This magic number is called hopf index and can be calculated for any smooth unit vector field 1. Intuitively, the linking number represents the number of times that each curve winds around the other. Hopf algebra that is invariant under the antipode is shown to yield a link invariant. However none give any insight into why the hopf invariant is useful.
Chapter 1 the hopf invariant introduction let f be a map of the sphere s 3 onto the sphere. The linking number is always an integer, but may be positive or negative depending on the orientation of the two curves. The asymptotic hopf invariant and its applications. The hopf invariant in topology and algebra andrew ranicki. This correspondence has been previously employed to measure the. Total linking numbers of torus links and klein links. Under the hopf map, the trajectories followed by the inverse images of any two vectors on s2 elucidate the topological invariant as a linking number lin t3. In the 1930s the hopf invariant map was proved to be an isomorphism h. It turns out that any two component link with linking number 5 can be put. In section 2 we discuss the construction of torus links and klein links. In this construction hopf exploited the number systems given by the complex numbers, the quaternions, and the cayley octonions, and provided an invariantgeneralizing the linking numberby which to detect them.
The hopf invariant gives the linking number of two disjoint manifolds and is a genuine topological invariant 7. The scienti c and technological potential of 3d topological solitons can be appreciated by considering the recent advances in the studies of their twodimensional counterparts. Topological defect with nonzero hopf invariant in yang. Using the inner structure of the topological tensor current, the relationship between hopf invariant and the linking numbers of the higher dimensional knots can be constructed. An invariant of a homotopy class of mappings of topological spaces. Hopf proved that the hopf map has hopf invariant 1, and therefore is not nullhomotopic. Moreover the o3 nlsm with the hopf term was canonically quantized 4 and the cp1 model with the hopf term 59, which can be related with the o3 nlsm via the hopf map projection from s3 to s2, was also canonically.
S 2 and let us suppose it to be simplicia1 relative to some triangulations of s2 and s 3 if q and r are any two points in the interior of a 2simplex of s 2, f. Using the compatibility of the suspension and the ltration on p 1, one can see that the stable hopf invariant gives a lower bound on the sphere of origin for stable elements, and when the lower bound is achieved it detects the hopf invariant. It was an important problem of topology to determine for which dimensions the hopf invariant was one. A primer of hopf algebras 3 basis, and the multiplication in gis extended to kgby linearity. It was later shown that the homotopy group is the infinite cyclic group generated by. I i want to describe a generalization of the hopf invariant to more general maps than just s3. We can prove 2 from property 4th of kauffman bracket. S 2 and let us suppose it to be simplicia1 relative to some triangulations of s2 and s 3 if q and r are any two points in the interior of a 2simplex of s 2, f 9 and fl r are 1cycles of the complex s 3 see 9 2 1. S2n which depended only on the homotopy class of the map. As in 5, we will refer to the total linking number as simply linking number. Finally, we give some remarks about the hopf invariant. Then the hopf invariant is defined as the linking coefficient.
In 1931 heinz hopf used clifford parallels to construct the hopf map. Theorem a the pairwise linking numbers p, qand rof the link lare equal to the degrees of its characteristic map g l on the 2dimensional coordinate subtori of t3, while twice milnors invariant for lis equal to pontryagins invariant for g l. The asymptotic hopf invariant and its applications springerlink. In particular, he discovered an in variant on maps s2n 1. Triple linking numbers, ambiguous hopf invariants and integral formulas theorem a. Here we give a simple visualization of the hopf mapping. Hopf algebras and invariants of 3manifolds louis h. The links pictured in figure 2 have linking number 5. It is unique invariant of oriented links that satis. There are several clear expositions giving the definition of the hopf invariant including the wikipedia article in the link of this post. The topologically nontrivial hopf mapping is characterized by the hopf invariant h. In the minimal energy case, if we normalize such that. Threecomponent links in the 3dimensional sphere were classified up to link homotopy by john milnor in his senior thesis, published in 1954. This number is obviously an invariant but is not always easy to compute.
The hopf invariant one theorem states that the only maps of hopf invariant one, h. S2, which is particularly useful in the classi cation of manifolds with nontrivial fundamental group i the generalized hopf invariant involves the modern algebraic theory of symmetric and quadratic forms on chain complexes over a ring with involution. How ktheory solves the hopf invariant one problem sean pohorence introduction it is now well known that hopf invariant one maps s2 n1. S1 s2 are determined up to homotopy by the degrees pf,qf,rf. The linking number is defined for two linked circles. The hopf invariant is the linking number of any two of these. Under a hopf mapping, the preimage of a point on s 2 is a circle in s 3. In algebraic topology, the cup product is a farreaching algebraic generalization of the linking number, with the massey products being the algebraic analogs for. In mathematics, the linking number is a numerical invariant that describes the linking of two closed curves in threedimensional space. Introduction to the steenrod squares and the hopf invariant. The original proof that the only maps of hopf invariant one are the hopf constructions on the four normed division algebras is due to. The hopf invariant hh is welldefined as the integer linking number of two oriented curves 1a, h h1b, where a,b. We relate the ktheory definition to the cohomology definition using the chern character in chap.
S only occur when nis equal to 1, 2, 4, or 8 though, for the remainder of this note we will ignore the case n 1. A rigorous numerical algorithm for computing the linking. Whiteheads integral formula for the hopf invariant, adapted to maps of the 3torus to the 2sphere, together with a formula for the fundamental solution of the scalar laplacian on the 3torus as a fourier series in three variables, to provide an explicit integral formula for nu, and hence for mu. Tables of knots are arranged by the crossing number. Note that the linking number is a homotopical, rather than homological, invariant. If this link is built from rope or beads, then one quickly finds that unlinking the two is impossible. Frank adams, on the nonexistence of elements of hopf invariant one, ann. Z of the restrictions of f to the 2dimensional subtori and by an ambiguous hopf invariant. The treatment of the hopf invariant of the hopf map. Hence a famous open question in the 1950s was for which maps. If is a closed equipped manifold and if, then the characteristic stiefelwhitney number of the normal bundle is the same as the hopf invariant of the mapping that is a representative of the class of equipped cobordisms of. Also let t 5, t 0 5, denote the same links but where the orientation on the unknot is switched so that the linking number becomes 5, see figure 4.
A complete set of invariants is given by the pairwise linking numbers p, q and r of the components, and by the residue class of one further integer mu, the triple linking number of the title, which is welldefined modulo the greatest common divisor of. We also obtain a precise expression for hopf invariant from which one can relate the hopf invariant with the linking number and self linking numbers of knots. Whitehead an expression of hopfs invariant as an integral, proc. Hopf invariant, higher dimensional knot, linking number. When p, q and r are all zero, the mu and nuinvariants are ordinary integers.
The pairwise linking numbers p, q and r of the link l are equal to the degrees of its characteristic map g l on the 2dimensional coordinate subtori of t3, while twice milnors invariant for l is equal to pontryagins. We view this as a natural extension of the familiar fact that the linking number of a 2component link is the degree of an associated map of the 2torus to the 2sphere. In particular, we present the elementary atiyah proof of the nonexistence of elements of hopf invariant 1. American mathematical society volume 8, number 1, january 1983. The triple linking number is an ambiguous hopf invariant. In section 2 we discuss the construction of torus links and. In the 30s hopf studied homotopy classes of maps between spaces and started with a simple case. When p, qand rare all zero, the and invariants are.
Remark 1 milnors invariant, typically denoted 123, is descriptive of a single threecomponent. The second invariant is an intrinsically floquet z 2 invariant, and represents a condensed matter realization of the topology underlying the witten anomaly in particle physics. With the link on the right the situation is less clear. Triple linking numbers, ambiguous hopf invariants and. We give a ktheory definition of the hopf invariant of maps s 2n. This is our reconstruction of hennings invariant 6 in an intrinsically unoriented context. The hopf invariant can also be defined in terms of the stiefel numbers cf.
Our algorithm is designed to handle spatially distributed data points directly. The classical hopf invariant distinguishes among the homotopy classes of continuous mappings from the threesphere to the twosphere and is equal to the linking number of the two curves that are the preimages of any two regular points of the twosphere. The hopf invariant in particular is a homotopy invariant of map between spheres. A complete set of invariants is given by the pairwise linking numbers p, q and r of the components, and by the residue class of one further integer mu, the triple linking number of the title, which is welldefined modulo the greatest common. The 3sphere is a fourdimensional object and it is difficult to imagine. This linking number is the socalled hopf index topological invariant, q, characterizing the topology of the 3d topological solitons. This is because what we have in mind as input data is the data coming. In quantum mechanics, the riemann sphere is known as the bloch sphere, and the hopf fibration describes the topological structure of a quantum mechanical twolevel system or qubit. We will explore the linking numbers for two classes of links. Then the hopf invariant is defined as the linking coefficient of the.
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