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Operads and knot spaces
 J. Amer. Math. Soc
"... Let Em denote the space of embeddings of the interval I = [−1, 1] in the cube I m with endpoints and tangent vectors at those endpoints fixed on opposite faces of the cube, equipped with a homotopy through immersions to the unknot – see Definition 5.1. By Proposition 5.17, Em is homotopy equivalent ..."
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Cited by 24 (2 self)
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Let Em denote the space of embeddings of the interval I = [−1, 1] in the cube I m with endpoints and tangent vectors at those endpoints fixed on opposite faces of the cube, equipped with a homotopy through immersions to the unknot – see Definition 5.1. By Proposition 5.17, Em is homotopy equivalent to Emb(I, I m) × ΩImm(I, I m). In [28], McClure and Smith define a cosimplicial object O • associated
Quadrisecants give new lower bounds for the ropelength of a knot
, 2005
"... Using the existence of a special quadrisecant line, we show the ropelength of any nontrivial knot is at least 15.66. This improves the previously known lower bound of 12. Numerical experiments have found a trefoil with ropelength less than 16.372, so our new bounds are quite sharp. ..."
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Cited by 10 (4 self)
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Using the existence of a special quadrisecant line, we show the ropelength of any nontrivial knot is at least 15.66. This improves the previously known lower bound of 12. Numerical experiments have found a trefoil with ropelength less than 16.372, so our new bounds are quite sharp.
Alternating Quadrisecants of Knots
, 2004
"... A knot is a simple closed curve in R³. A secant line is a straight line which intersects the knot in at least two distinct places. Trisecant, quadrisecant and quintisecant lines are straight lines which intersect a knot in at least three, four and five distinct places, respectively. It is clear that ..."
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Cited by 8 (4 self)
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A knot is a simple closed curve in R³. A secant line is a straight line which intersects the knot in at least two distinct places. Trisecant, quadrisecant and quintisecant lines are straight lines which intersect a knot in at least three, four and five distinct places, respectively. It is clear that any closed curve has secants. A little thought will reveal that nontrivial knots must have trisecants, but they do not necessarily have quintisecants. The relationship between knots and quadrisecants is not so immediately clear. In 1933, E. Pannwitz proved that nontrivial generic polygonal knots have at least one quadrisecant. In 1994, G. Kuperberg showed that all (nontrivial tame) knots have at least one quadrisecant. Quadrisecants come in three basic types. These are distinguished by comparing the orders of the four points along the knot and along the quadrisecant line. These three types are labeled simple, flipped and alternating. It turns out that alternating quadrisecants capture the knottedness of a knot. The Main Theorem shows that every nontrivial tame knot in R³ has an alternating quadrisecant. This result refines the previous work about quadrisecants and gives greater geometric insight into knots. The Main Theorem provides new proofs to two previously known theorems about the total curvature and second hull of knotted curves. Moreover, essential alternating quadrisecants may be used to dramatically improve the known lower bounds on the ropelength of thick knots.
Lie coalgebras and rational homotopy theory
"... In this paper we complete the picture of Lie and commutative algebra and coalgebra models for rational spaces by giving a new, combinatorially rich Lie coalgebra model. Consider the following square of categories and functors between them. ..."
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Cited by 5 (2 self)
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In this paper we complete the picture of Lie and commutative algebra and coalgebra models for rational spaces by giving a new, combinatorially rich Lie coalgebra model. Consider the following square of categories and functors between them.
A NEW COHOMOLOGICAL FORMULA FOR HELICITY IN R 2k+1 REVEALS THE EFFECT OF A DIFFEOMORPHISM ON HELICITY
, 903
"... 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 sixdimensional integral, it is widely useful in the physics of fluids. For a divergencefree field tangent to the boundary of a domain in 3space, helicity is known t ..."
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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 sixdimensional integral, it is widely useful in the physics of fluids. For a divergencefree field tangent to the boundary of a domain in 3space, helicity is known to be invariant under volumepreserving diffeomorphisms of the domain that are homotopic to the identity. We give a new construction of helicity for closed (k + 1)forms on a domain in (2k + 1)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 fourdimensional integral. We provide a BiotSavart 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 helicitypreserving diffeomorphisms on a given domain, finding new helicitypreserving diffeomorphisms on the twoholed solid torus and proving that there are no new helicitypreserving 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 R 2k+1. 1.
HOMOTOPY APPROXIMATIONS TO THE SPACE OF KNOTS, FEYNMAN DIAGRAMS, AND A CONJECTURE OF SCANNELL AND SINHA
, 2006
"... ABSTRACT. Scannell and Sinha considered a spectral sequence to calculate the rational homotopy groups of spaces of long knots in R n, for n ≥ 4. At the end of the paper they conjecture that when n is odd, the terms on the antidiagonal at the E 2 stage precisely give the space of primitive Feynman di ..."
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ABSTRACT. Scannell and Sinha considered a spectral sequence to calculate the rational homotopy groups of spaces of long knots in R n, for n ≥ 4. At the end of the paper they conjecture that when n is odd, the terms on the antidiagonal at the E 2 stage precisely give the space of primitive Feynman diagrams related to the theory of Vassiliev invariants. In this paper we prove that conjecture. This has the application that the path components of the terms of the Taylor tower for the space of long knots in R 3 are in onetoone correspondence with quotients of the module of Feynman diagrams, even though the Taylor tower does not actually converge. This provides strong evidence that the stages of the Taylor tower give rise to universal Vassiliev knot invariants in each degree. 1.
The topology of spaces of knots
, 2003
"... We present two models for the space of knots which have endpoints at fixed boundary points in a manifold with boundary, one model defined as an inverse limit of mapping spaces and another which is cosimplicial. At the geometric heart of these constructions is the evaluation map, used elsewhere for e ..."
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We present two models for the space of knots which have endpoints at fixed boundary points in a manifold with boundary, one model defined as an inverse limit of mapping spaces and another which is cosimplicial. At the geometric heart of these constructions is the evaluation map, used elsewhere for example to define linking number and BottTaubes integrals. Our models are weakly homotopy equivalent to the corresponding knot spaces when the dimension of the ambient manifold is greater than three. There are spectral sequences with identifiable E¹ terms which converge to their cohomology and homotopy groups.
FultonMacPherson compactification, cyclohedra, and the polygonal pegs problem
, 810
"... The cyclohedron Wn, known also as the BottTaubes polytope, arises both as the polyhedral realization of the poset of all cyclic bracketings of the word x1x2... xn and as an essential part of the FultonMacPherson compactification of the configuration space of n distinct, labelled points on the circ ..."
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The cyclohedron Wn, known also as the BottTaubes polytope, arises both as the polyhedral realization of the poset of all cyclic bracketings of the word x1x2... xn and as an essential part of the FultonMacPherson compactification of the configuration space of n distinct, labelled points on the circle S 1. The “polygonal pegs problem ” asks whether every simple, closed curve in the plane or in the higher dimensional space admits an inscribed polygon of a given shape. We develop a new approach to the polygonal pegs problem based on the FultonMacPherson (AxelrodSinger, Kontsevich) compactification of the configuration space of (cyclically) ordered nelement subsets in S 1. Among the new results obtained by this method are proofs of Grünbaum’s conjecture about inscribed affine regular hexagons in smooth Jordan curves and the conjecture of Hadwiger about inscribed parallelograms in smooth, simple, closed curves in the 3space. 1
Homotopy Approximations To The Space Of Knots, Feynman Diagrams,
"... Scannell and Sinha considered a spectral sequence to calculate the rational homotopy groups of spaces of long knots in R , for n 4. At the end of the paper they conjecture that when n is odd, the terms on the antidiagonal at the E stage precisely give the space of primitive Feynman diagra ..."
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Scannell and Sinha considered a spectral sequence to calculate the rational homotopy groups of spaces of long knots in R , for n 4. At the end of the paper they conjecture that when n is odd, the terms on the antidiagonal at the E stage precisely give the space of primitive Feynman diagrams related to the theory of Vassiliev invariants. In this paper we prove that conjecture. This has the application that the path components of the terms of the Taylor tower for the space of long knots in R are in onetoone correspondence with quotients of the module of Feynman diagrams, even though the Taylor tower does not actually converge. This provides strong evidence that the stages of the Taylor tower give rise to universal Vassiliev knot invariants in each degree.