THE SIMPLEST HYPERBOLIC KNOTS

1999 ◽  
Vol 08 (03) ◽  
pp. 279-297 ◽  
Author(s):  
PATRICK J. CALLAHAN ◽  
JOHN C. DEAN ◽  
JEFFREY R. WEEKS
Keyword(s):  
Ad Hoc ◽  
Crossing Number ◽  
Hyperbolic Manifolds ◽  
Torus Knots ◽  
Initial Observation ◽  
Knot Complements ◽  
Standard Notion ◽  
Dehn Fillings

While the crossing number is the standard notion of complexity for knots, the number of ideal tetrahedra required to construct the complement provides a natural alternative. We determine which hyperbolic manifolds with 6 or fewer ideal tetrahedra are knot complements, and explicitly describe the corresponding knots in the 3-sphere. Thus, these 72 knots are the simplest knots according to this notion of complexity. Many of these knots have the structure of twisted torus knots. The initial observation that led to the project was the abundance of knot complements with small Seifert-fibered Dehn fillings among the census manifolds. Since many of these knots have rather large crossing number they do not appear in the knot tables. Our methods, while ad hoc, yield some detailed information about the knot complements as well as the manifolds that arise from exceptional surgeries on these knots.

scholarly journals RIBBONLENGTH OF TORUS KNOTS

2008 ◽  
Vol 17 (01) ◽  
pp. 13-23 ◽  
Author(s):  
BROOKE KENNEDY ◽  
THOMAS W. MATTMAN ◽  
ROBERTO RAYA ◽  
DAN TATING
Keyword(s):  
Crossing Number ◽  
Regular Polygons ◽  
Torus Knots ◽  
Torus Knot

Using Kauffman's model of flat knotted ribbons, we demonstrate how all regular polygons of at least seven sides can be realized by ribbon constructions of torus knots. We calculate length to width ratios for these constructions thereby bounding the Ribbonlength of the knots. In particular, we give evidence that the closed (respectively, truncation) Ribbonlength of a (q + 1,q) torus knot is (2q + 1) cot (π/(2q + 1)) (respectively, 2q cot (π/(2q + 1))). Using these calculations, we provide the bounds c1 ≤ 2/π and c2 ≥ 5/3 cot π/5 for the constants c1 and c2 that relate Ribbonlength R(K) and crossing number C(K) in a conjecture of Kusner: c1 C(K) ≤ R(K) ≤ c2 C(K).

THE ADDITIVITY OF CROSSING NUMBERS

2004 ◽  
Vol 13 (07) ◽  
pp. 857-866 ◽  
Author(s):  
YUANAN DIAO
Keyword(s):  
Knot Theory ◽  
Difficult Problem ◽  
Crossing Number ◽  
The Other ◽  
Connected Sum ◽  
Torus Knots ◽  
Crossing Numbers ◽  
Weaker Conditions ◽  
Alternating Links ◽  
Alternating Link

It has long been conjectured that the crossing numbers of links are additive under the connected sum of links. This is a difficult problem in knot theory and has been open for more than 100 years. In fact, many questions of much weaker conditions are still open. For instance, it is not known whether Cr(K1#K2)≥Cr(K1) or Cr(K1#K2)≥Cr(K2) holds in general, here K1#K2 is the connected sum of K1 and K2 and Cr(K) stands for the crossing number of the link K. However, for alternating links K1 and K2, Cr(K1#K2)=Cr(K1)+Cr(K2) does hold. On the other hand, if K1 is an alternating link and K2 is any link, then we have Cr(K1#K2)≥Cr(K1). In this paper, we show that there exists a wide class of links over which the crossing number is additive under the connected sum operation. This class is different from the class of all alternating links. It includes all torus knots and many alternating links. Furthermore, if K1 is a connected sum of any given number of links from this class and K2 is a non-trivial knot, we prove that Cr(K1#K2)≥Cr(K1)+3.

scholarly journals REDUCING DEHN FILLINGS AND SMALL SURFACES

2005 ◽  
Vol 92 (1) ◽  
pp. 203-223 ◽  
Author(s):  
SANGYOP LEE ◽  
SEUNGSANG OH ◽  
MASAKAZU TERAGAITO
Keyword(s):  
Projective Plane ◽  
Klein Bottle ◽  
Hyperbolic Manifolds ◽  
The Other ◽  
Möbius Band ◽  
Dehn Fillings ◽  
Orientable Surfaces

In this paper we investigate the distances between Dehn fillings on a hyperbolic 3-manifold that yield 3-manifolds containing essential small surfaces including non-orientable surfaces. In particular, we study the situations where one filling creates an essential sphere or projective plane, and the other creates an essential sphere, projective plane, annulus, Möbius band, torus or Klein bottle, for all eleven pairs of such non-hyperbolic manifolds.

Klein slopes on hyperbolic 3-manifolds

2007 ◽  
Vol 143 (2) ◽  
pp. 419-447
Author(s):  
DANIEL MATIGNON ◽  
NABIL SAYARI
Keyword(s):  
Upper Bound ◽  
Klein Bottle ◽  
Hyperbolic Manifolds ◽  
Dehn Fillings

AbstractThis paper is devoted to 3-manifolds which admit two distinct Dehn fillings producing a Klein bottle.LetMbe a compact, connected and orientable 3-manifold whose boundary contains a 2-torusT. IfMis hyperbolic then only finitely many Dehn fillings alongTyield non-hyperbolic manifolds. We consider the situation where two distinct slopes γ1, γ2produce a Klein bottle. We give an upper bound for the distance Δ(γ1, γ2), between γ1and γ2. We show that there are exactly four hyperbolic manifolds for which Δ(γ1, γ2) > 4.

MANY CLASSICAL KNOT INVARIANTS ARE NOT VASSILIEV INVARIANTS

1994 ◽  
Vol 03 (01) ◽  
pp. 7-10 ◽  
Author(s):  
JOHN DEAN
Keyword(s):  
Crossing Number ◽  
Torus Knots ◽  
Knot Invariants ◽  
Vassiliev Invariants ◽  
Vassiliev Invariant ◽  
Bridge Number

We show that under twisting, a Vassiliev invariant of order n behaves like a polynomial of degree at most n. This greatly restricts the values that a Vassiliev invariant can take, for example, on the (2, m) torus knots. In particular, this implies that many classical numerical knot invariants such as the signature, genus, bridge number, crossing number, and unknotting number are not Vassiliev invariants.

scholarly journals EXAMPLES OF REDUCIBLE AND FINITE DEHN FILLINGS

2010 ◽  
Vol 19 (05) ◽  
pp. 677-694 ◽  
Author(s):  
SUNGMO KANG
Keyword(s):  
Hyperbolic Manifolds ◽  
Dehn Filling ◽  
Dehn Fillings

If a hyperbolic 3-manifold M admits a reducible and a finite Dehn filling, the distance between the filling slopes is known to be 1. This has been proved recently by Boyer, Gordon and Zhang. The first example of a manifold with two such fillings was given by Boyer and Zhang. In this paper, we give examples of hyperbolic manifolds admitting a reducible Dehn filling and a finite Dehn filling of every type: cyclic, dihedral, tetrahedral, octahedral and icosahedral.

scholarly journals THE NEXT SIMPLEST HYPERBOLIC KNOTS

2004 ◽  
Vol 13 (07) ◽  
pp. 965-987 ◽  
Author(s):  
ABHIJIT CHAMPANERKAR ◽  
ILYA KOFMAN ◽  
ERIC PATTERSON
Keyword(s):  
Crossing Number ◽  
Hyperbolic Manifolds ◽  
Jones Polynomials

We complete the project begun by Callahan, Dean and Weeks to identify all knots whose complements are in the SnapPea census of hyperbolic manifolds with seven or fewer tetrahedra. Many of these "simple" hyperbolic knots have high crossing number. We also compute their Jones polynomials.

scholarly journals Integrality of volumes of representations

2021 ◽  
Author(s):  
Michelle Bucher ◽  
Marc Burger ◽  
Alessandra Iozzi
Keyword(s):  
Finite Volume ◽  
Hyperbolic Manifolds ◽  
3 Dimensional ◽  
Definition Of ◽  
Dehn Fillings

AbstractLet M be an oriented complete hyperbolic n-manifold of finite volume. Using the definition of volume of a representation previously given by the authors in [3] we show that the volume of a representation $$\rho :\pi _1(M)\rightarrow \mathrm {Isom}^+({{\mathbb {H}}}^n)$$ ρ : π 1 ( M ) → Isom + ( H n ) , properly normalized, takes integer values if n is even and $$\ge 4$$ ≥ 4 . If M is not compact and 3-dimensional, it is known that the volume is not locally constant. In this case we give explicit examples of representations with volume as arbitrary as the volume of hyperbolic manifolds obtained from M via Dehn fillings.

scholarly journals Quivers for 3-manifolds: the correspondence, BPS states, and 3d $$ \mathcal{N} $$ = 2 theories

2020 ◽  
Vol 2020 (9) ◽  
Author(s):  
Piotr Kucharski
Keyword(s):  
Physical Interpretation ◽  
General Structure ◽  
Torus Knots ◽  
Knot Complements ◽  
Bps Spectrum ◽  
Bps States

Abstract We introduce and explore the relation between quivers and 3-manifolds with the topology of the knot complement. This idea can be viewed as an adaptation of the knots-quivers correspondence to Gukov-Manolescu invariants of knot complements (also known as FK or $$ \hat{Z} $$ Z ̂ ). Apart from assigning quivers to complements of T(2,2p+1) torus knots, we study the physical interpretation in terms of the BPS spectrum and general structure of 3d $$ \mathcal{N} $$ N = 2 theories associated to both sides of the correspondence. We also make a step towards categorification by proposing a t-deformation of all objects mentioned above.

scholarly journals A Menagerie of SU(2)-Cyclic 3-Manifolds

2021 ◽  
Author(s):  
Steven Sivek ◽  
Raphael Zentner
Keyword(s):  
Fundamental Group ◽  
Lens Space ◽  
Hyperbolic Manifolds ◽  
Cyclic Surgery ◽  
Dehn Fillings

Abstract We classify $SU(2)$-cyclic and $SU(2)$-abelian 3-manifolds, for which every representation of the fundamental group into $SU(2)$ has cyclic or abelian image, respectively, among geometric 3-manifolds that are not hyperbolic. As an application, we give examples of hyperbolic 3-manifolds that do not admit degree-1 maps to any Seifert Fibered manifold other than $S^3$ or a lens space. We also produce infinitely many one-cusped hyperbolic manifolds with at least four $SU(2)$-cyclic Dehn fillings, one more than the number of cyclic fillings allowed by the cyclic surgery theorem.

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