jonathan simon - Academia.edu (original) (raw)

Papers by jonathan simon

Proceedings of the American Mathematical Society, 1980

Proceedings of the American Mathematical Society, 1976

Modulo the conjecture that the complement of a prime knot in the 3-sphere is determined by its fu... more Modulo the conjecture that the complement of a prime knot in the 3-sphere is determined by its fundamental group, we show that at most finitely many mutually inequivalent knots can have homeomorphic complements.

Large bank holding companies (BHCs) are structured into intricate ownership hierarchies involving... more Large bank holding companies (BHCs) are structured into intricate ownership hierarchies involving hundreds or even thousands of legal entities. Each subsidiary in these hierarchies has its own legal form, assets, liabilities, managerial goals, and supervisory authorities. In the event of BHC default or insolvency, regulators may need to resolve the BHC and its constituent entities. Each entity individually will require some mix of cash infusion, outside purchase, consolidation with other subsidiaries, legal guarantees, and outright dissolution. The subsidiaries are not resolved in isolation, of course, but in the context of resolving the consolidated BHC at the top of the hierarchy. The number, diversity, and distribution of subsidiaries within the hierarchy can therefore significantly ease or complicate the resolution process. We use graph theory to develop a set of related metrics intended to assess the complexity BHC ownership. These proposed metrics focus on the graph quotient r...

Topology and its Applications, 2002

The Möbius energy of a knot is an energy functional for smooth curves based on an idea of self-re... more The Möbius energy of a knot is an energy functional for smooth curves based on an idea of self-repelling. If a knot has a thick tubular neighborhood, we would intuitively expect the energy to be low. In this paper, we give explicit bounds for energy in terms of the ropelength of the knot, i.e. the ratio of the length of a thickest tube to its radius.

Topology and its Applications, 1999

Topology and its Applications, 2007

We establish a new relationship between total curvature of knots and crossing number. If K is a s... more We establish a new relationship between total curvature of knots and crossing number. If K is a smooth knot in R 3 , R the cross-section radius of a uniform tube neighborhood K, L the arclength of K, and κ the total curvature of K, then crossing number of K < 4 L R κ. The proof generalizes to show that for smooth knots in R 3 , the crossing number, writhe, Möbius Energy, Normal Energy, and Symmetric Energy are all bounded by the product of total curvature and rope-length. One can construct knots in which the crossing numbers grow as fast as the (4/3) power of L R. Our theorem says that such families must have unbounded total curvature: If the total curvature is bounded, then the rate of growth of crossings with ropelength can only be linear. Our proof relies on fundamental lemmas about the total curvature of curves that are packed in certain ways: If a long smooth curve A with arclength L is contained in a solid ball of radius ρ, then the total curvature of K is at least proportional to L/ρ. If A connects concentric spheres of radii a ≥ 2 and b ≥ a + 1, by running from the inner sphere to the outer sphere and back again, then the total curvature of A is at least proportional to 1/ √ a.

Topology and its Applications, 1999

Classical knot theory studies one-dimensional filaments; in this paper we model knots as more phy... more Classical knot theory studies one-dimensional filaments; in this paper we model knots as more physically "real". e.g., made of some "rope" with nonLero thickness. A motivating question is: How much length of unit radius rope is needed to tie a nontrivial knot? For a smooth knot K, the "injectivity radius" R(I<) is the supremum of radii of embedded tubular neighborhoods. The "thickness" of Ii, a new measure of knot complexity, is the ratio of R(IY) to arc-length. We relate thickness to curvature. self-distance. distortion, and (for knot types) edge-number.

Contemporary Mathematics, 2002

What happens to knot theory when the knots, traditionally studied as purely one dimensional, comp... more What happens to knot theory when the knots, traditionally studied as purely one dimensional, completely flexible filaments, are given physical substance-in the form of thickness, rigidity, or some kind of self-repelling? Researchers have developed several measures of knot complexity, modeled on these kinds of physical "reality". We shall explore these ideas, see relations between different notions of complexity, and compare the "ideal" conformations of knots that arise. We also note that there are observed relationships between these measures of complexity and behavior of actual knotted DNA molecules. The development of "Physical Knot Theory" is characterized by growing understanding along with an ample supply of open questions. 1991 Mathematics Subject Classification. 57M25.

Transactions of the American Mathematical Society, 1971

If a polygonal knot K in the 3-sphere Sa does not separate an interpolating manifold S for K, the... more If a polygonal knot K in the 3-sphere Sa does not separate an interpolating manifold S for K, then S-K does not carry the first homology of either closed component of S3-S. It follows that most knots K with nontrivial interpolating manifolds have the property that a simply connected manifold cannot be obtained by removing a regular neighborhood of K from S3 and sewing it back differently.

Topology and its Applications, 1993

Invariants of theta-curves and other graphs in 3-space, Topology and its Applications 49 (1993) 1... more Invariants of theta-curves and other graphs in 3-space, Topology and its Applications 49 (1993) 193-216. Given a graph in 3-space, in general knotted, can one construct a surface containing the graph in some canonical way so that the embedding of the surface in space, or even the link type of its boundary, is an invariant of the knotted graph? We consider, in particular, surfaces that contain the graph as a spine and that are canonical in the sense of having trivial Seifert linking form. It turns out that &curves and &-graphs are the unique graphs for which this approach works.

Topology and its Applications, 1993

Buck, G. and J. Simon, Knots as dynamical systems, Topology and its Applications 51 (1993) 229924... more Buck, G. and J. Simon, Knots as dynamical systems, Topology and its Applications 51 (1993) 2299246. A knot is considered as an n-gon in Iw". Two potential energies for these PL knot conformations are found, mapping iws" + Iw (the configuration space to energy). These functions have the property that they "blow up" if the edges approach crossing, that is, as the knot changes type. Therefore the configuration space is divided into manifolds for each knot type by infinitely high potential walls. Therefore we have energy surfaces associated with each knot type. Descriptions of these surfaces are invariants of the knot type. For example the locations of and connections between the critical points are invariants of the knot type. In particular, the global minimum energy position for the knot could be said to be a canonical conformation of the knot. The existence and achievement of the global minimum is proved. It is shown that the energy surface for any knot of N vertices has a finite number of components. Also several topological properties of the configuration space of polygonal knots in [w3 are established. These invariants are in some sense computable. In a sequel paper, G. Buck and J. Orloff discuss computer simulations of the gradient flow of one of the energy functions.

Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2012

DNA, hair, shoelaces, vortex lines, rope, proteins, integral curves, thread, magnetic flux tubes,... more DNA, hair, shoelaces, vortex lines, rope, proteins, integral curves, thread, magnetic flux tubes, cosmic strings and extension cords; filaments come in all sizes and with diverse qualities. Filaments tangle, with profound results: DNA replication is halted, field energy is stored, polymer materials acquire their remarkable properties, textiles are created and shoes stay on feet. We classify entanglement patterns by the rate with which entanglement complexity grows with the length of the filament. We show which rates are possible and which are expected in arbitrary circumstances. We identify a fundamental phase transition between linear and nonlinear entanglement rates. We also find (perhaps surprising) relationships between total curvature, bending energy and entanglement.

L’Enseignement Mathématique, 2010

Journal of Knot Theory and Its Ramifications, 2008

The image of a polygonal knot K under a spherical inversion of ℝ3 ∪ ∞ is a simple closed curve ma... more The image of a polygonal knot K under a spherical inversion of ℝ3 ∪ ∞ is a simple closed curve made of arcs of circles, perhaps some line segments, having the same knot type as the mirror image of K. But suppose we reconnect the vertices of the inverted polygon with straight lines, making a new polygon [Formula: see text]. This may be a different knot type. For example, a certain 7-segment figure-eight knot can be transformed to a figure-eight knot, a trefoil, or an unknot, by selecting different inverting spheres. Which knot types can be obtained from a given original polygon K under this process? We show that for large n, most n-segment knot types cannot be reached from one initial n-segment polygon, using a single inversion or even the whole Möbius group. The number of knot types is bounded by the number of complementary domains of a certain system of round 2-spheres in ℝ3. We show the number of domains is at most polynomial in the number of spheres, and the number of spheres is ...

Journal of Knot Theory and Its Ramifications, 2006

We establish a fundamental connection between smooth and polygonal knot energies, showing that th... more We establish a fundamental connection between smooth and polygonal knot energies, showing that the Minimum Distance Energy for polygons inscribed in a smooth knot converges to the Möbius Energy of the smooth knot as the polygons converge to the smooth knot. For this to work, the polygons must converge in a "nice" way, and the energies must be correctly regularized. We determine an explicit error bound for the convergence.

At least one co-author has disclosed a financial relationship of potential relevance for this res... more At least one co-author has disclosed a financial relationship of potential relevance for this research. Further information is available online at http://www.nber.org/papers/w23755.ack NBER working papers are circulated for discussion and comment purposes. They have not been peer-reviewed or been subject to the review by the NBER Board of Directors that accompanies official NBER publications.

Large bank holding companies (BHCs) are structured into intricate ownership hierarchies involving... more Large bank holding companies (BHCs) are structured into intricate ownership hierarchies involving hundreds or even thousands of legal entities. Each subsidiary in these hierarchies has its own legal form, assets, liabilities, managerial goals, and supervisory authorities. In the event of BHC default or insolvency, regulators may need to resolve the BHC and its constituent entities. Each entity individually will require some mix of cash infusion, outside purchase, consolidation with other subsidiaries, legal guarantees, and outright dissolution. The subsidiaries are not resolved in isolation, of course, but in the context of resolving the consolidated BHC at the top of the hierarchy. The number, diversity, and distribution of subsidiaries within the hierarchy can therefore significantly ease or complicate the resolution process. We propose a set of related metrics intended to assess the complexity of the BHC ownership graph. These proposed metrics focus on the graph quotient relative to certain well identified partitions on the set of subsidiaries, such as charter type and regulatory jurisdiction. The intended measures are mathematically grounded, intuitively sensible, and easy to implement. We illustrate the process with a case study of one large U.S. BHC.

Proceedings of the American Mathematical Society, 1980

Proceedings of the American Mathematical Society, 1976

Modulo the conjecture that the complement of a prime knot in the 3-sphere is determined by its fu... more Modulo the conjecture that the complement of a prime knot in the 3-sphere is determined by its fundamental group, we show that at most finitely many mutually inequivalent knots can have homeomorphic complements.

Large bank holding companies (BHCs) are structured into intricate ownership hierarchies involving... more Large bank holding companies (BHCs) are structured into intricate ownership hierarchies involving hundreds or even thousands of legal entities. Each subsidiary in these hierarchies has its own legal form, assets, liabilities, managerial goals, and supervisory authorities. In the event of BHC default or insolvency, regulators may need to resolve the BHC and its constituent entities. Each entity individually will require some mix of cash infusion, outside purchase, consolidation with other subsidiaries, legal guarantees, and outright dissolution. The subsidiaries are not resolved in isolation, of course, but in the context of resolving the consolidated BHC at the top of the hierarchy. The number, diversity, and distribution of subsidiaries within the hierarchy can therefore significantly ease or complicate the resolution process. We use graph theory to develop a set of related metrics intended to assess the complexity BHC ownership. These proposed metrics focus on the graph quotient r...

Topology and its Applications, 2002

The Möbius energy of a knot is an energy functional for smooth curves based on an idea of self-re... more The Möbius energy of a knot is an energy functional for smooth curves based on an idea of self-repelling. If a knot has a thick tubular neighborhood, we would intuitively expect the energy to be low. In this paper, we give explicit bounds for energy in terms of the ropelength of the knot, i.e. the ratio of the length of a thickest tube to its radius.

Topology and its Applications, 1999

Topology and its Applications, 2007

We establish a new relationship between total curvature of knots and crossing number. If K is a s... more We establish a new relationship between total curvature of knots and crossing number. If K is a smooth knot in R 3 , R the cross-section radius of a uniform tube neighborhood K, L the arclength of K, and κ the total curvature of K, then crossing number of K < 4 L R κ. The proof generalizes to show that for smooth knots in R 3 , the crossing number, writhe, Möbius Energy, Normal Energy, and Symmetric Energy are all bounded by the product of total curvature and rope-length. One can construct knots in which the crossing numbers grow as fast as the (4/3) power of L R. Our theorem says that such families must have unbounded total curvature: If the total curvature is bounded, then the rate of growth of crossings with ropelength can only be linear. Our proof relies on fundamental lemmas about the total curvature of curves that are packed in certain ways: If a long smooth curve A with arclength L is contained in a solid ball of radius ρ, then the total curvature of K is at least proportional to L/ρ. If A connects concentric spheres of radii a ≥ 2 and b ≥ a + 1, by running from the inner sphere to the outer sphere and back again, then the total curvature of A is at least proportional to 1/ √ a.

Topology and its Applications, 1999

Classical knot theory studies one-dimensional filaments; in this paper we model knots as more phy... more Classical knot theory studies one-dimensional filaments; in this paper we model knots as more physically "real". e.g., made of some "rope" with nonLero thickness. A motivating question is: How much length of unit radius rope is needed to tie a nontrivial knot? For a smooth knot K, the "injectivity radius" R(I<) is the supremum of radii of embedded tubular neighborhoods. The "thickness" of Ii, a new measure of knot complexity, is the ratio of R(IY) to arc-length. We relate thickness to curvature. self-distance. distortion, and (for knot types) edge-number.

Contemporary Mathematics, 2002

What happens to knot theory when the knots, traditionally studied as purely one dimensional, comp... more What happens to knot theory when the knots, traditionally studied as purely one dimensional, completely flexible filaments, are given physical substance-in the form of thickness, rigidity, or some kind of self-repelling? Researchers have developed several measures of knot complexity, modeled on these kinds of physical "reality". We shall explore these ideas, see relations between different notions of complexity, and compare the "ideal" conformations of knots that arise. We also note that there are observed relationships between these measures of complexity and behavior of actual knotted DNA molecules. The development of "Physical Knot Theory" is characterized by growing understanding along with an ample supply of open questions. 1991 Mathematics Subject Classification. 57M25.

Transactions of the American Mathematical Society, 1971

If a polygonal knot K in the 3-sphere Sa does not separate an interpolating manifold S for K, the... more If a polygonal knot K in the 3-sphere Sa does not separate an interpolating manifold S for K, then S-K does not carry the first homology of either closed component of S3-S. It follows that most knots K with nontrivial interpolating manifolds have the property that a simply connected manifold cannot be obtained by removing a regular neighborhood of K from S3 and sewing it back differently.

Topology and its Applications, 1993

Invariants of theta-curves and other graphs in 3-space, Topology and its Applications 49 (1993) 1... more Invariants of theta-curves and other graphs in 3-space, Topology and its Applications 49 (1993) 193-216. Given a graph in 3-space, in general knotted, can one construct a surface containing the graph in some canonical way so that the embedding of the surface in space, or even the link type of its boundary, is an invariant of the knotted graph? We consider, in particular, surfaces that contain the graph as a spine and that are canonical in the sense of having trivial Seifert linking form. It turns out that &curves and &-graphs are the unique graphs for which this approach works.

Topology and its Applications, 1993

Buck, G. and J. Simon, Knots as dynamical systems, Topology and its Applications 51 (1993) 229924... more Buck, G. and J. Simon, Knots as dynamical systems, Topology and its Applications 51 (1993) 2299246. A knot is considered as an n-gon in Iw". Two potential energies for these PL knot conformations are found, mapping iws" + Iw (the configuration space to energy). These functions have the property that they "blow up" if the edges approach crossing, that is, as the knot changes type. Therefore the configuration space is divided into manifolds for each knot type by infinitely high potential walls. Therefore we have energy surfaces associated with each knot type. Descriptions of these surfaces are invariants of the knot type. For example the locations of and connections between the critical points are invariants of the knot type. In particular, the global minimum energy position for the knot could be said to be a canonical conformation of the knot. The existence and achievement of the global minimum is proved. It is shown that the energy surface for any knot of N vertices has a finite number of components. Also several topological properties of the configuration space of polygonal knots in [w3 are established. These invariants are in some sense computable. In a sequel paper, G. Buck and J. Orloff discuss computer simulations of the gradient flow of one of the energy functions.

Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2012

DNA, hair, shoelaces, vortex lines, rope, proteins, integral curves, thread, magnetic flux tubes,... more DNA, hair, shoelaces, vortex lines, rope, proteins, integral curves, thread, magnetic flux tubes, cosmic strings and extension cords; filaments come in all sizes and with diverse qualities. Filaments tangle, with profound results: DNA replication is halted, field energy is stored, polymer materials acquire their remarkable properties, textiles are created and shoes stay on feet. We classify entanglement patterns by the rate with which entanglement complexity grows with the length of the filament. We show which rates are possible and which are expected in arbitrary circumstances. We identify a fundamental phase transition between linear and nonlinear entanglement rates. We also find (perhaps surprising) relationships between total curvature, bending energy and entanglement.

L’Enseignement Mathématique, 2010

Journal of Knot Theory and Its Ramifications, 2008

The image of a polygonal knot K under a spherical inversion of ℝ3 ∪ ∞ is a simple closed curve ma... more The image of a polygonal knot K under a spherical inversion of ℝ3 ∪ ∞ is a simple closed curve made of arcs of circles, perhaps some line segments, having the same knot type as the mirror image of K. But suppose we reconnect the vertices of the inverted polygon with straight lines, making a new polygon [Formula: see text]. This may be a different knot type. For example, a certain 7-segment figure-eight knot can be transformed to a figure-eight knot, a trefoil, or an unknot, by selecting different inverting spheres. Which knot types can be obtained from a given original polygon K under this process? We show that for large n, most n-segment knot types cannot be reached from one initial n-segment polygon, using a single inversion or even the whole Möbius group. The number of knot types is bounded by the number of complementary domains of a certain system of round 2-spheres in ℝ3. We show the number of domains is at most polynomial in the number of spheres, and the number of spheres is ...

Journal of Knot Theory and Its Ramifications, 2006

We establish a fundamental connection between smooth and polygonal knot energies, showing that th... more We establish a fundamental connection between smooth and polygonal knot energies, showing that the Minimum Distance Energy for polygons inscribed in a smooth knot converges to the Möbius Energy of the smooth knot as the polygons converge to the smooth knot. For this to work, the polygons must converge in a "nice" way, and the energies must be correctly regularized. We determine an explicit error bound for the convergence.

At least one co-author has disclosed a financial relationship of potential relevance for this res... more At least one co-author has disclosed a financial relationship of potential relevance for this research. Further information is available online at http://www.nber.org/papers/w23755.ack NBER working papers are circulated for discussion and comment purposes. They have not been peer-reviewed or been subject to the review by the NBER Board of Directors that accompanies official NBER publications.

Large bank holding companies (BHCs) are structured into intricate ownership hierarchies involving... more Large bank holding companies (BHCs) are structured into intricate ownership hierarchies involving hundreds or even thousands of legal entities. Each subsidiary in these hierarchies has its own legal form, assets, liabilities, managerial goals, and supervisory authorities. In the event of BHC default or insolvency, regulators may need to resolve the BHC and its constituent entities. Each entity individually will require some mix of cash infusion, outside purchase, consolidation with other subsidiaries, legal guarantees, and outright dissolution. The subsidiaries are not resolved in isolation, of course, but in the context of resolving the consolidated BHC at the top of the hierarchy. The number, diversity, and distribution of subsidiaries within the hierarchy can therefore significantly ease or complicate the resolution process. We propose a set of related metrics intended to assess the complexity of the BHC ownership graph. These proposed metrics focus on the graph quotient relative to certain well identified partitions on the set of subsidiaries, such as charter type and regulatory jurisdiction. The intended measures are mathematically grounded, intuitively sensible, and easy to implement. We illustrate the process with a case study of one large U.S. BHC.