Haken spheres in the connected sum of two lens spaces

Diagrams and Reidemeister moves for links in a twisted S1-bundle over an unorientable surface are introduced. Using these diagrams, we compute the Kauffman Bracket Skein Module (KBSM) of ℝP3♯ℝP3. In particular, we show that it has torsion. We also present a new computation of the KBSM of S1 × S2 and the lens spaces L(p, 1).

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Haken spheres for genus two Heegaard splittings

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004117000718 ◽

2017 ◽

Vol 165 (3) ◽

pp. 563-572 ◽

Cited By ~ 1

Author(s):

SANGBUM CHO ◽

YUYA KODA

Keyword(s):

Heegaard Splitting ◽

Combinatorial Structure ◽

Lens Spaces ◽

Heegaard Splittings ◽

Precise Description ◽

Connected Sum ◽

Connected Sums ◽

Genus 2 ◽

Genus Two

AbstractA manifold which admits a reducible genus-2 Heegaard splitting is one of the 3-sphere, S2 × S1, lens spaces or their connected sums. For each of those splittings, the complex of Haken spheres is defined. When the manifold is the 3-sphere, S2 × S1 or a connected sum whose summands are lens spaces or S2 × S1, the combinatorial structure of the complex has been studied by several authors. In particular, it was shown that those complexes are all contractible. In this work, we study the remaining cases, that is, when the manifolds are lens spaces. We give a precise description of each of the complexes for the genus-2 Heegaard splittings of lens spaces. A remarkable fact is that the complexes for most lens spaces are not contractible and even not connected.

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Connect sum of lens spaces surgeries: application to Hin recombination

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004111000090 ◽

2011 ◽

Vol 150 (3) ◽

pp. 505-525 ◽

Cited By ~ 14

Author(s):

DOROTHY BUCK ◽

MAURO MAURICIO

Keyword(s):

Lens Space ◽

Lens Spaces ◽

Branched Covers ◽

Dehn Filling ◽

Connected Sum ◽

Hin Recombinase ◽

Complete Set ◽

Dehn Fillings ◽

Filling Problem ◽

Tangle Model

AbstractWe extend the tangle model, originally developed by Ernst and Sumners [18], to include composite knots. We show that, for any prime tangle, there are no rational tangle attachments of distance greater than one that first yield a 4-plat and then a connected sum of 4-plats. This is done by studying the corresponding Dehn filling problem via double branched covers. In particular, we build on results on exceptional Dehn fillings at maximal distance to show that if Dehn filling on an irreducible manifold gives a lens space and then a connect sum of lens spaces, the distance between the slopes must be one. We then apply our results to the action of the Hin recombinase on mutated sites. In particular, after solving the tangle equations for processive recombination, we use our work to give a complete set of solutions to the tangle equations modelling distributive recombination.

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REDUCIBLE DEHN SURGERY AND THE BRIDGE NUMBER OF A KNOT

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216509007038 ◽

2009 ◽

Vol 18 (04) ◽

pp. 493-504

Author(s):

NABIL SAYARI

Keyword(s):

Lens Spaces ◽

Homology Sphere ◽

Dehn Surgery ◽

Connected Sum ◽

Bridge Number

Let K be a knot in S3 and suppose that K(r) is a reducible manifold. Howie has proved that the number of connected summands is at most 3 [10, Corollary 5.3]. Furthermore, he showed that if K(r) is the connected sum of exactly three irreducible 3-manifolds, then K(r) = L(p1, q1)♯L(p2, q2)♯M, where L(p1, q1) and L(p2, q2) are a lens spaces and M is a ℤ-homology sphere. In this paper we study this specific case and we show that |r| ≤ (b-2)(b-1), for any knot with bridge number b.

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Rational cobordisms and integral homology

Compositio Mathematica ◽

10.1112/s0010437x20007320 ◽

2020 ◽

Vol 156 (9) ◽

pp. 1825-1845

Author(s):

Paolo Aceto ◽

Daniele Celoria ◽

JungHwan Park

Keyword(s):

Homology Group ◽

Floer Homology ◽

Lens Spaces ◽

Integral Homology ◽

Rational Homology ◽

Cobordism Class ◽

Connected Sum ◽

Knot Floer Homology ◽

Infinite Rank ◽

Homology Cobordism

We consider the question of when a rational homology $3$-sphere is rational homology cobordant to a connected sum of lens spaces. We prove that every rational homology cobordism class in the subgroup generated by lens spaces is represented by a unique connected sum of lens spaces whose first homology group injects in the first homology group of any other element in the same class. As a first consequence, we show that several natural maps to the rational homology cobordism group have infinite-rank cokernels. Further consequences include a divisibility condition between the determinants of a connected sum of $2$-bridge knots and any other knot in the same concordance class. Lastly, we use knot Floer homology combined with our main result to obstruct Dehn surgeries on knots from being rationally cobordant to lens spaces.

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