scholarly journals CONJUGATE GENERATORS OF KNOT AND LINK GROUPS

2010 ◽  
Vol 19 (07) ◽  
pp. 905-916
Author(s):  
JASON CALLAHAN
Keyword(s):  
Link Group ◽  
Whitehead Link ◽  
Trefoil Knot ◽  
Figure Eight Knot

This note establishes a stronger version of a conjecture of Reid and others in the arithmetic case: if two elements of equal trace (e.g. conjugate elements) generate an arithmetic two-bridge knot or link group, then the elements are parabolic (and hence peripheral). This includes the figure-eight knot and Whitehead link groups. Similarly, if two conjugate elements generate the trefoil knot group, then the elements are peripheral.

scholarly journals On SL(3, $${\varvec{\mathbb {C}}}$$ C )-representations of the Whitehead link group

2018 ◽  
Vol 202 (1) ◽  
pp. 81-101 ◽  
Author(s):  
Antonin Guilloux ◽  
Pierre Will
Keyword(s):  
Link Group ◽  
Whitehead Link

Untangle problems, with knot theory

2020 ◽  
pp. 73-80
Author(s):  
Susan D'Agostino
Keyword(s):  
Knot Theory ◽  
Closed Loop ◽  
The Other ◽  
Mathematics Students ◽  
Crossing Numbers ◽  
Trefoil Knot ◽  
Figure Eight Knot

“Untangle problems, with knot theory” offers a basic introduction to the mathematical subfield of knot theory, including the classification of knots by crossing numbers. A mathematical knot is a closed loop that may or may not be tangled. Two knots are considered the same if one may be manipulated into the other using easy-to-understand techniques. Readers learn to identify knots by crossing numbers and encounter numerous hand-drawn sketches of knots, including the trivial knot, trefoil knot, figure-eight knot, and more. Mathematics students and enthusiasts are encouraged to employ knot theory methods for untangling problems in mathematics or life by asking whether they have encountered the problem before. At the chapter’s end, readers may check their understanding by working on a problem. A solution is provided.

ON 2-ADJACENCY OF CLASSICAL PRETZEL KNOTS

2013 ◽  
Vol 22 (11) ◽  
pp. 1350066 ◽  
Author(s):  
ZHI-XIONG TAO
Keyword(s):  
Jones Polynomial ◽  
Conway Polynomial ◽  
Pretzel Knots ◽  
Trefoil Knot ◽  
Figure Eight Knot

For classical (3-strand) pretzel knots (including even type and odd type), we study their 2-adjacency using Conway polynomial and Jones polynomial. We show that only the trefoil knot and the figure-eight knot in these knots are 2-adjacent.

scholarly journals Universal Quantum Computing and Three-Manifolds

2018 ◽  
Author(s):  
Michel Planat ◽  
Raymond Aschheim ◽  
Marcelo Amaral ◽  
Klee Irwin
Keyword(s):  
Quantum Computing ◽  
Modular Group ◽  
Bloch Sphere ◽  
Borromean Rings ◽  
Single Qubit ◽  
Whitehead Link ◽  
Trefoil Knot ◽  
Universal Quantum ◽  
Dehn Fillings ◽  
Knots And Links

A single qubit may be represented on the Bloch sphere or similarly on the $3$-sphere $S^3$. Our goal is to dress this correspondence by converting the language of universal quantum computing (uqc) to that of $3$-manifolds. A magic state and the Pauli group acting on it define a model of uqc as a POVM that one recognizes to be a $3$-manifold $M^3$. E. g., the $d$-dimensional POVMs defined from subgroups of finite index of the modular group $PSL(2,\mathbb{Z})$ correspond to $d$-fold $M^3$- coverings over the trefoil knot. In this paper, one also investigates quantum information on a few \lq universal' knots and links such as the figure-of-eight knot, the Whitehead link and Borromean rings, making use of the catalog of platonic manifolds available on SnapPy. Further connections between POVMs based uqc and $M^3$'s obtained from Dehn fillings are explored.

scholarly journals Group Geometrical Axioms for Magic States of Quantum Computing

Mathematics ◽  
2019 ◽  
Vol 7 (10) ◽  
pp. 948 ◽  
Author(s):  
Michel Planat ◽  
Raymond Aschheim ◽  
Marcelo M. Amaral ◽  
Klee Irwin
Keyword(s):  
Quantum Computing ◽  
Fundamental Group ◽  
Free Group ◽  
Normal Closure ◽  
Trefoil Knot ◽  
Universal Quantum ◽  
Nontrivial Subgroup ◽  
Low Dimensional ◽  
Figure Eight Knot ◽  
Stabilizer States

Let H be a nontrivial subgroup of index d of a free group G and N be the normal closure of H in G. The coset organization in a subgroup H of G provides a group P of permutation gates whose common eigenstates are either stabilizer states of the Pauli group or magic states for universal quantum computing. A subset of magic states consists of states associated to minimal informationally complete measurements, called MIC states. It is shown that, in most cases, the existence of a MIC state entails the two conditions (i) N = G and (ii) no geometry (a triple of cosets cannot produce equal pairwise stabilizer subgroups) or that these conditions are both not satisfied. Our claim is verified by defining the low dimensional MIC states from subgroups of the fundamental group G = π 1 ( M ) of some manifolds encountered in our recent papers, e.g., the 3-manifolds attached to the trefoil knot and the figure-eight knot, and the 4-manifolds defined by 0-surgery of them. Exceptions to the aforementioned rule are classified in terms of geometric contextuality (which occurs when cosets on a line of the geometry do not all mutually commute).

scholarly journals Universal Quantum Computing and Three-Manifolds

2018 ◽  
Author(s):  
Michel Planat ◽  
Raymond Aschheim ◽  
Marcelo Amaral ◽  
Klee Irwin
Keyword(s):  
Quantum Computing ◽  
Bloch Sphere ◽  
Borromean Rings ◽  
Single Qubit ◽  
Whitehead Link ◽  
Trefoil Knot ◽  
Universal Quantum ◽  
Dehn Fillings ◽  
Knots And Links ◽  
Positive Operator Valued Measure

A single qubit may be represented on the Bloch sphere or similarly on the 3-sphere S3. Our goal is to dress this correspondence by converting the language of universal quantum computing (UQC) to that of 3-manifolds. A magic state and the Pauli group acting on it define a model of UQC as a positive operator-valued measure (POVM) that one recognizes to be a 3-manifold M3. More precisely, the d-dimensional POVMs defined from subgroups of finite index of the modular group PSL(2, Z) correspond to d-fold M3- coverings over the trefoil knot. In this paper, one also investigates quantum information on a few ‘universal’ knots and links such as the figure-of-eight knot, the Whitehead link and Borromean rings, making use of the catalog of platonic manifolds available on the software SnapPy. Further connections between POVMs based UQC and M3’s obtained from Dehn fillings are explored.

Geometric conemanifold structures on 𝕋p/q, the result of p/q surgery in the left-handed trefoil knot 𝕋

2015 ◽  
Vol 24 (12) ◽  
pp. 1550057 ◽  
Author(s):  
María Teresa Lozano ◽  
José María Montesinos-Amilibia
Keyword(s):  
Singular Set ◽  
Geometric Structures ◽  
Quaternion Algebras ◽  
Interesting Phenomenon ◽  
The Core ◽  
Left Handed ◽  
Trefoil Knot ◽  
Figure Eight Knot

As an example of the transitions between some of the eight geometries of Thurston, investigated in [M. T. Lozano and J. M. Montesinos-Amilibia, On the degeneration of some 3-manifold geometries via unit groups of quaternion algebras RACSAM 109 (2015) 669–715], we study the geometries supported by the conemanifolds obtained by surgery on the trefoil knot with singular set the core of the surgery. The geometric structures are explicitly constructed. The most interesting phenomenon is the transition from SL(2, ℝ)-geometry to S3-geometry through Nil-geometry. A plot of the different geometries is given, in the spirit of the analogous plot of Thurston for the geometries supported by surgeries on the figure-eight knot.

scholarly journals Universal Quantum Computing and Three-Manifolds

Symmetry ◽  
2018 ◽  
Vol 10 (12) ◽  
pp. 773 ◽  
Author(s):  
Michel Planat ◽  
Raymond Aschheim ◽  
Marcelo Amaral ◽  
Klee Irwin
Keyword(s):  
Quantum Computing ◽  
Bloch Sphere ◽  
Borromean Rings ◽  
Single Qubit ◽  
Whitehead Link ◽  
Trefoil Knot ◽  
Universal Quantum ◽  
Dehn Fillings ◽  
Knots And Links ◽  
Positive Operator Valued Measure

A single qubit may be represented on the Bloch sphere or similarly on the 3-sphere S 3 . Our goal is to dress this correspondence by converting the language of universal quantum computing (UQC) to that of 3-manifolds. A magic state and the Pauli group acting on it define a model of UQC as a positive operator-valued measure (POVM) that one recognizes to be a 3-manifold M 3 . More precisely, the d-dimensional POVMs defined from subgroups of finite index of the modular group P S L ( 2 , Z ) correspond to d-fold M 3 - coverings over the trefoil knot. In this paper, we also investigate quantum information on a few ‘universal’ knots and links such as the figure-of-eight knot, the Whitehead link and Borromean rings, making use of the catalog of platonic manifolds available on the software SnapPy. Further connections between POVMs based UQC and M 3 ’s obtained from Dehn fillings are explored.

2-Adjacency between knots

2015 ◽  
Vol 24 (11) ◽  
pp. 1550054 ◽  
Author(s):  
Zhi-Xiong Tao
Keyword(s):  
Symmetric Relation ◽  
Jones Polynomials ◽  
Trefoil Knot ◽  
Figure Eight Knot ◽  
Homfly Polynomials

We study the properties of a knot K to be 2-adjacent to another knot W by analyzing their Conway polynomials, Jones polynomials and Homfly polynomials and give some very useful conditions. We discuss whether each pair of knots can be 2-adjacent to each other, i.e. whether 2-adjacency is a symmetric relation. We discuss also whether the trivial knot, the trefoil knot and the figure-eight knot can be 2-adjacent to any knot in Rolfsen's table and the opposite cases, except for 934 it is not decided whether it is 2-adjacent to 41. Finally, we give some examples to answer I. Torisu's problem partly and etc.

2-Adjacency between some special knots and pretzel knot

2019 ◽  
Vol 28 (10) ◽  
pp. 1950066
Author(s):  
Zhi-Xiong Tao
Keyword(s):  
Number Theory ◽  
Elementary Number Theory ◽  
Elementary Number ◽  
Early Results ◽  
Pretzel Knots ◽  
Trefoil Knot ◽  
Figure Eight Knot

We study 2-adjacency between a classical (3-strand) pretzel knot and the trefoil knot or the figure-eight knot by using the early results about classical pretzel knots and their polynomials and elementary number theory. We show that except for the trefoil knot or the figure-eight knot, a nontrivial classical pretzel knot is not 2-adjacent to either of them, and vice versa.

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