912 research outputs found
On possible existence of HOMFLY polynomials for virtual knots
Virtual knots are associated with knot diagrams, which are not obligatory
planar. The recently suggested generalization from N=2 to arbitrary N of the
Kauffman-Khovanov calculus of cycles in resolved diagrams can be
straightforwardly applied to non-planar case. In simple examples we demonstrate
that this construction preserves topological invariance -- thus implying the
existence of HOMFLY extension of cabled Jones polynomials for virtual knots and
links.Comment: 12 page
Robust spatial memory maps encoded in networks with transient connections
The spiking activity of principal cells in mammalian hippocampus encodes an
internalized neuronal representation of the ambient space---a cognitive map.
Once learned, such a map enables the animal to navigate a given environment for
a long period. However, the neuronal substrate that produces this map remains
transient: the synaptic connections in the hippocampus and in the downstream
neuronal networks never cease to form and to deteriorate at a rapid rate. How
can the brain maintain a robust, reliable representation of space using a
network that constantly changes its architecture? Here, we demonstrate, using
novel Algebraic Topology techniques, that cognitive map's stability is a
generic, emergent phenomenon. The model allows evaluating the effect produced
by specific physiological parameters, e.g., the distribution of connections'
decay times, on the properties of the cognitive map as a whole. It also points
out that spatial memory deterioration caused by weakening or excessive loss of
the synaptic connections may be compensated by simulating the neuronal
activity. Lastly, the model explicates functional importance of the
complementary learning systems for processing spatial information at different
levels of spatiotemporal granularity, by establishing three complementary
timescales at which spatial information unfolds. Thus, the model provides a
principal insight into how can the brain develop a reliable representation of
the world, learn and retain memories despite complex plasticity of the
underlying networks and allows studying how instabilities and memory
deterioration mechanisms may affect learning process.Comment: 24 pages, 10 figures, 4 supplementary figure
Multistrand Eigenvalue conjecture and Racah symmetries
Racah matrices of quantum algebras are of great interest at present time.
These matrices have a relation with -matrices, which are much
simpler than the Racah matrices themselves. This relation is known as the
eigenvalue conjecture. In this paper we study symmetries of Racah matrices
which follow from the eigenvalue conjecture for multistrand braids.Comment: 6 pages, 4 figure
Evolution method and HOMFLY polynomials for virtual knots
Following the suggestion of arXiv:1407.6319 to lift the knot polynomials for
virtual knots and links from Jones to HOMFLY, we apply the evolution method to
calculate them for an infinite series of twist-like virtual knots and
antiparallel 2-strand links. Within this family one can check topological
invariance and understand how differential hierarchy is modified in virtual
case. This opens a way towards a definition of colored (not only cabled) knot
polynomials, though problems still persist beyond the first symmetric
representation.Comment: 28 page
Explicit examples of DIM constraints for network matrix models
Dotsenko-Fateev and Chern-Simons matrix models, which describe Nekrasov
functions for SYM theories in different dimensions, are all incorporated into
network matrix models with the hidden Ding-Iohara-Miki (DIM) symmetry. This
lifting is especially simple for what we call balanced networks. Then, the Ward
identities (known under the names of Virasoro/W-constraints or loop equations
or regularity condition for qq-characters) are also promoted to the DIM level,
where they all become corollaries of a single identity.Comment: 46 page
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