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Lee Rudolph

Publications and source records attributed to Lee Rudolph.

17 recordsLinked to original sources
arXiv2015

Some 3-dimensional transverse C-links (Constructions of higher-dimensional C-links, I)

By use of a variety of techniques (most based on constructions of quasipositive knots and links, some old and others new), many smooth 3-manifolds are realized as transverse intersections of complex surfaces in complex 3-space with strictly pseudoconvex 5-spheres. These manifolds not only inherit interesting intrinsic structures (eg, they have canonical Stein-fillable contact structures), they also have extrinsic structures of a knot-theoretical nature (eq, the 3-sphere arises in infinitely many distinct ways). This survey is not comprehensive; a number of questions are left open for future work.

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arXiv2009

Whitehead's Integral Formula, Isolated Critical Points, and the Enhancement of the Milnor Number

J. H. C. Whitehead gave an elegant integral formula for the Hopf invariant H(p) of a smooth map p from the 3-sphere to the 2-sphere. Given an open book structure b on the 3-sphere (or, essentially equivalently, an isolated critical point of a map F from 4-space to the plane), Whitehead's formula can be "integrated along the fibers" to express H(p) as the integral of a certain 1-form over the circle. In case p is geometrically related to b (or F) -- for instance, if p is the map (one component of the fiberwise generalized Gauss map of F) whose Hopf invariant lambda(K) is the "enhancement of the Milnor number" of the fibered link K in the 3-sphere associated to F (or b), previously studied by the author and others -- it might be hoped that this 1-form has geometric significance. This note makes that hope somewhat more concrete, in the form of several speculations and questions.

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arXiv2004

Algebraic functions and closed braids

This article was originally published in Topology 22 (1983). The present hyperTeXed redaction includes references to post-1983 results as Addenda, and corrects a few typographical errors. (See math.GT/0411115 for a more comprehensive overview of the subject as it appears 21 years later.)

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arXiv2004

Knot theory of complex plane curves

The primary objects of study in the ``knot theory of complex plane curves'' are C-links: links (or knots) cut out of a 3-sphere in the complex plane by complex plane transverse and totally tangential. Transverse C-links are naturally oriented. There are many natural classes of examples: links of singularities; links at infinity; links of divides, free divides, tree divides, and graph divides; and--most generally--quasipositive links. Totally tangential C-links are unoriented but naturally framed; they turn out to be precisely the real-analytic Legendrian links, and can profitably be investigated in terms of certain closely associated transverse C-links. The knot theory of complex plane curves is attractive not only for its own internal results, but also for its intriguing relationships and interesting contributions elsewhere in mathematics. Within low-dimensional topology, related subjects include braids, concordance, polynomial invariants, contact geometry, fibered links and open books, and Lefschetz pencils. Within low-dimensional algebraic and analytic geometry, related subjects include embeddings and injections of the complex line in the complex plane, line arrangements, Stein surfaces, and Hilbert's 16th problem.

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arXiv2003

Constructions of Morse maps for knots and links, and upper bounds on the Morse-Novikov number

The Morse-Novikov number MN(L) of an oriented link L in the 3-sphere is the minimum number of critical points of a Morse map from the complement of L in the 3-sphere to the circle representing the class of a Seifert surface for L (e.g., the Morse-Novikov number of L is zero if and only if L is fibered). We develop various constructions of Morse maps (Milnor maps, Stallings twists, splicing along a link which is a closed braid with respect to a Morse map, Murasugi sums, cutting a Morse map along an arc on a page) and use them to bound Morse-Novikov numbers from above in terms of other knot and link invariants (free genus, crossing number, braid index, wrapping genus and layered wrapping genus).

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arXiv2002

Isolated critical points of mappings from $\mathbf{R}^4$ to $\mathbf{R}^2$ and a natural splitting of the Milnor number of a classical fibered link. Part I: Basic theory; examples

From a fibered link in the 3-sphere may be constructed a field of not everywhere tangent 2-planes; when the fibered link is the link of an isolated critical point of a map from 4-space to the plane, the plane field is essentially the field of kernels of the derivative of the map. Homotopically, such a plane field determines two integers. I show that the sum of these integers is the Milnor number of the fibered link. Taking the mirror image of a link exchanges the integers. Various examples are computed. It is noted (proof given elsewhere) that these integers are not determined by the algebraic monodromy (or Seifert form) of the fibered link.

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arXiv2002

Nœuds non concordants à un C-bord

An oriented link L in a 3-sphere S in complex 2-space is a C-boundary if it bounds a piece of algebraic curve in the 4-ball bounded by S. Using Kronheimer and Mrowka's proof of the Thom Conjecture, we construct many oriented knots which are not concordant to a C-boundary. We use the two-variable HOMFLY polynomial to give an obstruction to a knot's being a C-boundary in a strictly pseudoconvex S. We make several conjectures.

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arXiv2001

Quasipositive Annuli (Constructions of Quasipositive Knots and Links, IV)

[Original abstract (1992):] The modulus of quasipositivity q(K) of a knot K was introduced as a tool in the knot theory of complex plane curves, and can be applied to Legendrian knot theory in symplectic topology. It has also, however, a straightforward characterization in ordinary knot theory: q(K) is the supremum of the integers f such that the framed knot (K;f) embeds non-trivially on a fiber surface of a positive torus link. Geometric constructions show that -\infty < q(K), calculations with link polynomials that q(K) < \infty. The present paper aims to provide sharper lower bounds (by optimizing the geometry with positive plats) and more readily calculated upper bounds (by modifying known link polynomials), and so to compute q(K) for various classes of knots, such as positive closed braids (for which q(K) = μ(K)-1) and most positive pretzels. As an aside, it is noted that a recent result of Kronheimer & Mrowka implies that q(K) < 0 if K is slice. [Additional abstract (December 2001):] A 1995 paper gives a proof that q(K) equals TB(K), the maximal Thurston-Bennequin invariant of K. Thus the bounds for, and calculations of q(K) derived in this paper are equally bounds for, or calculations of, TB(K). Similar (sometimes sharper) results for TB(K) have been derived more recently by a number of researchers, using a variety of different methods.

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arXiv2001

Murasugi sums of Morse maps to the circle, Morse-Novikov numbers, and free genus of knots

Murasugi sums can be defined as readily for Morse maps to the circle of (arbitrary) link complements in the 3-sphere as for fibrations over the circle of (fibered) link complements in the 3-sphere. As one application, I show that if a knot K has free genus m, then there is a Morse map from its complement to the circle (representing the relative homology class of a Seifert surface for K) with no more than 4m critical points.

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arXiv2001

Some knot theory of complex plane curves

This survey of some of the more topological aspects of the placement problem for complex curves in complex surfaces was originally published in L'Enseignement Mathematique 29 (1983). The present LaTeXed redaction corrects several typographical errors without, I hope, introducing new ones; some minor emendations and references to post-1983 results have been added as footnotes, but I have not made a thoroughgoing effort to update the text. Anyone having further information -- for instance, on the current status of questions incorrectly described as still open -- is encouraged to let me know!

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arXiv2001

A non-ribbon plumbing of fibered ribbon knots

A closer look at an example introduced by Livingston & Melvin and later studied by Miyazaki shows that a plumbing of two fibered ribbon knots (along their fiber surfaces) may be algebraically slice yet not ribbon.

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arXiv2000

Some fibered and non-fibered links at infinity of hyperbolic complex line arrangements

Let F be R or C, d the dimension of F over R. Denote by P(F) either the affine plane A(F) or the hyperbolic plane H(F) over F. An arrangement L of k lines in P(F) (pairwise non-parallel in the hyperbolic case) has a link at infinity K(L) comprising k unknotted (d-1)-spheres in the (2d-1)-sphere, whose topology reflects the combinatorics of L `at infinity'. The class of links at infinity of affine F-line arrangements is properly included in the class of links at infinity of hyperbolic F-line arrangements. Many links at infinity of (essentially non-affine) connected hyperbolic C-line arrangements are far from being fibered; the proof is a direct construction, using ``Legendrian inscription''. In contrast, if the (affine or hyperbolic) R-line arrangement L is connected, then its complexification (an affine or hyperbolic C-line arrangement) has a fibered link at infinity; the proof uses A'Campo's divides.

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arXiv2000

Hopf plumbing, arborescent Seifert surfaces, baskets, espaliers, and homogeneous braids

Four constructions of Seifert surfaces - Hopf plumbing, arborescent plumbing, basketry, and T-bandword handle decomposition - are described, and some interrelationships found, e.g.: arborescent Seifert surfaces are baskets; Hopf-plumbed baskets are precisely homogeneous T-bandword surfaces. A Seifert surface is Hopf-plumbed if it is a 2-disk D or if it can be constructed by plumbing a positive or negative Hopf annulus A(O,-1) or A(O,1) to a Hopf-plumbed surface along a proper arc. A Seifert surface is a basket if it is D or it can be constructed by plumbing an n-twisted unknotted annulus A(O,n) to a basket along a proper arc in D. A Seifert surface is arborescent if it is D, or it is A(O,n), or it can be constructed by plumbing A(O,n) to an arborescent Seifert surface along a transverse arc of an annulus plumband. Every arborescent Seifert surface is a basket. A tree T embedded in the complex plane C determines a set of generators for a braid group. An espalier is a tree in the closed lower halfplane with vertices on the real line R. If T is an espalier then words b in the T-generators correspond nicely to T-bandword surfaces S(b). (For example, if I is an espalier with an edge from p to p+1 for p=1,...,n-1, then the I-generators of the n-string braid group are the standard generators; when Seifert's algorithm is applied to the closed braid diagram of a word in those generators, the result is an I-bandword surface.) Theorem. For any espalier T, a T-bandword surface S(b) is a Hopf-plumbed basket iff b is homogeneous iff S(b) is a fiber surface iff S(b) is connected and incompressible. A Hopf-plumbed basket S (for instance, an arborescent fiber surface) is isotopic to a homogeneous T-bandword surface for some espalier T.

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arXiv2000

Quasipositivity as an obstruction to sliceness

For an oriented link $L \subset S^3 = \Bd\!D^4$, let $χ_s(L)$ be the greatest Euler characteristic $χ(F)$ of an oriented 2-manifold $F$ (without closed components) smoothly embedded in $D^4$ with boundary $L$. A knot $K$ is {\it slice} if $χ_s(K)=1$. Realize $D^4$ in $\C^2$ as $\{(z,w):|z|^2+|w|^2\le1\}$. It has been conjectured that, if $V$ is a nonsingular complex plane curve transverse to $S^3$, then $χ_s(V\cap S^3)=χ(V\cap D^4)$. Kronheimer and Mrowka have proved this conjecture in the case that $V\cap D^4$ 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 $\Pscr(-3,5,7)$; all knots obtained from a positive trefoil $O\{2,3\}$ by iterated untwisted positive doubling). As a sidelight, I give an optimal counterexample to the ``topologically locally-flat Thom conjecture''.

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arXiv2000

Quasipositive pretzels

A necessary and sufficient condition for an oriented pretzel surface to be quasipositive yields an estimate for the slice genus of the boundary of an arbitrary oriented pretzel surface.

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arXiv1999

Positive links are strongly quasipositive

Let S(D) be the surface produced by applying Seifert's algorithm to the oriented link diagram D. I prove that if D has no negative crossings then S(D) is a quasipositive Seifert surface, that is, S(D) embeds incompressibly on a fiber surface plumbed from positive Hopf annuli. This result, combined with the truth of the `local Thom Conjecture', has various interesting consequences; for instance, it yields an easily-computed estimate for the slice euler characteristic of the link L(D) (where D is arbitrary) that extends and often improves the `slice-Bennequin inequality' for closed-braid diagrams; and it leads to yet another proof of the chirality of positive and almost positive knots.

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