Kauffman International Symposium on Sustainable Mathematics Applications - Topics

SYMPOSIA

Kauffman International Symposium (8th Intl. Symp. on Sustainable Mathematics Applications)

Click here to see the Chair special introduction for this symposium

SYMPOSIUM and ROUND TABLE TOPICS

The symposium will be covering but is not limited to:

Topological foundations of physical and mathematical knowledge.

Topology.
Graph Theory.
Knot Theory.
Knot theory of embedded graphs.
Rope Magic.
Quantum Information Theory.
Foundations of mathematics and physics.
Epistemological foundations of knowledge.
Elements of form, distinction, discrimination.
Low dimensional topology and knot theory.
Relationships topology in three and four dimensions with mathematical physics and natural science.

Higher dimensional knotting and exotic structures on higher dimensional manifolds.
Influence of invariants of knots in three space on exotic structures on high dimensional manifolds.
Generalizations of branched covering constructions.
Chirality and irregular branched coverings of knots in the three sphere.
Topology of Brieskorn varieties and links of algebraic singularities.
Constructions for non-standard differentiable structures on spheres and manifolds.
Links of exotic differentiable structures with physics.
Brieskorn manfolds in the light of their use by the physicists Pham and Tulio Regge.
Knot products, branched fibrations, exotic structures.
Alexander polynomial and Jones polynomial (and Khovanov homology) for knots in three space in relation to Casson handles and exotic structures on four-manifolds.
Band passing and Arf invariant of knots in three dimensions.
Knot cobordism and concordance.
Signatures of knots and branched coverings.
Casson-Gordon invariants.
Quandles, finite and infinite.

Knot logic.
Knots and lambda calculus.
Indicative shift, lambda calculus and Goedelian self-reference.
Recursive forms.
Eigenform.
Recursive distinguishing.
Mathematics of autopoiesis and eigenform.

State summation models for the Alexander - Conway polynomial.
Formal Knot Theory state summation for Alexander-Conway polynomial.
Use of Formal Knot Theory (FKT) states in Heegaard-Floer homology for knots and links.
Mock Alexander polynomials.
Kauffman bracket polynomial state model for the Jones polynomial.
Kauffman bracket as a partition function for the construction of knot invariants.
Relationship of the Kauffman bracket with the Potts model in statistical mechanics.
Relationship of Kauffman bracket with coloring problems for graphs and with the four-color problem.
Penrose coloring evaluations for coloring or planar trivalent graphs.
Penrose-Kauffman polynomials for generalized coloring of arbitrary trivalent graphs.
Vector cross product (SO(3) Lie algebra) reformulation of the four color theorem.
Quantum invariants of knots and links.
Braided tensor categories.
Kauffman two-variable polynomial.
Homflypt polynomial.
Hopf algebras and quantum invariants of knots and links.
Drinfeld double construction.
Invariants of three manifolds via integrals on a finite dimensional Hopf algebra.
Kuperberg invariants of three manifolds.
Yang-Baxter equation.
Tensor networks, Penrose abstract tensors and quantum invariants.

Proofs of the Tait conjectures for the topological invariance of number of crossings for reduced alternating link projections.
Generalized Tait conjectures.

Virtual knot theory and virtual knots with unit Jones polynomial.
Welded knot theory.
Flat virtual knot theory.
Free virtual knot theory.
Multi-virtual knot theory.
Bracket polynomial, arrow polynomial, index polynomials and other invariants for virtual knot theory.
Generalized Penrose-Kauffman polynomials for multi-virtual knot theory.
Generalized quandles for multi-virtual knot theory.
State structure of the Kauffman bracket
Use of bracket state structure in Khovanov Homology.
Khovanov homology for virtual knots and links.
Rasmussen invariant for virtual knots and links.
Applications of virtual Khovanov homology for knots and links in three dimensional projective space.
Determination of the 4-ball genus of virtual knots.
Determination of the reconnection numbers for knotted vortices using Khovanov Homology and the Rasmussen invariant.

Knot diagrams and their checkerboard graphs.
Reidemeister moves translated to series-parallel, loop and pendant moves on checkerboard graphs.
Translation of Kirchoff electrical theory from graphs to knots and links.
Electrical conductivity invariants of knots and links.

Classification of rational tangles and rational knots.
Hard unknots and collapsing tangles.
Dynamics of knots under self-repulsion and examples of hard unknots via tangles.
DNA recombination modeled via tangle theory.

Virtual knot theory and checkerboard graphs.
Spinning constructions for 1-knots and 2-knots.
Fiberwise equivalence of welded knots.
Spinning constructions and generalizations of Zeeman’s fibration theorem using branched fibrations.
Higher dimensional band passing.

Knotoids in the sphere and in the plane.
Knotoids in surfaces.
Open knotted long chain molecules and relationships with protein folding.
Chirality of knotoids.
Formal Knot Theory state summation as generalized to Mock Alexander Polynomials for knotoids, linkoids and knots in thickened surfaces.

Temperley-Lieb Recoupling Theory and its applications to invariants of knots, links, three and four manifolds, and to topological quantum computing via non-abelian anyons.
The Fibonacci model (Kitaev) for topological quantum computing.
Meanders and projectors in Temperley Lieb algebra.

Representations of the Artin braid group related to the structure of Majorana Fermions and Clifford Algebras.
Braiding of Majorana Fermions in relation to the Dirac equation.

Nilpotent solutions of the Dirac equation in the sense of Peter Rowlands, applied to Majorana Fermions.

Reconnection and chirality in knotted vortices and for knots in liquid crystals.
Knotting, Chirality and Sustainability in Meta-Materials and Meta-Systems.

Diagrammatic systems for form and logic.
Structure of replication in logic and biology and topology.
Meanders and projectors in Temperley Lieb algebra as models of replication.
Recursive distinguishing models of replication.
Formal structure of recursion and self-reference.
Fractals.
Quaternionic fractals.
Relation of Sign and Space in the sense of Charles Sanders Peirce and George Spencer-Brown.
Square roots and higher roots of negation from the point of view of Laws of Form.
Modal logics and Heyting algebras in terms of Laws of Form.
Quaternions and Cayley Dickson Constructions in terms of Laws of Form.

The Combinatorial Hierarchy of F. Parker-Rhodes, Clive Kilmister, Ted Bastin and Pierre Noyes.
Theory of Indistinguishables.
Foundations of discrete physics.
Non-Commutative worlds and discrete physics.
Reformulation of discrete calculus in terms of commutators.
Reformulation of classical and quantum mechanics in non-commutative worlds.
Non-commutative geometry.
Quantum Knots and Mosaic Knots.
Quantum Lattice knots in three dimensions.
Teleportation topology.
Entanglement, topology, Heyting algebras and ER=EPR.
Entanglement and quantum information.
Q-deformed spin networks and anionic topological quantum computing.
Topological computing with Majorana Fermions.
Distinctions and K-Calculus in special relativity.
Preons, braid topology and representations of elementary particles.