Three-Dimensional Link Theory and Invariants of Plane Curve Singularities. (AM-110)

Eisenbud, David; Neumann, Walter D.

doi:10.1515/9781400881925

Cited by 308 publications

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“…[15,11,4,23,17], etc. ), are highly atypical transverse C-links (for instance, while the unknot O is the only slice knot which is a link of a singularity [11] or a link at infinity [23], many nontrivial slice knots are transverse C-links [19]).…”

Section: Quasipositivitymentioning

confidence: 99%

Quasipositivity as an obstruction to sliceness

Rudolph¹

1993

Bull. Amer. Math. Soc.

188

192

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Abstract. For an oriented link L C S3 = dD4 , let Xs(L) be the greatest Euler characteristic x(F) of an oriented 2-manifold F (without closed components) smoothly embedded in D4 with boundary L . A knot K is slice if Xs(K) = 1 . Realize D4 in C2 as {(z, w) : \z\2 + \w\2 < 1} . It has been conjectured that, if V is a nonsingular complex plane curve transverse to S3 , then Xs(VnS3) = ^(KnZ)4). Kronheimer and Mrowka have proved this conjecture in the case that V n D4 is the Milnor fiber of a singularity. I explain how this seemingly special case implies both the general case and the "slice-Bennequin inequality" for braids. As applications, I show that various knots are not slice (e.g., pretzel knots like ^(-3, 5,7); all knots obtained from a positive trefoil 0{2, 3} by iterated untwisted positive doubling). As a sidelight, I give an optimal counterexample to the "topologically locally-flat Thorn conjecture". A BRIEF HISTORY OF SLICENESSA link is a compact 1-manifold without boundary L (i.e., finite union of simple closed curves) smoothly embedded in the 3-sphere S3 ; a knot is a link with one component. If S3 is realized in R4 as, say, the unit sphere, then a natural way to construct links is to intersect suitable two-dimensional subsets X c R4 with S3 ; one may then ask how constraints on X are reflected in constraints on the link X r\S3.For instance, Fox and Milnor (c. 1960) considered, in effect, the case that X is a smooth 2-sphere intersecting S3 transversally; at Moise's suggestion, Fox [5] adopted the adjective slice to describe the knots and links X nS3 so constructed. Fox and Milnor [6] gave a criterion for a knot A" to be slice: its Alexander polynomial Ak(í) £ Z[t, t~x] must have the form F(t)F(rx). This shows that, for instance, the two trefoil knots 0{2, ±3} are not slice (since A0{2,±3} = t~x -l+t is not of the form F(t)F(t~x) ), but it says nothing about the two granny knots 0{2, 3}#0{2, 3}, 0{2, -3}#0{2, -3} (indeed, both granny knots share the Alexander polynomial (r-1 -1 + i)2 with the square

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Section: Quasipositivitymentioning

confidence: 99%

Quasipositivity as an obstruction to sliceness

Rudolph¹

1993

Bull. Amer. Math. Soc.

188

192

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“…Γ is given in Figure 3. This type of graphs is used to codify the plumbing graph manifolds [5] which will be constructed in the following section, where it will be clear why weights of plumbing graph are called Euler numbers. In Figure 3 the Euler numbers and form a straight line or chain structure.…”

Section: Block Matrix Representation For a Graph P γ Of Tree Typementioning

confidence: 99%

“…Thus our plumbing diagrams will always have the pairwise coprime weights around each node and correspond to -homology spheres [5].…”

Section: Block Matrix Representation For a Graph P γ Of Tree Typementioning

confidence: 99%

“…On each of these manifolds, there exists a Seifert fibration with unnormalized Seifert invariants ( ) is its rational Euler number, the well known topological invariant of a Bh-sphere. The topological operation known as plumbing [5] is used to past together Bh-spheres according to plumbing diagrams (Figure 4) which can be transformed in plumbing graphs ( Figure 3). This type of graphs and their Laplacian block matrices is the main object of attention in this article.…”

Section: Introductionmentioning

confidence: 99%

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Block Matrix Representation of a Graph Manifold Linking Matrix Using Continued Fractions

Lopez¹,

Ефремов²,

Magdaleno³

2014

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We consider the block matrices and 3-dimensional graph manifolds associated with a special type of tree graphs. We demonstrate that the linking matrices of these graph manifolds coincide with the reduced matrices obtained from the Laplacian block matrices by means of Gauss partial diagonalization procedure described explicitly by W. Neumann. The linking matrix is an important topological invariant of a graph manifold which is possible to interpret as a matrix of coupling constants of gauge interaction in Kaluza-Klein approach, where 3-dimensional graph manifold plays the role of internal space in topological 7-dimensional BF theory. The Gauss-Neumann method gives us a simple algorithm to calculate the linking matrices of graph manifolds and thus the coupling constants matrices.

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“…coincides with the classical Alexander polynomial in several variables of the algebraic link C ∩ S 3 ε ⊂ S 3 ε (S 3 ε is the sphere of small radius ε centred at the origin in C 2 ): see [10]. We prefer to keep the name in the general case.…”

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confidence: 99%