# Achiral 1-Cusped Hyperbolic 3-Manifolds Not Coming from Amphicheiral Null-homologous Knot Complements

@article{Ichihara2017Achiral1H, title={Achiral 1-Cusped Hyperbolic 3-Manifolds Not Coming from Amphicheiral Null-homologous Knot Complements}, author={Kazuhiro Ichihara and In Dae Jong and Kouki Taniyama}, journal={Lobachevskii Journal of Mathematics}, year={2017}, volume={39}, pages={1353-1361} }

It is experimentally known that achiral hyperbolic 3-manifolds are quite sporadic at least among those with small volume, while we can find plenty of them as amphicheiral knot complements in the 3-sphere. In this paper, we show that there exist infinitely many achiral 1-cusped hyperbolic 3- manifolds not homeomorphic to any amphicheiral null-homologous knot complement in any closed achiral 3-manifold.

#### References

SHOWING 1-10 OF 16 REFERENCES

Dehn Filling of the "Magic" 3-manifold

- Mathematics
- 2002

We classify all the non-hyperbolic Dehn fillings of the complement of the chain-link with 3 components, conjectured to be the smallest hyperbolic 3-manifold with 3 cusps. We deduce the classification… Expand

On the knot complement problem for non-hyperbolic knots

- Mathematics
- 2010

Abstract This paper explicitly provides two exhaustive and infinite families of pairs ( M , k ) , where M is a lens space and k is a non-hyperbolic knot in M, which produces a manifold homeomorphic… Expand

Cosmetic surgery on knots

- Mathematics
- 1999

This paper concerns the Dehn surgery construction, especially those Dehn surgeries leaving the manifold unchanged. In particular, we describe an oriented 1–cusped hyperbolic 3–manifold X with a pair… Expand

Cosmetic surgeries on knots in $S^3$

- Mathematics
- 2010

Two Dehn surgeries on a knot are called {\it purely cosmetic}, if they yield manifolds that are homeomorphic as oriented manifolds. Suppose there exist purely cosmetic surgeries on a knot in $S^3$,… Expand

Cosmetic banding on knots and links

- Mathematics
- 2016

We present various examples of cosmetic bandings on knots and links, that is, bandings on knots and links leaving their types unchanged. As a byproduct, we give a hyperbolic knot which admits exotic… Expand

A note on Jones polynomial and cosmetic surgery

- Mathematics
- 2016

We show that two Dehn surgeries on a knot $K$ never yield manifolds that are homeomorphic as oriented manifolds if $V_K''(1)\neq 0$ or $V_K'''(1)\neq 0$. As an application, we verify the cosmetic… Expand

Symmetries of spatial graphs and Simon invariants

- Mathematics
- 2007

An ordered and oriented 2-component link L in the 3-sphere is said to be achiral if it is ambient isotopic to its mirror image ignoring the orientation and ordering of the components. Kirk-Livingston… Expand

Cosmetic surgery and the $SL(2,\mathbb{C})$ Casson invariant for two-bridge knots

- Mathematics
- 2016

We consider the cosmetic surgery problem for two-bridge knots in the 3-sphere. It is seen that all the two-bridge knots at most 9 crossings other than $9_{27} = S(49,19)=C[2,2,-2,2,2,-2]$ admits no… Expand

Problems in low-dimensional topology

- Mathematics
- 1997

Four-dimensional topology is in an unsettled state: a great deal is known, but the largest-scale patterns and basic unifying themes are not yet clear. Kirby has recently completed a massive review of… Expand

Computation of Gordian Distances and H2-Gordian Distances of Knots

- Mathematics
- 2015

One of most complicated problems in knot theory is the computation ofunknotting number. Hass, Lagarias and Pippenger proved that the unknotting problem isNP. In this paper we discuss the question can… Expand