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Register a new userAlexander Arhangelskii
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Olga Sipacheva
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This is the opening article of the abstract book of conference Set-Theoretic Topology and Topological Algebra in honor of professor Alexander Arhangelskii on the occasion of his 80th birthday held in 2018 at Moscow State University.
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Conway-normalized Alexander polynomial of ribbon knots depend only on their ribbon diagrams. Here ribbon diagram means a ribbon spanning the ribbon knot marked with the information of singularities. We further give an algorithm to calculate Alexander polynomials of ribbon knots from their ribbon diagrams.
In November 2014 Alexander Grothendieck passed away at the age of 86. There is no doubt that he was one of the greatest and most innovative mathematicians of the 20th century. After a bitter childhood, his meteoric ascent started in the Cartan Seminar in Paris, it led to a breakthrough while he worked in Sao Paulo, and to the Fields Medal. He introduced numerous new concepts and techniques, which were involved in the groundbreaking solutions to long-standing problems. However, dramatic changes were still ahead of him. In recent years hardly anybody knew where he was living, and even if he was still alive; he had withdrawn to a modest life in isolation. Also beyond his achievements in mathematics, Grothendieck was an extraordinary person. This is a tribute of his fascinating life.
Twisted Alexander invariants of knots are well-defined up to multiplication of units. We get rid of this multiplicative ambiguity via a combinatorial method and define normalized twisted Alexander invariants. We then show that the invariants coincide with sign-determined Reidemeister torsion in a normalized setting, and refine the duality theorem. We further obtain necessary conditions on the invariants for a knot to be fibered, and study behavior of the highest degrees of the invariants.
The augmentation variety of a knot is the locus, in the 3-dimensional coefficient space of the knot contact homology dg-algebra, where the algebra admits a unital chain map to the complex numbers. We explain how to express the Alexander polynomial of a knot in terms of the augmentation variety: it is the exponential of the integral of a ratio of two partial derivatives. The expression is derived from a description of the Alexander polynomial as a count of Floer strips and holomorphic annuli, in the cotangent bundle of Euclidean 3-space, stretching between a Lagrangian with the topology of the knot complement and the zero-section, and from a description of the boundary of the moduli space of such annuli with one positive puncture.
We study generalizations of a classical link invariant -- the multivariable Alexander polynomial -- to tangles. The starting point is Archibalds tMVA invariant for virtual tangles which lives in the setting of circuit algebras, and whose target space has dimension that is exponential in the number of strands. Using the Hodge star map and restricting to tangles without closed components, we define a reduction of the tMVA to an invariant rMVA which is valued in matrices with Laurent polynomial entries, and so has a much more compact target space. We show the rMVA has the structure of a metamonoid morphism and is further equivalent to a tangle invariant defined by Bar-Natan. This invariant also reduces to the Gassner representation on braids and has a partially defined trace operation for closing open strands of a tangle.
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