scholarly journals Quantum Topology

10.4171/qt ◽  
2020 ◽  
Keyword(s):  
Quantum Topology

scholarly journals THE NON-COMMUTATIVE A-POLYNOMIAL OF TWIST KNOTS

2010 ◽  
Vol 19 (12) ◽  
pp. 1571-1595 ◽  
Author(s):  
STAVROS GAROUFALIDIS ◽  
XINYU SUN
Keyword(s):  
Difference Equation ◽  
Recursion Relation ◽  
Jones Polynomial ◽  
Minimal Order ◽  
Quantum Topology ◽  
And Topology ◽  
Jones Polynomials ◽  
Creative Telescoping ◽  
Colored Jones Polynomials ◽  
Telescoping Method

The purpose of the paper is two-fold: to introduce a multivariable creative telescoping method, and to apply it in a problem of Quantum Topology: namely the computation of the non-commutative A-polynomial of twist knots. Our multivariable creative telescoping method allows us to compute linear recursions for sums of the form [Formula: see text] given a recursion relation for [Formula: see text] and the hypergeometric kernel c(n, k). As an application of our method, we explicitly compute the non-commutative A-polynomial for twist knots with -15 and 15 crossings. The non-commutative A-polynomial of a knot encodes the monic, linear, minimal order q-difference equation satisfied by the sequence of colored Jones polynomials of the knot. Its specialization to q = 1 is conjectured to be the better-known A-polynomial of a knot, which encodes important information about the geometry and topology of the knot complement. Unlike the case of the Jones polynomial, which is easily computable for knots with 50 crossings, the A-polynomial is harder to compute and already unknown for some knots with 12 crossings.

Quantum Topology

1980 ◽  
pp. 57-61
Author(s):  
R. F. W. Bader
Keyword(s):  
Quantum Topology

High-dimensional quantum state manipulation and tracking

2020 ◽  
pp. 2050045
Author(s):  
Mevludin Licina
Keyword(s):  
Probability Distribution ◽  
Quantum State ◽  
Critical Points ◽  
Theoretical Framework ◽  
Quantum Systems ◽  
Quantum States ◽  
High Dimensional ◽  
Quantum Topology ◽  
Mathematical Representations ◽  
Framework Approach

Dynamical high-dimensional quantum states can be tracked and manipulated in many cases. Using a new theoretical framework approach of manipulating quantum systems, we will show how one can manipulate and introduce parameters that allow tracking and descriptive insight in the dynamics of states. Using quantum topology and other novel mathematical representations, we will show how quantum states behave in critical points when the shift of probability distribution introduces changes.

scholarly journals Planar diagrams for local invariants of graphs in surfaces

2020 ◽  
Vol 29 (01) ◽  
pp. 1950093
Author(s):  
Calvin McPhail-Snyder ◽  
Kyle A. Miller
Keyword(s):  
Sufficient Condition ◽  
Cubic Graphs ◽  
Quantum Topology ◽  
Ribbon Graphs ◽  
Spatial Graphs ◽  
Diagram Category ◽  
Local Invariants ◽  
Flow Polynomial ◽  
Local Topology ◽  
Planar Diagram

In order to apply quantum topology methods to nonplanar graphs, we define a planar diagram category that describes the local topology of embeddings of graphs into surfaces. These virtual graphs are a categorical interpretation of ribbon graphs. We describe an extension of the flow polynomial to virtual graphs, the [Formula: see text]-polynomial, and formulate the [Formula: see text] Penrose polynomial for non-cubic graphs, giving contraction–deletion relations. The [Formula: see text]-polynomial is used to define an extension of the Yamada polynomial to virtual spatial graphs, and with it we obtain a sufficient condition for non-classicality of virtual spatial graphs. We conjecture the existence of local relations for the [Formula: see text]-polynomial at squares of integers.

Hybrid QTAIM and electrostatic potential-based quantum topology phase diagrams for water clusters

2015 ◽  
Vol 17 (23) ◽  
pp. 15258-15273 ◽  
Author(s):  
Anmol Kumar ◽  
Shridhar R. Gadre ◽  
Xiao Chenxia ◽  
Xu Tianlv ◽  
Steven Robert Kirk ◽  
...  
Keyword(s):  
Charge Density ◽  
Phase Diagrams ◽  
Electrostatic Potential ◽  
Molecular Electrostatic Potential ◽  
Water Clusters ◽  
Scalar Fields ◽  
Electronic Charge ◽  
Electronic Charge Density ◽  
Quantum Topology

The topological diversity of sets of isomers of water clusters (W = H2O)n, 7 ≤ n ≤ 10, is analyzed employing the scalar fields of total electronic charge density ρ(r) and the molecular electrostatic potential (MESP).

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