KNOTS WITH DISTINCT PRIMITIVE/PRIMITIVE AND PRIMITIVE/SEIFERT REPRESENTATIVES

Berge introduced knots that are primitive/primitive with respect to the genus 2 Heegaard surface, F, in S3; surgery on such knots at the surface slope yields a lens space. Later Dean described a similar class of knots that are primitive/Seifert with respect to F; surgery on these knots at the surface slope yields a Seifert fibered space. Here we construct a two-parameter family of knots that have distinct primitive/Seifert embeddings in F with the same surface slope, as well as a family of torus knots that have a primitive/primitive representative and a primitive/Seifert representative with the same surface slope.

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Additional cases of positive twisted torus knots

Journal of Knot Theory and Its Ramifications ◽

10.1142/s021821651750078x ◽

2017 ◽

Vol 26 (12) ◽

pp. 1750078

Author(s):

Evan Amoranto ◽

Brandy Doleshal ◽

Matt Rathbun

Keyword(s):

Sufficient Conditions ◽

Parameter Family ◽

Surface Slope ◽

Torus Knots ◽

Torus Knot ◽

Genus 2 ◽

Heegaard Surface

A twisted torus knot is a knot obtained from a torus knot by twisting adjacent strands by full twists. The twisted torus knots lie in [Formula: see text], the genus 2 Heegaard surface for [Formula: see text]. Primitive/primitive and primitive/Seifert knots lie in [Formula: see text] in a particular way. Dean gives sufficient conditions for the parameters of the twisted torus knots to ensure they are primitive/primitive or primitive/Seifert. Using Dean’s conditions, Doleshal shows that there are infinitely many twisted torus knots that are fibered and that there are twisted torus knots with distinct primitive/Seifert representatives with the same slope in [Formula: see text]. In this paper, we extend Doleshal’s results to show there is a four parameter family of positive twisted torus knots. Additionally, we provide new examples of twisted torus knots with distinct representatives with the same surface slope in [Formula: see text].

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FIBERED AND PRIMITIVE/SEIFERT TWISTED TORUS KNOTS

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216512501416 ◽

2013 ◽

Vol 22 (01) ◽

pp. 1250141 ◽

Cited By ~ 2

Author(s):

BRANDY GUNTEL DOLESHAL

Keyword(s):

Torus Knots ◽

Genus 2 ◽

Heegaard Surface

The twisted torus knots lie on the standard genus 2 Heegaard surface for S3, as do the primitive/primitive and primitive/Seifert knots. It is known that primitive/primitive knots are fibered, and that not all primitive/Seifert knots are fibered. Since there is a wealth of primitive/Seifert knots that are twisted torus knots, we consider the twisted torus knots to partially answer the question of which primitive/Seifert knots are fibered. A braid computation shows that a particular family of twisted torus knots is fibered, and that computation is then used to generalize the results of a previous paper by the author.

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The holographic conformal manifold of 3d $$ \mathcal{N} $$ = 2 S-fold SCFTs

Journal of High Energy Physics ◽

10.1007/jhep07(2021)221 ◽

2021 ◽

Vol 2021 (7) ◽

Author(s):

Nikolay Bobev ◽

Friðrik Freyr Gautason ◽

Jesse van Muiden

Keyword(s):

String Theory ◽

Three Dimensional ◽

Parameter Family ◽

Global Symmetry ◽

Maximal Supergravity ◽

All Solutions ◽

Operator Spectrum ◽

Type Iib String Theory ◽

Two Parameter

Abstract We employ a non-compact gauging of four-dimensional maximal supergravity to construct a two-parameter family of AdS4 J-fold solutions preserving $$ \mathcal{N} $$ N = 2 supersymmetry. All solutions preserve $$ \mathfrak{u} $$ u (1) × $$ \mathfrak{u} $$ u (1) global symmetry and in special limits we recover the previously known $$ \mathfrak{su} $$ su (2) × $$ \mathfrak{u} $$ u (1) invariant $$ \mathcal{N} $$ N = 2 and $$ \mathfrak{su} $$ su (2) × $$ \mathfrak{su} $$ su (2) invariant $$ \mathcal{N} $$ N = 4 J-fold solutions. This family of AdS4 backgrounds can be uplifted to type IIB string theory and is holographically dual to the conformal manifold of a class of three-dimensional S-fold SCFTs obtained from the $$ \mathcal{N} $$ N = 4 T [U(N)] theory of Gaiotto-Witten. We find the spectrum of supergravity excitations of the AdS4 solutions and use it to study how the operator spectrum of the three-dimensional SCFT depends on the exactly marginal couplings.

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