Lens Space Surgeries Along Certain 2-Component Links Related With Park’s rational Blow Down, and Reidemeister-Turaev Torsion

Doleshal

Hoffman

2014

Bol. Soc. Mat. Mex.

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 × 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.

Section: Doubly Primitive Knots Andmentioning

confidence: 99%

On manifolds with multiple lens space fillings

Doleshal

Hoffman

2014

Bol. Soc. Mat. Mex.

“…The knots on the right and left of Figure 1 are actually isotopic. Kadokami-Yamada show that among the non-torus gofk knots this is the only one (up to homeomorphism) that admits two nontrivial lens space surgeries [31]. Along these lines, Berge shows there is a unique hyperbolic knot in the solid torus with two non-trivial lens space surgeries [10], and this gives rise to a single bgiv knot (up to homeomoprhism) having surgeries to both orientations of L(49, 18) [11].…”

Section: Simple Knotsmentioning

confidence: 99%

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

Communications in Analysis and Geometry

Buck

Lecuona

2016

We propose a classification of knots in S 1 × S 2 that admit a longitudinal surgery to a lens space. Any lens space obtainable by longitudinal surgery on some knots in S 1 × S 2 may be obtained from a Berge-Gabai knot in a Heegaard solid torus of S 1 × 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 × 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 × S 2 and reduce this conjecture to classifying the homology classes of knots in lens spaces admitting a longitudinal S 1 × S 2 surgery.

“…(4) If, say, s = ±1, then the knots K p,q r,±1 are in L(r, 1) and have non-trivial lens space surgeries. In particular, when {r, s} = {0, ±1} these knots form the family of knots in S 1 × S 2 called gofk by Baker-Buck-Lecuona [6] and called A m,n by Kadokami-Yamada [27].…”

Section: Remarkmentioning

confidence: 99%

“…2. When (r, s) = (0, 0), the knots K of knots in S 1 × S 2 called gofk by Baker-Buck-Lecuona [6] and are called A m,n by Kadokami-Yamada [26]. 5.…”

mentioning

confidence: 99%

A cabling conjecture for knots in lens spaces

2014

Bol. Soc. Mat. Mex.

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-Muñoz-Wu and Kang all fit this construction. As such, we propose a generalization of the cabling conjecture of González-Acuña-Short for knots in lens spaces. The main results, context, and conjecturesBanding the braid closure L = β of a closed 3-string braid β along a "level" framed arc a produces a two-bridge link L = β that is a plait closure of β and a dual framed Dedicated to the 70th birthday of Professor Fico González-Acuña.