A generalization of Milnor's μ-invariants to higher-dimensional link maps

Abstract. The 2-twist spun trefoil is an example of a sphere that is knotted in 4-dimensional space. A proof is given in this paper that this sphere is distinct from the same sphere with its orientation reversed. Our proof is based on a state-sum invariant for knotted surfaces developed via a cohomology theory of racks and quandles (also known as distributive groupoids).A quandle is a set with a binary operation -the axioms of which model the Reidemeister moves in classical knot theory. Colorings of diagrams of knotted curves and surfaces by quandle elements, together with cocycles of quandles, are used to define state-sum invariants for knotted circles in 3-space and knotted surfaces in 4-space.Cohomology groups of various quandles are computed herein and applied to the study of the state-sum invariants. Non-triviality of the invariants is proved for a variety of knots and links, and conversely, knot invariants are used to prove non-triviality of cohomology for a variety of quandles.

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“…Although we provided a diagrammatic definition and proof, this invariant has been known in different contexts; see [44,45,32] for example. …”

Section: Lemma the Numbers T (I J K) Are Invariants Of Isotopy Clamentioning

confidence: 99%

Quandle cohomology and state-sum invariants of knotted curves and surfaces

Carter

Jelsovsky

Kamada

et al. 2003

Trans. Amer. Math. Soc.

336

456

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“…This degree is computed by the Gauss integral. In fact, Koschorke has shown [26] that all Milnor linking numbers may be understood through evaluation maps from ðS 1 Þ k to C k ðR 3 Þ: For knots, the situation is more subtle because one is led to consider maps from the space of configurations on the knot, which is an open simplex, to C k ðR 3 Þ: The technical heart of the matter is what to do ''on the boundary'' of the configuration space, including what boundary to use in the first place. In [36] the fourth author showed that the appropriate boundary conditions are prescribed when one relates the calculus of embeddings to the evaluation map.…”

Section: Spaces Of Knots and Evaluation Mapsmentioning

confidence: 97%

New perspectives on self-linking

Budney

Conant

Scannell

et al. 2005

Advances in Mathematics

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We initiate the study of classical knots through the homotopy class of the nth evaluation map of the knot, which is the induced map on the compactified n-point configuration space. Sending a knot to its nth evaluation map realizes the space of knots as a subspace of what we call the nth mapping space model for knots. We compute the homotopy types of the first three mapping space models, showing that the third model gives rise to an integer-valued invariant. We realize this invariant in two ways, in terms of collinearities of three or four points on the knot, and give some explicit computations. We show this invariant coincides with the second coefficient of the Conway polynomial, thus giving a new geometric definition of the simplest finite-type invariant. Finally, using this geometric definition, we give some new applications of this invariant relating to quadrisecants in the knot and to complexity of polygonal and polynomial realizations of a knot. r

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“…For n-component homotopy Brunnian links in R 3 -meaning links that become trivial up to link homotopy when any single component is removed-and analogous links in higher dimensions, Koschorke 31 showed that the representation e is again faithful. This provided the first proof (up to sign) of our Theorem A for the case when the pairwise linking numbers are zero.…”

Section: Configuration Spacesmentioning

confidence: 99%

Generalized Gauss maps and integrals for three-component links: Toward higher helicities for magnetic fields and fluid flows

DeTurck

Gluck

Komendarczyk

et al. 2013

Journal of Mathematical Physics

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To each three-component link in the 3-sphere we associate a generalized Gauss map from the 3-torus to the 2-sphere, and show that the pairwise linking numbers and Milnor triple linking number that classify the link up to link homotopy correspond to the Pontryagin invariants that classify its generalized Gauss map up to homotopy. We view this as a natural extension of the familiar situation for two-component links in 3-space, where the linking number is the degree of the classical Gauss map from the 2-torus to the 2-sphere. The generalized Gauss map, like its prototype, is geometrically natural in the sense that it is equivariant with respect to orientationpreserving isometries of the ambient space, thus positioning it for application to physical situations. When the pairwise linking numbers of a three-component link are all zero, we give an integral formula for the triple linking number analogous to the Gauss integral for the pairwise linking numbers. This new integral is also geometrically natural, like its prototype, in the sense that the integrand is invariant under orientation-preserving isometries of the ambient space. Versions of this integral have been applied by Komendarczyk in special cases to problems of higher order helicity and derivation of lower bounds for the energy of magnetic fields. We have set this entire paper in the 3-sphere because our generalized Gauss map is easiest to present here, but in a subsequent paper we will give the corresponding maps and integral formulas in Euclidean 3-space. C 2013 American Institute of Physics.

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A generalization of Milnor's μ-invariants to higher-dimensional link maps

Cited by 35 publications

References 16 publications

Quandle cohomology and state-sum invariants of knotted curves and surfaces

Quandle cohomology and state-sum invariants of knotted curves and surfaces

New perspectives on self-linking

Generalized Gauss maps and integrals for three-component links: Toward higher helicities for magnetic fields and fluid flows

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