On the Knot Floer Homology of Twisted Torus Knots

Vafaee, Faramarz

doi:10.1093/imrn/rnu130

Cited by 30 publications

(25 citation statements)

References 22 publications

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“…Thus, since b δ + n ≡ (−qδ)(βq * ) ≡ βδ (mod p), there exists k 0 ∈ Z >0 such that b δ + n = βδ−k 0 p > 0. Thus (b δ + , k 0 ) provides a solution for (b, k) in (47), which, together with the relation b δ + ≡ pγ − qδ (mod g), implies (48) also holds for (b, k) = (b δ + , k 0 ), and so Y # (λ # ) is not an L-space. Conversely, suppose that Y # (λ # ) is not an L-space.…”

Section: Thus If We Setmentioning

confidence: 99%

“…Then there exist δ = δι(m 1 )+γι(l 1 ) ∈ D τ >0 (Y ) and (b, k) ∈ Z×Z >0 for which (47) and (48) hold. In particular, (47) implies bn < βδ and b > 0, while (47) and (48) together imply b ≡ b δ + (mod pg), requiring b ≥ b δ + . Thus βδ > bn ≥ b δ + n, implying β/n > b δ + /δ.…”

Section: Thus If We Setmentioning

confidence: 99%

“…• The twisted torus knots T (p, kp ± 1; 2, 1) studied by Vafaee [47] • Cables of L-space braids [21] • Satellites where the pattern knot is a Berge-Gabai knot and the companion is an L-space braid [22]…”

Section: 7mentioning

confidence: 99%

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Floer simple manifolds and L-space intervals

Rasmussen

2017

Advances in Mathematics

120

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Abstract. An oriented three-manifold with torus boundary admits either no L-space Dehn filling, a unique L-space filling, or an interval of L-space fillings. In the latter case, which we call "Floer simple," we construct an invariant which computes the interval of Lspace filling slopes from the Turaev torsion and a given slope from the interval's interior. As applications, we give a new proof of the classification of Seifert fibered L-spaces over S 2 , and prove a special case of a conjecture of Boyer and Clay [6] about L-spaces formed by gluing three-manifolds along a torus.

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Section: Thus If We Setmentioning

confidence: 99%

Section: Thus If We Setmentioning

confidence: 99%

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Floer simple manifolds and L-space intervals

Rasmussen

2017

Advances in Mathematics

120

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“…The following theorem due to Vafaee [Vaf13] provides a class of twisted torus knots which are known to be L-space knots. In [CW13], Clay and Watson apply Theorem 5 to certain subfamilies of the Lspace knots specified in Theorem 11.…”

Section: Introductionmentioning

confidence: 99%

Non-left-orderable surgeries on twisted torus knots

Christianson

Goluboff

Hamann

et al. 2016

Proc. Amer. Math. Soc.

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Abstract. Boyer, Gordon, and Watson [BGW13] have conjectured that an irreducible rational homology 3-sphere is an L-space if and only if its fundamental group is not left-orderable. Since large classes of L-spaces can be produced from Dehn surgery on knots in S 3 , it is natural to ask what conditions on the knot group are sufficient to imply that the quotient associated to Dehn surgery is not left-orderable. Clay and Watson develop a criterion for determining the left-orderability of this quotient group in [CW13] and use it to verify the conjecture for surgeries on certain L-space twisted torus knots. We generalize a recent theorem of Ichihara and Temma [IT14] to provide another such criterion. We then use this new criterion to generalize the results of Clay and Watson and to verify the conjecture for a much broader class of L-space twisted torus knots.

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“…In general, it is difficult to calculate HF K explicitly. However, we might refer to [1,24] for a combinatorial method for the calculation of the above Heegaard Floer homology.…”

Section: The Khovanov Cohomology Of Kanenobu Knotsmentioning

confidence: 99%

A Recursive Formula for the Khovanov Cohomology of Kanenobu Knots

Lei¹,

Zhang²

2017

Bulletin of the Korean Mathematical Society

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Abstract. Kanenobu has given infinite families of knots with the same HOMFLY polynomial invariant but distinct Alexander module structure. In this paper, we give a recursive formula for the Khovanov cohomology of all Kanenobu knots K(p, q), where p and q are integers. The result implies that the rank of the Khovanov cohomology of K(p, q) is an invariant of p + q. Our computation uses only the basic long exact sequence in knot homology and some results on homologically thin knots.

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On the Knot Floer Homology of Twisted Torus Knots

Cited by 30 publications

References 22 publications

Floer simple manifolds and L-space intervals

Floer simple manifolds and L-space intervals

Non-left-orderable surgeries on twisted torus knots

A Recursive Formula for the Khovanov Cohomology of Kanenobu Knots

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