the study of knotted DNA and of the enzymes which affect it. The helicity of a vector field defined on a domain in 3-space is the standard measure of the extent to which the field lines wrap and coil around one another; it plays important roles in fluid dynamics and plasma physics. The Biot-Savart operator associates with each current distribution on a given domain the restriction of its magnetic field to that domain. When the domain is simply connected, the divergence-free fields which are tangent to the boundary and which minimize energy for given helicity provide models for stable force-free magnetic fields in space and laboratory plasmas; these fields appear mathematically as the extreme eigenfields for an appropriate modification of the Biot-Savart operator. Information about these fields can be converted into bounds on the writhing number of a given piece of DNA. The purpose of this paper is to reveal new properties of the Biot-Savart operator which are useful in these applications. © 2001 American Institute of Physics. [ABSTRACT FROM AUTHOR]

The helicity of a smooth vector field defined on a domain in three-space is the standard measure of the extent to which the field lines wrap and coil around one another. It plays important roles in fluid mechanics, magnetohydrodynamics, and plasma physics. The isoperimetric problem in this setting is to maximize helicity among all divergence-free vector fields of given energy, defined on and tangent to the boundary of all domains of given volume in three-space. The Biot-Savart operator starts with a divergence-free vector field defined on and tangent to the boundary of a domain in three-space, regards it as a distribution of electric current, and computes its magnetic field. Restricting the magnetic field to the given domain, we modify it by subtracting a gradient vector field so as to keep it divergence-free while making it tangent to the boundary of the domain. The resulting operator, when extended to the L[sup 2] completion of this family of vector fields, is compact and self-adjoint, and thus has a largest eigenvalue, whose corresponding eigenfields are smooth by elliptic regularity. The isoperimetric problem for this modified Biot-Savart operator is to maximize its largest eigenvalue among all domains of given volume in three-space. The curl operator, when restricted to the image of the modified Biot-Savart operator, is its inverse, and the isoperimetric problem for this restriction of the curl is to minimize its smallest positive eigenvalue among all domains of given volume in three-space. These three isoperimetric problems are equivalent to one another. In this paper, we will derive the first variation formulas appropriate to these problems, and use them to constrain the nature of any possible solution. For example, suppose that the vector field V, defined on the compact, smoothly bounded domain Ω, maximizes helicity among all divergence-free vector fields of given nonzero energy, defined on and tangent to the boundary of all such domains of g... [ABSTRACT FROM AUTHOR]

This paper presents a mathematically complete derivation of the minimum-energy divergence-free vector fields of fixed helicity, defined on and tangent to the boundary of solid balls and spherical shells. These fields satisfy the equation ∇xV=λV, where λ is the eigenvalue of curl having smallest nonzero absolute value among such fields. It is shown that on the ball the energy minimizers are the axially symmetric spheromak fields found by Woltjer and Chandrasekhar-Kendall, and on spherical shells they are spheromak-like fields. The geometry and topology of these minimum-energy fields, as well as of some higher-energy eigenfields, are illustrated. © 2000 American Institute of Physics. [ABSTRACT FROM AUTHOR]