Boundary slopes for Montesinos knots

Hatcher, Allen; Oertel, Ulrich

doi:10.1016/0040-9383(89)90005-0

Cited by 82 publications

(205 citation statements)

References 9 publications

Supporting

Mentioning

202

Contrasting

Order By: Relevance

“…Similarly to the direct proofs of Lemmas 7,8,9, it is easy to show that the expressions in subscripts under the other two lines, (155) and (157), also are equal to this subscript. But the varieties given by the main parts (without subscripts) of pictures (155)-(157) and (154) satisfy condition (34).…”

Section: Lemma 8 the Chain (131) Is Equal To Zeromentioning

confidence: 99%

“…Finally, we get that the sum of four F-blocks indicated in the left-hand sides of equalities (10), (9), (12), and (13) provides a homology in σ 1 between the cycle (8) and the vice-maximal cell of σ 1 taken with some coefficient λ ∈ Z 2 . Denote this sum by γ 1 .…”

Section: Problemmentioning

confidence: 99%

“…Example 5. If ∆ is given by the sum (8), then the chain A is the first summand in (8), and A ↑ is indicated in the left-hand side of (9). The filtration of the chain A − ∂A ↑ is then strictly lower than that of A.…”

Section: 2mentioning

confidence: 99%

“…We obtain that the cycle (8) is homologous inσ 1 to the sum of the second and third summands of the right-hand sides of formulas (9) and (10). The sum of these second (respectively, third) summands can be depicted by the first (respectively, the second) summand in the next formula:…”

Section: Introductionmentioning

confidence: 99%

“…The latter condition is depicted by the "broken arrow" as in the left-hand side of (9). In a similar way, we try to span the second summand in (8) by the variety depicted in the left-hand side of (10).…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Untitled

2005

Trans. Moscow Math. Soc.

View full text Add to dashboard Cite

Abstract. We construct a homology algebraic algorithm for computing combinatorial formulas of all finite degree knot invariants. Its input is an arbitrary weight system, i.e., a virtual principal part of a finite degree invariant, and the output is either a proof of the fact that this weight system actually does not correspond to any knot invariant or an effective description of some invariant with this principal part, i.e., a finite collection of easily described singular chains of full dimension in the space of spatial curves such that the value of this invariant on any generic knot is equal to the sum of multiplicities of these chains in a neighborhood of the knot. (In examples calculated by now, the former possibility never occurred.) This algorithm is formally realized over Z 2 , but its generalization to the case of arbitrary coefficients is just a technical task. The algorithm is based on the study of a complex of chains in the space of smooth curves in the three-dimensional space with a fixed flag of directions, and also in the discriminant variety of this space of curves.

show abstract

Section: Lemma 8 the Chain (131) Is Equal To Zeromentioning

confidence: 99%

Section: Problemmentioning

confidence: 99%

Section: 2mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Untitled

2005

Trans. Moscow Math. Soc.

View full text Add to dashboard Cite

show abstract

Incompressible punctured tori in the complements of alternating knots

Patton

1995

Math. Ann.

View full text Add to dashboard Cite

Mathematics Subject Classification (1991): 7M25 (57N10) In some recent work, William Menasco and Morwen Thistlethwaite proved that the only alternating knots which allow incompressible planar surfaces with boundary to be properly embedded in their complements are the (2,p)-torus knots [M-T1]. This also answers the question of which alternating knots we can do a Dehn surgery on and get a reducible manifold. An obvious question to ask next would be which alternating knots will allow an incompressible punctured toms to be properly embedded in their complements. This question is of interest because from the combined works of Jaco-Shalen, Johannsen, and Thurston we know that Haken manifolds decompose along incompressible tori into manifolds which are either Seifert fibred spaces or hyperbolic manifolds. So we would like to find all of these alternating knots and explicitly describe the manifolds obtained by a surgery along the boundary slope of the incompressible punctured toms.We approach this problem by first using a result in [M-T1] to give us a bound on a diagrammatic quantity called the twist-crossing number. We then show that all of the diagrams satisfying this bound are either 2-bridge knots or Montesinos knots having only three rational tangles. The work of Hatcher and Thurston allows us to determine exactly which 2-bridge knots admit incompressible genus one surfaces in their complements and exactly what these surfaces must look like [H-T]. We then find which of these knots allow more than one distinct such surface. The work of Hatcher and Oertel [H-O] provides us with a structure for finding exactly which alternating Montesinos knots admit an incompressible genus one surface in their complements and, again, we can get an explicit picture of the surface. I then show which manifolds arise from surgeries on the boundary slopes of these surfaces in the case of 2-bridge knots.

show abstract

Boundary slopes of knots

Culler¹,

Shalen²

1999

Comment. Math. Helv.

View full text Add to dashboard Cite

Mathematics Subject Classification (1991). Primary 57M40, 57M25; Secondary 57M50, 57N10. Abstract.The results in this paper show that simple connectivity of a 3-manifold is reflected in the behavior of essential surfaces in exteriors of knots in the manifold. A corollary of the main theorem is that any non-trivial knot, with irreducible complement, in a homotopy 3-sphere must have two boundary slopes that differ by at least 2. This statement is false for knots in a homology 3-sphere. The main theorem itself applies more generally to knots in closed orientable 3-manifolds with cyclic fundamental group.

show abstract

Boundary slopes for Montesinos knots

Cited by 82 publications

References 9 publications

Untitled

Untitled

Incompressible punctured tori in the complements of alternating knots

Boundary slopes of knots

Contact Info

Product

Resources

About