scholarly journals Cosmetic crossing conjecture for genus one knots with non-trivial Alexander polynomial

2021 ◽  
pp. 1
Author(s):  
Tetsuya Ito
Keyword(s):  
Alexander Polynomial ◽  
Genus One

scholarly journals TWISTED ALEXANDER POLYNOMIALS OF 2-BRIDGE KNOTS

2013 ◽  
Vol 22 (01) ◽  
pp. 1250138 ◽  
Author(s):  
JIM HOSTE ◽  
PATRICK D. SHANAHAN
Keyword(s):  
Alexander Polynomial ◽  
Torus Knots ◽  
Alexander Polynomials ◽  
Twisted Alexander Polynomial ◽  
Genus One

We investigate the twisted Alexander polynomial of a 2-bridge knot associated to a Fox coloring. For several families of 2-bridge knots, including but not limited to, torus knots and genus-one knots, we derive formulae for these twisted Alexander polynomials. We use these formulae to confirm a conjecture of Hirasawa and Murasugi for these knots.

scholarly journals Adjoint twisted Alexander polynomials of genus one two-bridge knots

2016 ◽  
Vol 25 (11) ◽  
pp. 1650065 ◽  
Author(s):  
Anh T. Tran
Keyword(s):  
Alexander Polynomial ◽  
Explicit Formulas ◽  
Reidemeister Torsion ◽  
Alexander Polynomials ◽  
Twisted Alexander Polynomial ◽  
Genus One

We give explicit formulas for the adjoint twisted Alexander polynomial and nonabelian Reidemeister torsion of genus one two-bridge knots.

scholarly journals THE ALEXANDER POLYNOMIAL OF (1,1)-KNOTS

2006 ◽  
Vol 15 (09) ◽  
pp. 1119-1129 ◽  
Author(s):  
A. CATTABRIGA
Keyword(s):  
Fundamental Group ◽  
Alexander Polynomial ◽  
Lens Spaces ◽  
Cyclic Covering ◽  
Genus One

In this paper we investigate the Alexander polynomial of (1,1)-knots, which are knots lying in a 3-manifold with genus one at most, admitting a particular decomposition. More precisely, we study the connections between the Alexander polynomial and a polynomial associated to a cyclic presentation of the fundamental group of an n-fold strongly-cyclic covering branched over the knot K, which we call the n-cyclic polynomial of K. In this way, we generalize to all (1,1)-knots, with the only exception of those lying in S2×S1, a result obtained by Minkus for 2-bridge knots and extended by the author and M. Mulazzani to the case of (1,1)-knots in S3. As corollaries some properties of the Alexander polynomial of knots in S3 are extended to the case of (1,1)-knots in lens spaces.

scholarly journals Minimal genus and fibering of canonical surfaces via disk decomposition

2014 ◽  
Vol 17 (1) ◽  
pp. 77-108 ◽  
Author(s):  
A. Stoimenow
Keyword(s):  
Alexander Polynomial ◽  
Minimal Genus ◽  
Genus 2 ◽  
Knot Diagrams ◽  
Genus One

AbstractThis paper contains some applications of the description of knot diagrams by genus, and Gabai’s methods of disk decomposition. We show that there exists no genus one knot of canonical genus 2, and that canonical genus 2 fiber surfaces realize almost every Alexander polynomial only finitely many times (partially confirming a conjecture of Neuwirth).

Adjoint twisted Alexander polynomial of twisted Whitehead links

2018 ◽  
Vol 27 (04) ◽  
pp. 1850026
Author(s):  
Hoang-An Nguyen ◽  
Anh T. Tran
Keyword(s):  
Knot Theory ◽  
Three Dimensional ◽  
Adjoint Action ◽  
Alexander Polynomial ◽  
Reidemeister Torsion ◽  
Alexander Polynomials ◽  
Whitehead Link ◽  
Twisted Alexander Polynomial ◽  
Genus One

The adjoint twisted Alexander polynomial has been computed for twist knots [A. Tran, Twisted Alexander polynomials with the adjoint action for some classes of knots, J. Knot Theory Ramifications 23(10) (2014) 1450051], genus one two-bridge knots [A. Tran, Adjoint twisted Alexander polynomials of genus one two-bridge knots, J. Knot Theory Ramifications 25(10) (2016) 1650065] and the Whitehead link [J. Dubois and Y. Yamaguchi, Twisted Alexander invariant and nonabelian Reidemeister torsion for hyperbolic three dimensional manifolds with cusps, Preprint (2009), arXiv:0906.1500 ]. In this paper, we compute the adjoint twisted Alexander polynomial and nonabelian Reidemeister torsion of twisted Whitehead links.

scholarly journals Sequence of a Coxiella endosymbiont of the tick Amblyomma nuttalli suggests a pattern of convergent genome reduction in the Coxiella genus

2020 ◽  
Author(s):  
Tiago Nardi ◽  
Emanuela Olivieri ◽  
Edward Kariuki ◽  
Davide Sassera ◽  
Michele Castelli
Keyword(s):  
Coxiella Burnetii ◽  
Phylogenetic Analyses ◽  
Genome Reduction ◽  
Metabolic Profiles ◽  
The Novel ◽  
Bacterial Symbionts ◽  
Human Pathogen ◽  
Host Genus ◽  
Supportive Role ◽  
Genus One

Abstract Ticks require bacterial symbionts for the provision of necessary compounds that are absent in their hematophagous diet. Such symbionts are frequently vertically transmitted and, most commonly, belong to the Coxiella genus, which also includes the human pathogen Coxiella burnetii. This genus can be divided in four main clades, presenting partial but incomplete co-cladogenesis with the tick hosts. Here we report the genome sequence of a novel Coxiella, endosymbiont of the African tick Amblyomma nuttalli, and the ensuing comparative analyses. Its size (~1 Mb) is intermediate between symbionts of Rhipicephalus species and other Amblyomma species. Phylogenetic analyses show that the novel sequence is the first genome of the B clade, the only one for which no genomes were previously available. Accordingly, it allows to draw an enhanced scenario of the evolution of the genus, one of parallel genome reduction of different endosymbiont lineages, which are now at different stages of reduction from a more versatile ancestor. Gene content comparison allows to infer that the ancestor could be reminiscent of Coxiella burnetii. Interestingly, the convergent loss of mismatch repair could have been a major driver of such reductive evolution. Predicted metabolic profiles are rather homogenous among Coxiella endosymbionts, in particular vitamin biosynthesis, consistently with a host-supportive role. Concurrently, similarities among Coxiella endosymbionts according to host genus and despite phylogenetic unrelatedness hint at possible host-dependent effects.

scholarly journals Mazur’s rational torsion result for pointless genus one curves: examples

2021 ◽  
Vol 7 (1) ◽  
Author(s):  
Arjan Dwarshuis ◽  
Majken Roelfszema ◽  
Jaap Top
Keyword(s):  
Algebraic Geometry ◽  
Elliptic Curves ◽  
Rational Point ◽  
Torsion Points ◽  
Arbitrary Genus ◽  
Mazur’S Theorem ◽  
Genus One

AbstractThis note reformulates Mazur’s result on the possible orders of rational torsion points on elliptic curves over $$\mathbb {Q}$$ Q in a way that makes sense for arbitrary genus one curves, regardless whether or not the curve contains a rational point. The main result is that explicit examples are provided of ‘pointless’ genus one curves over $$\mathbb {Q}$$ Q corresponding to the torsion orders 7, 8, 9, 10, 12 (and hence, all possibilities) occurring in Mazur’s theorem. In fact three distinct methods are proposed for constructing such examples, each involving different in our opinion quite nice ideas from the arithmetic of elliptic curves or from algebraic geometry.

scholarly journals Two-loop superstring five-point amplitudes. Part II. Low energy expansion and S-duality

2021 ◽  
Vol 2021 (2) ◽  
Author(s):  
Eric D’Hoker ◽  
Carlos R. Mafra ◽  
Boris Pioline ◽  
Oliver Schlotterer
Keyword(s):  
Moduli Space ◽  
Effective Interaction ◽  
Type Ii ◽  
Leading Order ◽  
Effective Interactions ◽  
Heterotic String Theory ◽  
Genus Two ◽  
Genus One ◽  
Graph Function ◽  
Point Amplitude

Abstract In an earlier paper, we constructed the genus-two amplitudes for five external massless states in Type II and Heterotic string theory, and showed that the α′ expansion of the Type II amplitude reproduces the corresponding supergravity amplitude to leading order. In this paper, we analyze the effective interactions induced by Type IIB superstrings beyond supergravity, both for U(1)R-preserving amplitudes such as for five gravitons, and for U(1)R-violating amplitudes such as for one dilaton and four gravitons. At each order in α′, the coefficients of the effective interactions are given by integrals over moduli space of genus-two modular graph functions, generalizing those already encountered for four external massless states. To leading and sub-leading orders, the coefficients of the effective interactions D2ℛ5 and D4ℛ5 are found to match those of D4ℛ4 and D6ℛ4, respectively, as required by non-linear supersymmetry. To the next order, a D6ℛ5 effective interaction arises, which is independent of the supersymmetric completion of D8ℛ4, and already arose at genus one. A novel identity on genus-two modular graph functions, which we prove, ensures that up to order D6ℛ5, the five-point amplitudes require only a single new modular graph function in addition to those needed for the four-point amplitude. We check that the supergravity limit of U(1)R-violating amplitudes is free of UV divergences to this order, consistently with the known structure of divergences in Type IIB supergravity. Our results give strong consistency tests on the full five-point amplitude, and pave the way for understanding S-duality beyond the BPS-protected sector.

scholarly journals There are genus one curves of every index over every infinite, finitely generated field

2019 ◽  
Vol 2019 (749) ◽  
pp. 65-86
Author(s):  
Pete L. Clark ◽  
Allan Lacy
Keyword(s):  
Elliptic Curve ◽  
Abelian Variety ◽  
Finitely Generated ◽  
Weil Theorem ◽  
Genus One

Abstract We show that a nontrivial abelian variety over a Hilbertian field in which the weak Mordell–Weil theorem holds admits infinitely many torsors with period any given n>1 that is not divisible by the characteristic. The corresponding statement with “period” replaced by “index” is plausible but open, and it seems much more challenging. We show that for every infinite, finitely generated field K, there is an elliptic curve E_{/K} which admits infinitely many torsors with index any given n>1 .

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