Narrow Results

In 1990, John Berge described several families of knots in the three-dimensional sphere which have non-trivial Dehn surgeries yielding lens spaces. We study a subfamily of them from the view point of resolution of singularity of complex curves and surfaces, Kirby calculus in topology of four-dimensional manifolds and A'Campo's divide knot theory.

A knot in a lens is said to be spinal if it can be isotoped on a standard spine (e.g. in ℝP³, spinal knots bound a Möbius band). We prove that a Dehn surgery on a non-spinal knot in a lens space cannot produce 𝕊³. With a view to study the Dehn surgeries that produce lens spaces, the main part is devoted to finding an obstruction for a standard spine to be minimal. We consider the intersection graphs coming from a standard spine and an arbitrary surface. This obstruction is given by the existence of a generalized Scharlemann cycle.

By obtaining surgery descriptions of knots which lie on the genus one fiber of the trefoil or figure eight knot, we show that these include hyperbolic knots with arbitrarily large volume. These knots admit lens space surgeries and form two families of Berge knots. By way of tangle descriptions we also obtain surgery descriptions for these knots on minimally twisted chain links.

Using Kirby calculus, we explicitly pass from Berge's R-R descriptions of ten families of knots with lens space surgeries to surgery descriptions on the minimally twisted five chain link (MT5C). Since the MT5C admits a strong inversion, we also give the corresponding tangle descriptions.

Let K be a knot in S³ and suppose that K(r) is a reducible manifold. Howie has proved that the number of connected summands is at most 3 [10, Corollary 5.3]. Furthermore, he showed that if K(r) is the connected sum of exactly three irreducible 3-manifolds, then K(r) = L(p₁, q₁)♯L(p₂, q₂)♯M, where L(p₁, q₁) and L(p₂, q₂) are a lens spaces and M is a ℤ-homology sphere. In this paper we study this specific case and we show that |r| ≤ (b-2)(b-1), for any knot with bridge number b.

This paper contains two remarks about the application of the $d$-invariant in Heegaard Floer homology and Donaldson's diagonalization theorem to knot theory. The first is the equivalence of two obstructions they give to a 2-bridge knot being smoothly slice. The second carries out a suggestion by Stefan Friedl to replace the use of Heegaard Floer homology by Donaldson's theorem in the proof of the main result of [J. E. Greene, Lattices, graphs, and Conway mutation, Invent. Math.192(3) (2013) 717–750] concerning Conway mutation of alternating links.

Taking advantage of a numerical invariant, we visualize a distribution of rational homology 3-spheres on a plane via the Casson–Walker–Lescop (CWL) invariant and observe several aspects of the distribution. In particular, we study the characteristics of the distribution of lens spaces as a fundamental family of rational homology 3-spheres with a way to yield a family of estimation for the Dedekind sum. The CWL invariant captures the finiteness of lens space surgeries along knots. According to the finiteness, for example, the CWL invariant determines possible lens spaces as the results of integral surgeries along a knot $K$ with $a_1(K)=12$.

In the paper, we give an equivalent description of the lens space $L(p,q)$ with $p$ prime in terms of any corresponding Heegaard diagrams as follows: Let $M$ be a closed orientable 3-manifold with ${\pi }_1(M)\ne 1,$ and $U{\cup }_{F}V$ a Heegaard splitting of genus $n$ for $M$ with an associated Heegaard diagram $(U;J_1,\dots ,J_n)$. Assume $p$ is a prime integer. Then $M$ is homeomorphic to the lens space $L(p,q)$ if and only if there exists an embedding $f:U\hookrightarrow L(p,q)$ such that $L=\{f(J_1),\dots ,f(J_n)\}$ bounds a complete system of surfaces for $W=\overline{L(p,q)\backslash f(U)}$.

The main results characterize Legendrian negative torus knots in every universally tight contact structure on $L(p,q)$. Together with Onaran and Min’s results, we obtain that in every universally tight contact structure on $L(p,q)$, Legendrian torus knots are determined by its classical invariants up to Legendrian isotopy. The main method is splitting $L(p,q)$ by convex Heegaard decomposition.

We extend the Jones polynomial for links in S³ to links in L(p, q), p>0. Specifically, we show that the (2, ∞)-skein module of L(p, q) is free with [p/2]+1 generators. In the case of S¹×S² the skein module is infinitely generated.

We define invariants < L)₋₁ and F_L(A) for links L in Σ × R where Σ is a surface by using the idea of Kauffman bracket. We compute < L)₋₁ in case of Σ = P², a projective plane. In case of Σ = T², a torus, we show examples of computations of F_L(A) for a certain class of links L in T² × R related to genus 1 Heegaard diagrams of lens spaces and a relation among them. This relation implies that F_L(A) is a topological invariant for lens spaces in a certain sense.

In this paper, we define invariants for primitive Legendrian knots in lens spaces L(p, q), q ≠ 1. The main invariant is a differential graded algebra which is computed from a labeled Lagrangian projection of the pair (L(p, q), K). This invariant is formally similar to a DGA defined by Sabloff which is an invariant for Legendrian knots in smooth S¹-bundles over Riemann surfaces. The second invariant defined for K ⊂ L(p, q) takes the form of a DGA enhanced with a free cyclic group action and can be computed from a cyclic cover of the pair (L(p, q), K).

In this paper, we classify the smooth orientation preserving cyclic $p$-group actions on the real projective space $ℝ{\mathbb{P}}^3$ up equivalence, where two actions are equivalent if their images are conjugate in the group of self-diffeomorphisms. We view $ℝ{\mathbb{P}}^3$ as the lens space $L(2,1)$. We show that any such action on $ℝ{\mathbb{P}}^3$ is conjugate to a standard action explicitly defined, and we identify the quotient spaces of these actions. In addition, we enumerate the equivalence classes.

The Turaev-Viro-Ocneanu invariant of three manifolds is a generalization of the Turaev-Viro invariant of three manifolds, and is derived from the paragroup of the subfactor. We investigate the E₆ Turaev-Viro-Ocneanu invariant derived from the E₆ subfactor. We give a formula for the invariant of the lens space L(p, 1). The value of the E₆ Turaev-Viro-Ocneanu invariant can be a complex number, in contrast with the fact that the value of the Turaev-Viro invariant must be a real number because of the relation "Turaev-Viro invariant" = |"Reshetikhin-Turaev invariant"|². Thus the E₆ Turaev-Viro-Ocneanu invariant can distinguish the lens space L(p,1) from the same one with reversed orientation L(p, p- 1) for p = 12k + 3,12k + 4,12k + 8,12k + 9 while the Turaev-Viro invariant can not.