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A Generalization of the Lightbulb Theorem and Pl I-equivalence of Links
| Content Provider | Semantic Scholar |
|---|---|
| Author | Litherland, Rick |
| Copyright Year | 2010 |
| Abstract | By the "lightbulb theorem" I mean the result that a knot of S1 in S1 X S2 which meets some S2 factor in a single transverse point is isotopic to an S1 factor. We prove an analogous result for knots of S" in S" X S2, and apply it to answer a question of Rolfsen concerning PL I-equivalence of links. Introduction. In [9], Rolfsen asked the following Question. Do there exist links L = Lx U • • • UL^ and U = L\ U • • • UL' of «-spheres in an (n + 2)-manifold M such that L and L' are I-equivalent and, for each i = l,...,p, Lj and L\ are concordant knots, yet L fails to be concordant to L"! (This refers to the PL category; I-equivalence is the relation that results when concordances are not required to be locally flat.) The question arises because Theorem 3 of [9] asserts that there are no such links in S"+2. The proof of that theorem shows that the answer is no if n is even (since the knot concordance group is zero in even dimensions). We shall show that there are examples for every odd n. The case n = 1 is easily described. We take M = S1 X S2. Let x and y be two points of S2, set L, = L\ = S1 X {x} and L2 = S1 X {y}, and let L'2 be the result of locally tying a trefoil in L2. Then L = Lx U L2 and L' = L[ U L'2 satisfy the desired conditions. In fact L2 and L'2 are ambient isotopic; this is a special case of what is sometimes known as the lightbulb theorem. To construct examples for greater values of n, we would like to replace S1 by S" throughout (and the trefoil by some nonslice knot of S"). To see that this produces links with the right properties, we prove in §1 a higher-dimensional version of the lightbulb theorem. For this it seems to be necessary to work in the smooth category. A little triangulation theory gives results for PL case, and hence our examples, in §2. For us, a knot~ of M in TV will mean a submanifold of N isomorphic (i.e., diffeomorphic or PL homeomorphic, as appropriate) to M, rather than an embedding of M in TV. (In the PL case, the submanifold is to be locally flat.) All manifolds will be oriented, and all isomorphisms between manifolds will be orientation preserving. If Mx and M2 are submanifolds of N and h: N -* N is an isomorphism, the statement h(Mx) = M2 means that h(Mx) and M2 are equal as oriented manifolds, i.e., that h \ Mx: Mx -* M2 is also orientation preserving. Received by the editors August 8, 1985. 1980 Mathematics Subject Classification. Primary 57Q45. ©1986 American Mathematical Society 0002-9939/86 $1.00 + $.25 per page |
| File Format | PDF HTM / HTML |
| Alternate Webpage(s) | http://www.ams.org/journals/proc/1986-098-02/S0002-9939-1986-0854046-2/S0002-9939-1986-0854046-2.pdf |
| Language | English |
| Access Restriction | Open |
| Subject Keyword | Arabic numeral 0 Concordance (publishing) Dimensions Embedding Existential quantification Generalization (Psychology) Graph isomorphism Knot (unit) Mathematics Subject Classification Maxwell Myxovirus Resistance Proteins NP-equivalent Transverse wave Turing completeness newton triangulation |
| Content Type | Text |
| Resource Type | Article |
