500 research outputs found
Knot Theory: from Fox 3-colorings of links to Yang-Baxter homology and Khovanov homology
This paper is an extended account of my "Introductory Plenary talk at Knots
in Hellas 2016" conference We start from the short introduction to Knot Theory
from the historical perspective, starting from Heraclas text (the first century
AD), mentioning R.Llull (1232-1315), A.Kircher (1602-1680), Leibniz idea of
Geometria Situs (1679), and J.B.Listing (student of Gauss) work of 1847. We
spend some space on Ralph H. Fox (1913-1973) elementary introduction to diagram
colorings (1956). In the second section we describe how Fox work was
generalized to distributive colorings (racks and quandles) and eventually in
the work of Jones and Turaev to link invariants via Yang-Baxter operators, here
the importance of statistical mechanics to topology will be mentioned. Finally
we describe recent developments which started with Mikhail Khovanov work on
categorification of the Jones polynomial. By analogy to Khovanov homology we
build homology of distributive structures (including homology of Fox colorings)
and generalize it to homology of Yang-Baxter operators. We speculate, with
supporting evidence, on co-cycle invariants of knots coming from Yang-Baxter
homology. Here the work of Fenn-Rourke-Sanderson (geometric realization of
pre-cubic sets of link diagrams) and Carter-Kamada-Saito (co-cycle invariants
of links) will be discussed and expanded.
Dedicated to Lou Kauffman for his 70th birthday.Comment: 35 pages, 31 figures, for Knots in Hellas II Proceedings, Springer,
part of the series Proceedings in Mathematics & Statistics (PROMS
Knot invariants in lens spaces
In this survey we summarize results regarding the Kauffman bracket, HOMFLYPT,
Kauffman 2-variable and Dubrovnik skein modules, and the Alexander polynomial
of links in lens spaces, which we represent as mixed link diagrams. These
invariants generalize the corresponding knot polynomials in the classical case.
We compare the invariants by means of the ability to distinguish between some
difficult cases of knots with certain symmetries
HOMFLY and superpolynomials for figure eight knot in all symmetric and antisymmetric representations
Explicit answer is given for the HOMFLY polynomial of the figure eight knot
in arbitrary symmetric representation R=[p]. It generalizes the old
answers for p=1 and 2 and the recently derived results for p=3,4, which are
fully consistent with the Ooguri-Vafa conjecture. The answer can be considered
as a quantization of the \sigma_R = \sigma_{[1]}^{|R|} identity for the
"special" polynomials (they define the leading asymptotics of HOMFLY at q=1),
and arises in a form, convenient for comparison with the representation of the
Jones polynomials as sums of dilogarithm ratios. In particular, we construct a
difference equation ("non-commutative A-polynomial") in the representation
variable p. Simple symmetry transformation provides also a formula for
arbitrary antisymmetric (fundamental) representation R=[1^p], which also passes
some obvious checks. Also straightforward is a deformation from HOMFLY to
superpolynomials. Further generalizations seem possible to arbitrary Young
diagrams R, but these expressions are harder to test because of the lack of
alternative results, even partial.Comment: 14 page
Boolean Dynamics with Random Couplings
This paper reviews a class of generic dissipative dynamical systems called
N-K models. In these models, the dynamics of N elements, defined as Boolean
variables, develop step by step, clocked by a discrete time variable. Each of
the N Boolean elements at a given time is given a value which depends upon K
elements in the previous time step.
We review the work of many authors on the behavior of the models, looking
particularly at the structure and lengths of their cycles, the sizes of their
basins of attraction, and the flow of information through the systems. In the
limit of infinite N, there is a phase transition between a chaotic and an
ordered phase, with a critical phase in between.
We argue that the behavior of this system depends significantly on the
topology of the network connections. If the elements are placed upon a lattice
with dimension d, the system shows correlations related to the standard
percolation or directed percolation phase transition on such a lattice. On the
other hand, a very different behavior is seen in the Kauffman net in which all
spins are equally likely to be coupled to a given spin. In this situation,
coupling loops are mostly suppressed, and the behavior of the system is much
more like that of a mean field theory.
We also describe possible applications of the models to, for example, genetic
networks, cell differentiation, evolution, democracy in social systems and
neural networks.Comment: 69 pages, 16 figures, Submitted to Springer Applied Mathematical
Sciences Serie
Homeostatic dysregulation proceeds in parallel in multiple physiological systems
Abstract: An increasing number of aging researchers believes that multisystem physiological dysregulation may be a key biological mechanism of aging, but evidence of this has been sparse. Here, we used biomarker data on nearly 33 000 individuals from four large datasets to test for the presence of multi-system dysregulation. We grouped 37 biomarkers into six a priori groupings representing physiological systems (lipids, immune, oxygen transport, liver function, vitamins, and electrolytes), then calculated dysregulation scores for each system in each individual using statistical distance. Correlations among dysregulation levels across systems were generally weak but significant. Comparison of these results to dysregulation in arbitrary ‘systems’ generated by random grouping of biomarkers showed that a priori knowledge effectively distinguished the true systems in which dysregulation proceeds most independently. In other words, correlations among dysregulation levels were higher using arbitrary systems, indicating that only a priori systems identified distinct dysregulation processes. Additionally, dysregulation of most systems increased with age and significantly predicted multiple health outcomes including mortality, frailty, diabetes, heart disease, and number of chronic diseases. The six systems differed in how well their dysregulation scores predicted health outcomes and age. These findings present the first unequivocal demonstration of integrated multi-system physiological dysregulation during aging, demonstrating that physiological dysregulation proceeds neither as a single global process nor as a completely independent process in different systems, but rather as a set of system-specific processes likely linked through weak feedback effects. These processes – probably many more than the six measured here – are implicated in aging
Super-A-polynomials for Twist Knots
We conjecture formulae of the colored superpolynomials for a class of twist
knots where p denotes the number of full twists. The validity of the
formulae is checked by applying differentials and taking special limits. Using
the formulae, we compute both the classical and quantum super-A-polynomial for
the twist knots with small values of p. The results support the categorified
versions of the generalized volume conjecture and the quantum volume
conjecture. Furthermore, we obtain the evidence that the Q-deformed
A-polynomials can be identified with the augmentation polynomials of knot
contact homology in the case of the twist knots.Comment: 22+16 pages, 16 tables and 5 figures; with a Maple program by Xinyu
Sun and a Mathematica notebook in the ancillary files linked on the right; v2
change in appendix B, typos corrected and references added; v3 change in
section 3.3; v4 corrections in Ooguri-Vafa polynomials and quantum
super-A-polynomials for 7_2 and 8_1 are adde
Perfect k-Colored Matchings and (k+2)-Gonal Tilings
We derive a simple bijection between geometric plane perfect matchings on 2n points in convex position and triangulations on n+2 points in convex position. We then extend this bijection to monochromatic plane perfect matchings on periodically k-colored vertices and (k+2)-gonal tilings of convex point sets. These structures are related to a generalization of Temperley–Lieb algebras and our bijections provide explicit one-to-one relations between matchings and tilings. Moreover, for a given element of one class, the corresponding element of the other class can be computed in linear time
Recommended from our members
Change escalation processes and complex adaptive systems: from incremental reconfigurations to discontinuous restructuring
This study examines when “incremental” change is likely to trigger “discontinuous” change, using the lens of complex adaptive systems theory. Going beyond the simulations and case studies through which complex adaptive systems have been approached so far, we study the relationship between incremental organizational reconfigurations and discontinuous organizational restructurings using a large-scale database of U.S. Fortune 50 industrial corporations. We develop two types of escalation process in organizations: accumulation and perturbation. Under ordinary conditions, it is perturbation rather than the accumulation that is more likely to trigger subsequent discontinuous change. Consistent with complex adaptive systems theory, organizations are more sensitive to both accumulation and perturbation in conditions of heightened disequilibrium. Contrary to expectations, highly interconnected organizations are not more liable to discontinuous change. We conclude with implications for further research, especially the need to attend to the potential role of managerial design and coping when transferring complex adaptive systems theory from natural systems to organizational systems
The Genetics of Adaptation for Eight Microvirid Bacteriophages
Theories of adaptive molecular evolution have recently experienced significant expansion, and their predictions and assumptions have begun to be subjected to rigorous empirical testing. However, these theories focus largely on predicting the first event in adaptive evolution, the fixation of a single beneficial mutation. To address long-term adaptation it is necessary to include new assumptions, but empirical data are needed for guidance. To empirically characterize the general properties of adaptive walks, eight recently isolated relatives of the single-stranded DNA (ssDNA) bacteriophage φX174 (family Microviridae) were adapted to identical selective conditions. Three of the eight genotypes were adapted in replicate, for a total of 11 adaptive walks. We measured fitness improvement and identified the genetic changes underlying the observed adaptation. Nearly all phages were evolvable; nine of the 11 lineages showed a significant increase in fitness. However, fitness plateaued quickly, and adaptation was achieved through only three substitutions on average. Parallel evolution was rampant, both across replicates of the same genotype as well as across different genotypes, yet adaptation of replicates never proceeded through the exact same set of mutations. Despite this, final fitnesses did not vary significantly among replicates. Final fitnesses did vary significantly across genotypes but not across phylogenetic groupings of genotypes. A positive correlation was found between the number of substitutions in an adaptive walk and the magnitude of fitness improvement, but no correlation was found between starting and ending fitness. These results provide an empirical framework for future adaptation theory
- …