4. The Dehn Surgery Formula

1992 ◽  
pp. 81-94
Keyword(s):  
Dehn Surgery ◽  
Surgery Formula

scholarly journals Hyperbolic tunnel-number-one knots with Seifert-fibered Dehn surgeries

2020 ◽  
Vol 29 (11) ◽  
pp. 2050075
Author(s):  
Sungmo Kang
Keyword(s):  
Dehn Surgery ◽  
Surface Slope ◽  
Addition Formula ◽  
Closed Curves ◽  
Tunnel Number ◽  
Genus Two ◽  
Surgery Formula ◽  
Tunnel Number One Knots

Suppose [Formula: see text] and [Formula: see text] are disjoint simple closed curves in the boundary of a genus two handlebody [Formula: see text] such that [Formula: see text] (i.e. a 2-handle addition along [Formula: see text]) embeds in [Formula: see text] as the exterior of a hyperbolic knot [Formula: see text] (thus, [Formula: see text] is a tunnel-number-one knot), and [Formula: see text] is Seifert in [Formula: see text] (i.e. a 2-handle addition [Formula: see text] is a Seifert-fibered space) and not the meridian of [Formula: see text]. Then for a slope [Formula: see text] of [Formula: see text] represented by [Formula: see text], [Formula: see text]-Dehn surgery [Formula: see text] is a Seifert-fibered space. Such a construction of Seifert-fibered Dehn surgeries generalizes that of Seifert-fibered Dehn surgeries arising from primtive/Seifert positions of a knot, which was introduced in [J. Dean, Small Seifert-fibered Dehn surgery on hyperbolic knots, Algebr. Geom. Topol. 3 (2003) 435–472.]. In this paper, we show that there exists a meridional curve [Formula: see text] of [Formula: see text] (or [Formula: see text]) in [Formula: see text] such that [Formula: see text] intersects [Formula: see text] transversely in exactly one point. It follows that such a construction of a Seifert-fibered Dehn surgery [Formula: see text] can arise from a primitive/Seifert position of [Formula: see text] with [Formula: see text] its surface-slope. This result supports partially the two conjectures: (1) any Seifert-fibered surgery on a hyperbolic knot in [Formula: see text] is integral, and (2) any Seifert-fibered surgery on a hyperbolic tunnel-number-one knot arises from a primitive/Seifert position whose surface slope corresponds to the surgery slope.

scholarly journals Computations of Turaev-Viro-Ocneanu Invariants of 3-Manifolds from Subfactors

2003 ◽  
Vol 12 (04) ◽  
pp. 543-574 ◽  
Author(s):  
Nobuya Sato ◽  
Michihisa Wakui
Keyword(s):  
Lens Spaces ◽  
Dehn Surgery ◽  
Canonical Isomorphism ◽  
Linear Isomorphism ◽  
Surgery Formula ◽  
Sector Theory

In this paper, we establish a rigorous correspondence between the two tube algebras, that one comes from the Turaev-Viro-Ocneanu TQFT introduced by Ocneanu and another comes from the sector theory introduced by Izumi, and construct a canonical isomorphism between the centers of the two tube algebras, which is a conjugate linear isomorphism preserving the products of the two algebras and commuting with the actions of SL(2, Z). Via this correspondence and the Dehn surgery formula, we compute Turaev-Viro-Ocneanu invariants from several subfactors for basic 3-manifolds including lens spaces and Brieskorn 3-manifolds by using Izumi's data written in terms of sectors.

scholarly journals A Dehn Surgery Formula for Walker Invariant on a Link

1994 ◽  
Vol 17 (2) ◽  
pp. 359-372 ◽  
Author(s):  
Katsuhiro ISHIBE
Keyword(s):  
Dehn Surgery ◽  
Surgery Formula

Dehn surgeries on some classical links

2011 ◽  
Vol 54 (1) ◽  
pp. 33-45 ◽  
Author(s):  
Alberto Cavicchioli ◽  
Fulvia Spaggiari ◽  
Agnese Ilaria Telloni
Keyword(s):  
Dehn Surgery ◽  
Fundamental Groups ◽  
Branched Coverings ◽  
Borromean Rings

AbstractWe consider orientable closed connected 3-manifolds obtained by performing Dehn surgery on the components of some classical links such as Borromean rings and twisted Whitehead links. We find geometric presentations of their fundamental groups and describe many of them as 2-fold branched coverings of the 3-sphere. Finally, we obtain some topological applications on the manifolds given by exceptional surgeries on hyperbolic 2-bridge knots.

scholarly journals Borromean surgery formula for the Casson invariant

2008 ◽  
Vol 8 (2) ◽  
pp. 787-801
Author(s):  
Jean-Baptiste Meilhan
Keyword(s):  
Surgery Formula

scholarly journals The shape of hyperbolic Dehn surgery space

2008 ◽  
Vol 12 (2) ◽  
pp. 1033-1090 ◽  
Author(s):  
Craig Hodgson ◽  
Steven Kerckhoff
Keyword(s):  
Dehn Surgery

CREATING KLEIN BOTTLES BY SURGERY ON KNOTS

2001 ◽  
Vol 10 (05) ◽  
pp. 781-794 ◽  
Author(s):  
MASAKAZU TERAGAITO
Keyword(s):  
Upper Bound ◽  
Klein Bottle ◽  
Dehn Surgery ◽  
The Creation ◽  
Genus One

In the present paper, we will study the creation of Klein bottles by Dehn surgery on knots in the 3-sphere, and we will give an upper bound for slopes creating Klein bottles for non-cabled knots by using the genera of knots. In particular, it is shown that if a Klein bottle is created by Dehn surgery on a genus one knot then the knot is a doubled knot. As a corollary, we obtain that genus one, cross-cap number two knots are doubled knot.

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