Lens Space Surgeries Research Articles

The Poincaré homology sphere, lens space surgeries, and some knots with tunnel number two

We exhibit an infinite family of knots in the Poincare homology sphere with tunnel number 2 that have a lens space surgery. Notably, these knots are not doubly primitive and provide counterexamples to a few conjectures. In the appendix, it is shown that hyperbolic knots in the Poincare homology sphere with a lens space surgery has either no symmetries or just a single strong involution.

On the Alexander polynomial of lens space knots

Ozsváth and Szabó proved the property that coefficients of the Alexander polynomial of any lens space knot are either ±1 or 0 and the non-zero coefficients are alternating. Combining the formulas of the Alexander polynomial of lens space knots by Kadokami and Yamada, and Ichihara, Saito and Teragaito, we refine Ozsváth and Szabó's property as the existence of simple curves included in a region in R2. The curves have no end-points and include just 1-component in a region. This can be much useful for studying the distribution of non-zero coefficients of the Alexander polynomial of the lens space knot. For example, we can investigate the location of the second, third and fourth non-zero coefficients. We extract a new invariant α-index. This invariant is an important factor to determine Alexander polynomials of lens space knots. We give a criterion for a lens space surgery that the Alexander polynomial is the same as a (2,r)-torus knot and a lens space surgery with small genus and so on. As well as lens space knots in S3, we also deal with lens space knots in homology spheres whose surgery duals are simple (1,1)-knots.

On LMO invariant constraints for cosmetic surgery and other surgery problems for knots in $S^3$

We use the LMO invariant to find constraints for a knot to admit a purely or reflectively cosmetic surgery. We also get a constraint for knots to admit a Lens space surgery, and some information for characterizing slopes.

On hyperbolic knots in S3 with exceptional surgeries at maximal distance

Baker showed that 10 of the 12 classes of Berge knots are obtained by surgery on the minimally twisted 5-chain link. In this article we enumerate all hyperbolic knots in S^3 obtained by surgery on the minimally twisted 5 chain link that realize the maximal known distances between slopes corresponding to exceptional (lens, lens), (lens, toroidal), (lens, Seifert fibred spaces) pairs. In light of Baker's work, the classification in this paper conjecturally accounts for 'most' hyperbolic knots in S^3 realizing the maximal distance between these exceptional pairs. All examples obtained in our classification are realized by filling the magic manifold. The classification highlights additional examples not mentioned in Martelli and Petronio's survey of the exceptional fillings on the magic manifold. Of particular interest, is an example of a knot with two lens space surgeries that is not obtained by filling the Berge manifold.

Berge duals and universally tight contact structures

Dehn surgery on a knot determines a dual knot in the surgered manifold, the core of the filling torus. We consider duals of knots in S3 that have a lens space surgery. Each dual supports a contact structure. We show that if a universally tight contact structure is supported, then the dual is in the same homology class as the dual of a torus knot.

Four-dimensional manifolds constructed by lens space surgeries of distinct types

A framed knot with an integral coefficient determines a simply-connected 4-manifold by 2-handle attachment. Its boundary is a 3-manifold obtained by Dehn surgery along the framed knot. For a pair of such Dehn surgeries along distinct knots whose results are homeomorphic, it is a natural problem: Determine the closed 4-manifold obtained by pasting the 4-manifolds along their boundaries. We study pairs of lens space surgeries along distinct knots whose lens spaces (i.e. the resulting lens spaces of the surgeries) are orientation-preservingly or -reversingly homeomorphic. In the authors’ previous work, we treated with the case both knots are torus knots. In this paper, we focus on the case where one is a torus knot and the other is a Berge’s knot Type VII or VIII, in a genus one fiber surface. We determine the complete list (set) of such pairs of lens space surgeries and study the closed 4-manifolds constructed as above. The list consists of six sequences. All framed links and handle calculus are indexed by integers.

Bounds on alternating surgery slopes

We show that if $p/q$-surgery on a nontrivial knot $K$ yields the branched double cover of an alternating knot or link, then $|p/q|\leq 4g(K)+3$. This generalises a bound for lens space surgeries first established by Rasmussen. We also show that all surgery coefficients yielding the double branched cover of an alternating knot or link must be contained in an interval of width two and this full range can be realised only if $K$ is a cable knot. The work of Greene and Gibbons shows that if $S^3_{p/q}(K)$ bounds a sharp 4-manifold $X$, then the intersection form of $X$ takes the form of a changemaker lattice. We extend this to show that the intersection form is determined uniquely by the knot $K$, the slope $p/q$ and the Betti number $b_2(X)$.

A distribution of rational homology 3-spheres captured by the CWL invariant Phase 1

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 [Formula: see text] with [Formula: see text].

On the geography and botany of knot Floer homology

This paper explores two questions: (1) Which bigraded groups arise as the knot Floer homology of a knot in the three-sphere? (2) Given a knot, how many distinct knots share its Floer homology? Regarding the first, we show there exist bigraded groups satisfying all previously known constraints of knot Floer homology which do not arise as the invariant of a knot. This leads to a new constraint for knots admitting lens space surgeries, as well as a proof that the rank of knot Floer homology detects the trefoil knot. For the second, we show that any non-trivial band sum of two unknots gives rise to an infinite family of distinct knots with isomorphic knot Floer homology. We also prove that the fibered knot with identity monodromy is strongly detected by its knot Floer homology, implying that Floer homology solves the word problem for mapping class groups of surfaces with non-empty boundary. Finally, we survey some conjectures and questions and, based on the results described above, formulate some new ones.

Some knots in $S^1 \times S^2$ with lens space surgeries

We propose a classification of knots in S^1 x S^2 that admit a longitudinal surgery to a lens space. Any lens space obtainable by longitudinal surgery on some knots in S^1 x S^2 may be obtained from a Berge-Gabai knot in a Heegaard solid torus of S^1 x S^2, as observed by Rasmussen. We show that there are yet two other families of knots: those that lie on the fiber of a genus one fibered knot and the `sporadic' knots. All these knots in S^1 x S^2 are both doubly primitive and spherical braids. This classification arose from generalizing Berge's list of doubly primitive knots in S^3, though we also examine how one might develop it using Lisca's embeddings of the intersection lattices of rational homology balls bounded by lens spaces as a guide. We conjecture that our knots constitute a complete list of doubly primitive knots in S^1 x S^2 and reduce this conjecture to classifying the homology classes of knots in lens spaces admitting a longitudinal S^1 x S^2 surgery.

A cabling conjecture for knots in lens spaces

Closed $$3$$ -string braids admit many bandings to two-bridge links. By way of the Montesinos Trick, this allows us to construct infinite families of knots in the connected sum of lens spaces $$L(r,1) \# L(s,1)$$ that admit a surgery to a lens space for all pairs of integers $$(r,s)$$ except $$(0,0)$$ . These knots are typically hyperbolic. We also demonstrate that the previously known two families of examples of hyperbolic knots in non-prime manifolds with lens space surgeries of Eudave-Munoz–Wu and Kang all fit this construction. As such, we propose a generalization of the cabling conjecture of Gonzalez-Acuna–Short for knots in lens spaces.

On manifolds with multiple lens space fillings

An irreducible \(3\)-manifold with torus boundary either is a Seifert fibered space or admits at most three lens space fillings according to the Cyclic Surgery Theorem. We examine the sharpness of this theorem by classifying the non-hyperbolic manifolds with more than one lens space filling, classifying the hyperbolic manifolds obtained by filling of the Minimally Twisted 5 Chain complement that have three lens space fillings, showing that the doubly primitive knots in \(S^3\) and \(S^1 \times S^2\) have no unexpected extra lens space surgery, and showing that the Figure Eight Knot Sister Manifold is the only non-Seifert fibered manifold with a properly embedded essential once-punctured torus and three lens space fillings.

LENS SPACE SURGERIES ALONG CERTAIN 2-COMPONENT LINKS RELATED WITH PARK’S RATIONAL BLOW DOWN, AND REIDEMEISTER-TURAEV TORSION

AbstractWe study lens space surgeries along two different families of 2-component links, denoted by${A}_{m, n} $and${B}_{p, q} $, related with the rational homology$4$-ball used in J. Park’s (generalized) rational blow down. We determine which coefficient$r$of the knotted component of the link yields a lens space by Dehn surgery. The link${A}_{m, n} $yields a lens space only by the known surgery with$r= mn$and unexpectedly with$r= 7$for$(m, n)= (2, 3)$. On the other hand,${B}_{p, q} $yields a lens space by infinitely many$r$. Our main tool for the proof are the Reidemeister-Turaev torsions, that is, Reidemeister torsions with combinatorial Euler structures. Our results can be extended to the links whose Alexander polynomials are same as those of${A}_{m, n} $and${B}_{p, q} $.

Dehn surgery, rational open books and knot Floer homology

By recent results of Baker--Etnyre--Van Horn-Morris, a rational open book decomposition defines a compatible contact structure. We show that the Heegaard Floer contact invariant of such a contact structure can be computed in terms of the knot Floer homology of its (rationally null-homologous) binding. We then use this description of contact invariants, together with a formula for the knot Floer homology of the core of a surgery solid torus, to show that certain manifolds obtained by surgeries on bindings of open books carry tight contact structures. Possible applications to lens space surgeries are discussed.

The classification of rational subtangle replacements between rational tangles

A natural generalization of a crossing change is a rational subtangle replacement (RSR). We characterize the fundamental situation of the rational tangles obtained from a given rational tangle via RSR, building on work of Berge and Gabai, and determine the sites where these RSR may occur. In addition we also determine the sites for RSR distance at least two between 2‐bridge links. These proofs depend on the geometry of the branched double cover. Furthermore, we classify all knots in lens spaces whose exteriors are generalized Seifert fibered spaces and their lens space surgeries, extending work of Darcy and Sumners. This work is in part motivated by the common biological situation of proteins cutting, rearranging and resealing DNA segments, effectively performing RSR on DNA “tangles”. 57M27

Surgery obstructions from Khovanov homology

For a 3-manifold with torus boundary admitting an appropriate involution, we show that Khovanov homology provides obstructions to certain exceptional Dehn fillings. For example, given a strongly invertible knot in S 3, we give obstructions to lens space surgeries, as well as obstructions to surgeries with finite fundamental group. These obstructions are based on homological width in Khovanov homology, and in the case of finite fundamental group depend on a calculation of the homological width for a family of Montesinos links.

Knots with the Lens Space Surgery

In this paper we construct an infinite family of hyperbolic (1, 1)–knots with two parameters, and show that some of them admit exceptional Dehn surgery such as lens space surgery, Seifert surgery, and toroidal surgery. Furthermore, we give simple examples of hyperbolic (1, 1)–knots which admit two toroidal surgeries at distance four such that both toroidal surgeries do not create Klein bottles.

EXCEPTIONAL SURGERIES ON CERTAIN (1, 1)-KNOTS

We determine certain exceptional surgeries on a 3-parametric family of hyperbolic 1-bridge genus one knots ((1, 1)-knots, in short). In p articular, we show that such knots admit two infinite series of lens space surgeries. Our work is related to a nice paper of Teragaito [16], since we represent his toroidal manifolds as 2-fold coverings of the 3-sphere branched over well-specified link s.

Knots which admit a surgery with simple knot Floer homology groups

The importance of studying knots inside rational homology spheres which have simple knot Floer homology came up in the study of the Berge conjecture using techniques from Heegaard Floer homology by Hedden [7], Rasmussen [14] and Baker, Grigsby and Hedden [1]. By definition, a knot K inside a rational homology sphere X has simple knot Floer homology if the rank of b HFK.X;K/ is equal to the rank of bHF.X / (see Oszvath and Szabo [11; 10] and Rasmussen [15] for the background on Heegaard Floer homology and knot Floer homology). The Berge conjecture concerning knots in S which admit a lens space surgery may almost be reduced to showing that a knot inside a lens space with simple knot Floer homology is simple.

A NOTE ON LENS SPACE SURGERIES: ORDERS OF FUNDAMENTAL GROUPS VERSUS SEIFERT GENERA

Let K be a non-trivial knot in the 3-sphere with a lens space surgery and L(p, q) a lens space obtained by a Dehn surgery on K. We study a relationship between the order p of the fundamental group of L(p, q) and the Seifert genus g of K. Considering certain infinite families of knots with lens space surgeries, the following estimation is suggested as a conjecture: [Formula: see text] except for (g, p) = (5, 19).