Lee Rudolph

Bio: Lee Rudolph is an academic researcher from Clark University. The author has contributed to research in topics: Fibered knot & Milnor number. The author has an hindex of 23, co-authored 58 publications receiving 2023 citations. Previous affiliations of Lee Rudolph include Columbia University & Brandeis University.

Quasipositivity as an obstruction to sliceness

Abstract: For an oriented link L ⊂ S 3 = ∂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 slice if χ S (K) = 1. Realize D 4 in C 2 as {(z, w): |z| 2 + |w| 2 ≤ 1}. It has been conjectured that, if V is a nonsingular complex plane curve transverse to S 3 , then χ S (V ∩ S 3 ) = χ(V ∩ D 4 ). Kronheimer and Mrowka have proved this conjecture in the case that V ∩ 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 P(−3, 5, 7); all knots obtained from a positive trefoil O{2, 3} by iterated untwisted positive doubling)

Embeddings of the line in the plane.

Abstract: In their paper of this title [1], Abhyankar and Moh have shown that (up to algebraic automorphism) the only embedding of the line in the plane, in characteristic 0, is the Standard one. For the complex numbers, this is a strong sort of unknotting theorem. In this paper, using only some reasonably elementary theory of knots, I give a new proof of the theorem of Abhyankar and Moh; this proof is intrinsically topological, and does not appear merely to be a translation of their original proof using \"high-school algebra\".

Braided surfaces and seifert ribbons for closed braids

Abstract: Apositive band in the braid groupB n is a conjugate of one of the standard generators; a negative band is the inverse of a positive band. Using the geometry of the configuration space, a theory of bands andbraided surfaces is developed. Each representation of a braid as a product of bands yields a handle decomposition of aSeifert ribbon bounded by the corresponding closed braid; and up to isotopy all Seifert ribbons occur in this manner. Thus,band representations provide a convenient calculus for the study of ribbon surfaces. For instance, from a band representation, a Wirtinger presentation of the fundamental group of the complement of the associated Seifert ribbon inD 4 can be immediately read off, and we recover a result of T. Yajima (and D. Johnson) that every Wirtinger-presentable group appears as such a fundamental group. In fact, we show that every such group is the fundamental group of a Stein manifold, and so that there are finite homotopy types among the Stein manifolds which cannot (by work of Morgan) be realized as smooth affine algebraic varieties.

An obstruction to sliceness via contact geometry and “classical” gauge theory

Abstract: In recent work (beginning with [8]) Kronheimer and Mrowka have developed the formidable machinery of "gauge theory for embedded surfaces", and deployed it against a host of problems of the general form, "What is the least genus of a smooth surface representing a specified homology class in a given 4-manifold?" One of their results (the "local Thom Conjecture") can be used [19] to derive a lower bound (the "slice-Bennequin inequality", hereinafter sBi) for the Murasuoi (or slice) genus of a knot K c S3--that is, the smallest genus of a smooth, oriented surface in D 4 with boundary K. A knot is slice if it has Murasugi genus 0, so in particular sBi provides an obstruction to sliceness. Though Bennequin originally conjectured sBi in the context [1] of an investigation into 3-dimensional contact geometry, to date sBi has not been proved using exclusively 3-dimensional methods. (Such methods did success- fully establish [1] the "Bennequin inequality" pure and simple, which sBi generalizes. It appears they can also be used [I 1] to answer the "question of Milnor" on unknotting number, achieved 4-dimensionally - using the local Thom Conjecture - in [8].) The following corollary of sBi will be proved in w independently of sBi, by combining ideas from dimensions 3 (contact geometry) and 4 ("classical" gauge theory [4], earlier and comparatively simpler than [8]): ifa knot K has non-negative maximal Thurston-Bennequin invariant TB(K), then K is not slice. Though TB (K) is defined analytically, an alternative combinatorial definition (recalled in w allows one to show TB(K) > 0 (and sometimes to calculate TB(K)) for many interesting knots K; in particular, results of [19] on non- sliceness of certain iterated doubled knots are recovered quickly and easily. Research partially supported by CNRS.

The geometry of biological time , by A. T. Winfree. Pp 544. DM68. Corrected Second Printing 1990. ISBN 3-540-52528-9 (Springer)

Abstract: 1980 Preface * 1999 Preface * 1999 Acknowledgements * Introduction * 1 Circular Logic * 2 Phase Singularities (Screwy Results of Circular Logic) * 3 The Rules of the Ring * 4 Ring Populations * 5 Getting Off the Ring * 6 Attracting Cycles and Isochrons * 7 Measuring the Trajectories of a Circadian Clock * 8 Populations of Attractor Cycle Oscillators * 9 Excitable Kinetics and Excitable Media * 10 The Varieties of Phaseless Experience: In Which the Geometrical Orderliness of Rhythmic Organization Breaks Down in Diverse Ways * 11 The Firefly Machine 12 Energy Metabolism in Cells * 13 The Malonic Acid Reagent ('Sodium Geometrate') * 14 Electrical Rhythmicity and Excitability in Cell Membranes * 15 The Aggregation of Slime Mold Amoebae * 16 Numerical Organizing Centers * 17 Electrical Singular Filaments in the Heart Wall * 18 Pattern Formation in the Fungi * 19 Circadian Rhythms in General * 20 The Circadian Clocks of Insect Eclosion * 21 The Flower of Kalanchoe * 22 The Cell Mitotic Cycle * 23 The Female Cycle * References * Index of Names * Index of Subjects

A polynomial invariant for knots via von Neumann algebras

Abstract: Thus, the trivial link with n components is represented by the pair (l ,n), and the unknot is represented by (si$2 * * • s n i , n) for any n, where si, $2, • • • > sn_i are the usual generators for Bn. The second example shows that the correspondence of (b, n) with b is many-to-one, and a theorem of A. Markov [15] answers, in theory, the question of when two braids represent the same link. A Markov move of type 1 is the replacement of (6, n) by (gbg~, n) for any element g in Bn, and a Markov move of type 2 is the replacement of (6, n) by (6s J 1 , n-hl). Markov's theorem asserts that (6, n) and (c, ra) represent the same closed braid (up to link isotopy) if and only if they are equivalent for the equivalence relation generated by Markov moves of types 1 and 2 on the disjoint union of the braid groups. Unforunately, although the conjugacy problem has been solved by F. Garside [8] within each braid group, there is no known algorithm to decide when (6, n) and (c, m) are equivalent. For a proof of Markov's theorem see J. Birman's book [4]. The difficulty of applying Markov's theorem has made it difficult to use braids to study links. The main evidence that they might be useful was the existence of a representation of dimension n — 1 of Bn discovered by W. Burau in [5]. The representation has a parameter t, and it turns out that the determinant of 1-(Burau matrix) gives the Alexander polynomial of the closed braid. Even so, the Alexander polynomial occurs with a normalization which seemed difficult to predict.

4-manifolds and Kirby calculus

Abstract: 4-manifolds: Introduction Surfaces in 4-manifolds Complex surfaces Kirby calculus: Handelbodies and Kirby diagrams Kirby calculus More examples Applications: Branched covers and resolutions Elliptic and Lefschetz fibrations Cobordisms, $h$-cobordisms and exotic ${\mathbb{R}}^{4,}$s Symplectic 4-manifolds Stein surfaces Appendices: Solutions Notation, important figures Bibliography Index.

The Jacobian conjecture: Reduction of degree and formal expansion of the inverse

Abstract: Introduction I. The Jacobian Conjecture 1. Statement of the Jacobian Problem; first observations 2. Some history of the Jacobian Conjecture 3. Faulty proofs 4. The use of stabilization and of formal methods II. The Reduction Theorem 1. Notation 2. Statement of the Reduction Theorem 3. Reduction to degree 3 4. Proof of the Reduction Theorem 5. r-linearization and unipotent reduction 6. Nilpotent rank 1 Jacobians III. The Formal Inverse 1. Notation 2. Abhyankar's Inversion Formula 3. The terms Gj 4. The tree expansion G\ = lT(\/a(T))lfPTf 5. Calculations References

Polynomial Automorphisms and the Jacobian Conjecture

Abstract: In this paper we give an update survey of the most important results concerning the Jacobian conjecture: several equivalent descriptions are given and various related conjectures are discussed. At the end of the paper, we discuss the recent counter-examples, in all dimensions greater than two, to the Markus-Yamabe conjecture (Global asymptotic Jacobian conjecture). Resume Dans ce papier nous presentons un rapport actualise sur les resultats les plus importants concernant la conjecture Jacobienne : plusieurs formulations equivalentes et diverses conjectures connexes sont considerees. A la fin du papier, nous donnons les contre-exemples recents, en toute dimension plus grande que deux, a la conjecture de Markus-Yamabe.