Narrow Results

We study an exact asymptotic behavior of the Witten–Reshetikhin–Turaev SU(2) invariant for the Brieskorn homology spheres Σ(p₁, p₂, p₃) by use of properties of the modular form following a method proposed by Lawrence and Zagier. Key observation is that the invariant coincides with a limiting value of the Eichler integral of the modular form with weight 3/2. We show that the Casson invariant is related to the number of the Eichler integrals which do not vanish in a limit τ → N ∈ ℤ. Correspondingly there is a one-to-one correspondence between the non-vanishing Eichler integrals and the irreducible representation of the fundamental group, and the Chern–Simons invariant is given from the Eichler integral in this limit. It is also shown that the Ohtsuki invariant follows from a nearly modular property of the Eichler integral, and we give an explicit form in terms of the L-function.

Recently, there had been a great deal of interest in obtaining and describing all kinds of knots in links in hydrodynamics, electrodynamics, non-Abelian gauge field theories and gravity. Although knots and links are observables of the Chern–Simons (CS) functional, the dynamical conditions for their generation lie outside the scope of the CS theory. The nontriviality of dynamical generation of knotted structures is caused by the fact that the complements of all knots/links, say, in S³ are 3-manifolds which have positive, negative or zero curvature. The ability to curve the ambient space is thus far attributed to masses. The mass theorem of general relativity requires the ambient 3-manifolds to be of nonnegative curvature. Recently, we established that, in the absence of boundaries, complements of dynamically generated knots/links are represented by 3-manifolds of nonnegative curvature. This fact opens the possibility to discuss masses in terms of dynamically generated knotted/linked structures. The key tool is the notion of knot/link concordance. The concept of concordance is a specialization of the concept of cobordism to knots and links. The logic of implementation of the concordance concept to physical masses results in new interpretation of Casson’s surgery formula in terms of the Regge trajectories. The latest thoroughly examined Chew–Frautschi (CF) plots associated with these trajectories demonstrate that the hadron mass spectrum for both mesons and baryons is nicely described by the data on the corresponding CF plots. The physics behind Casson’s surgery formula is similar but not identical to that described purely phenomenologically by Keith Moffatt in 1990. The developed topological treatment is fully consistent with available rigorous mathematical and experimentally observed results related to physics of hadrons.

We provide related Dehn surgery descriptions for rational homology spheres and a class of their regular finite cyclic covering spaces. As an application, we use the surgery descriptions to relate the Casson invariants of the covering spaces to that of the base space. Finally, we show that this places restrictions on the number of finite and cyclic Dehn fillings of the knot complements in the covering spaces beyond those imposed by Culler-Gordon-Luecke-Shalen and Boyer-Zhang.

In 1998, C. Lescop proved that any integral homology sphere with the Casson invariant zero can be obtained from S³ by surgery on a boundary link each component of which has a trivial Alexander polynomial. In this paper, we prove that for any integral homology sphere H, there exists an integer k such that H can be obtained from S³ by surgery on a boundary link each component of which has the Alexander polynomial 1+k(t^½-t^-½)².

Let P₃ be the group of pure 3-braids and the set of closed pure 3-braids. We calculate the μ-invariant of the Z₂-homology spheres obtained by integral Dehn surgery on elements of B, and the Casson invariant of the Z-homology spheres obtained by surgery on algebraically split, closed pure n-braids.

Let V be the standard solid torus in S³. Let K_{p, 2} be the (p, 2)-torus knot in V such that K_{p, 2} meets a meridian disk D of V in two points with the winding number zero and the 2-string tangle TK_{p, 2} obtained by cutting along D is a rational tangle. We compute the Casson invariant of the cyclic covering space of S³ branched over a satellite knot whose companion is any 2-bridge knot D(b₁,…,b_2m) and pattern is (V, K_{p, 2}).

We derive a simple closed formula for the SL₂(ℂ) Casson invariant for Seifert fibered homology 3-spheres using the correspondence between SL₂(ℂ) character varieties and moduli spaces of parabolic Higgs bundles of rank two. These results are then used to deduce the invariant for Dehn surgeries on twist knots by combining computations of the Culler-Shalen norms with the surgery formula for the SL₂(ℂ) Casson invariant.

Gauss diagram formulas are extensively used to study Vassiliev link invariants. Now we apply this approach to invariants of 3-manifolds, considering manifolds given by surgery on framed links in the 3-sphere. We study the lowest degree case — the celebrated Casson–Walker invariant of rational homology spheres. This paper is dedicated to a detailed treatment of 2-component links; a general case will be considered in a forthcoming paper. We present simple Gauss diagram formulas for the Casson–Walker invariant. This enables us to understand/separate its dependence on the unframed link and on the framings. We also obtain skein relations for the Casson–Walker invariant under crossing changes, and study its asymptotic behavior when framings tend to infinity. Finally, we present results of extensive computer calculations.

In this paper, we introduce a representation of knots and links in S³ by integral matrices and then give an explicit formula for the Casson invariant for integral homology 3-spheres obtained from S³ by Dehn surgery along the knots and links represented by the integral matrices in which either all entries are even or the entries of each row are the same odd number. As applications, we study the preimage of the Casson invariant for a given integer and also give formulas for the Casson invariants of some special classes of integral homology 3-spheres.

In this paper, we give a bound for the Δ-unknotting number of a Whitehead double in terms of the unknotting number and a certain integral invariant of its companion knot. As applications, we show that the Δ-unknotting number of m-twisted Whitehead doubles of certain knots does not remember its companion knot, and is equal to the twist number m. We also give possible Δ-unknotting number of m-twisted Whitehead doubles whose companions are knots with unknotting number 1, certain twist knots, amphicheiral knots, and positive knots.

Taubes proved that the Casson invariant of an integral homology 3-sphere equals half the Euler characteristic of its instanton Floer homology. We extend this result to all closed oriented 3-manifolds with positive first Betti number by establishing a similar relationship between the Lescop invariant of the manifold and its instanton Floer homology. The proof uses surgery techniques.

For Z/rZ-homology 3-spheres the Casson invariant is related to the quantum PSU(2) invariant at an rth root of unity with odd prime r. We prove a similar relationship between the Casson invariant and the quantum PSU(2) invariant at a fifth root of unity for a 3-manifold whose first Z/5Z-Betti number is positive.

We introduce the notion of weak slice for a homology 3-sphere or a knot in it in order to compute the Floer homology of topological imitations of a given homology 3-sphere. We obtain a qualitative result on the Floer homology. In particular, infinitely many hyperbolic homology 3-spheres with the same Floer homology are constructed as topological imitations of every given homology 3-sphere. The well-known connected sum and periodicity questions on the Floer homology are also discussed.

An open question is the possibility of defining an integer valued SU(3)-Casson invariant for integral homology 3-spheres which involves counting the irreducible portion of the non-degenerate (perturbed) moduli space of flat SU(3)-connections plus counter-terms associated to only the non-degenerate (perturbed) reducible portion of the moduli space. The obstruction to this is the non-trivial spectral flow of a family of twisted signature operators in 3-dimensions. The parallel U(2)-Seiberg–Witten theory also has a similiar obstruction but arising from the non-trivial spectral flow of a family of twisted Dirac operators. By taking the SU(3)-flat and U(2)-Seiberg–Witten equations simultaneously the obstructions can be made to cancel and an integer invariant is obtained.

We give a simple criterion for a Heegaard splitting to yield a Haken manifold. As a consequence, we construct many Haken manifolds, in particular homology spheres, with prescribed properties, namely Heegaard genus, Heegaard distance and Casson invariant.

Along the way we give simpler and shorter proofs of the existence of splittings with specified Heegaard distance, originally proven by Ido–Jang–Kobayashi, of the existence of hyperbolic manifolds with prescribed Casson invariant, originally due to Lubotzky–Maher–Wu, and of a result about subsurface projections of disc sets (for which we even get better constants), originally due to Masur–Schleimer.

An new invariant of a knot, called the zeta-function, is introduced. It is defined by counting the non-diagonalizable representations of the fundamental group into SL₂(𝔽_q), up to conjugacy, and forming a formal power series, similar to a zeta-function from algebraic geometry. The structure of this invariant in the general case is given. The special case of (2, n) torus knots is studied, and basic topological properties are noted.