arXiv subjects

Alexander Stoimenow

Publications and source records attributed to Alexander Stoimenow.

11 recordsLinked to original sources

arXiv2024

Minimal grid diagrams of the prime knots with crossing number 14 and arc index 13, 14

There are 46,972 prime knots with crossing number 14. Among them 19,536 are alternating and have arc index 16. Among the non-alternating knots, 17, 477, and 3,180 have arc index 10, 11, and 12, respectively. The remaining 23,762 have arc index 13 or 14. There are none with arc index smaller than 10 or larger than 14. We obtained 8,027 knots having arc index 13 and 15,735 knots having arc index 14. We show them by their minimal grid diagrams.

math.GT↗

arXiv2024

Minimal grid diagrams of the prime alternating knots with 13 crossings

A knot is a closed loop in space without self-intersection. Two knots are equivalent if there is a self homeomorphism of space bringing one onto the other. An arc presentation is an embedding of a knot in the union of finitely many half planes with a common boundary line such that each half plane contains a simple arc of the knot. The minimal number of such half planes among all arc presentations of a given knot is called the arc index of the knot. A knot is usually presented as a planar diagram with finitely many crossings of two strands where one of the strands goes over the other. A grid diagram is a planar diagram which is a non-simple rectilinear polygon such that vertical edges always cross over horizontal edges at all crossings. It is easily seen that an arc presentation gives rise to a grid diagram and vice versa. It is known that the arc index of an alternating knot is two plus its minimal crossing number. There are 4878 prime alternating knots with minimal crossing number 13. We obtained minimal arc presentations of them in the form of grid diagrams having 15 vertical segments. This is a continuation of the works on prime alternating knots of 11 crossings and 12 crossings.

math.GT↗

arXiv2022

Invariants of weakly successively almost positive links

As an extension of positive and almost positive diagrams and links, we study two classes of links we call successively almost positive and weakly successively almost positive links. We prove various properties of polynomial invariants and signatures of such links, extending previous results or answering open questions about positive or almost positive links. We discuss their minimal genus and fibering property and for the latter prove a fibering extension of Scharlemann-Thompson's theorem (valid for general links).

math.GT↗

arXiv2014

Everywhere Equivalent 3-Braids

A knot (or link) diagram is said to be everywhere equivalent if all the diagrams obtained by switching one crossing represent the same knot (or link). We classify such diagrams of a closed 3-braid.

math.GT↗

arXiv2012

Conduct and Correctness in Mathematical Publishing

"We risk sliding down toward the standards where the validity of action is decided by whether one can get away with it." (P. Doty) "We do not 'risk' sliding down toward such standards; we have reached them." (S. Lang) This is an essay in which I try to express my fear about the establishment of a culture of publishing, where no one is willing to take responsibility for the correctness of mathematics, and readers finding mistakes in published proofs are stamped as outcasts, because they are deemed to target the reputation of authors and journals.

math.HO↗

arXiv2011

Exchange moves and braid representations of links

We prove that under fairly general conditions an iterated exchange move gives infinitely many non-conjugate braids. As a consequence, every knot has infinitely many conjugacy classes of n-braid representations if and only if it has one admitting an exchange move.

math.GT↗

arXiv2009

Genus generators and the positivity of the signature

It is a conjecture that the signature of a positive link is bounded below by an increasing function of its negated Euler characteristic. In relation to this conjecture, we apply the generator description for canonical genus to show that the boundedness of the genera of positive knots with given signature can be algorithmically partially decided. We relate this to the result that the set of knots of canonical genus greater than or equal to n is dominated by a finite subset of itself in the sense of Taniyama's partial order.

math.GT↗

arXiv2009

The density of Lawrence-Krammer and non-conjugate braid representations of links

We use some Lie group theory and Budney's unitarization of the Lawrence-Krammer representation, to prove that for generic parameters of definite form the image of the representation (also on certain types of subgroups) is dense in the unitary group. This implies that, except possibly for closures of full-twist braids, all links have infinitely many conjugacy classes of braid representations on any non-minimal number of (and at least 4) strands.

math.GR↗

arXiv2009

Euclidean Mahler measure and twisted links

If the twist numbers of a collection of oriented alternating link diagrams are bounded, then the Alexander polynomials of the corresponding links have bounded euclidean Mahler measure (see Definition 1.2). The converse assertion does not hold. Similarly, if a collection of oriented link diagrams, not necessarily alternating, have bounded twist numbers, then both the Jones polynomials and a parametrization of the 2-variable Homflypt polynomials of the corresponding links have bounded Mahler measure.

math.GT↗

arXiv2008

Mutation and the colored Jones polynomial

We show examples of knots with the same polynomial invariants and hyperbolic volumes, with variously coinciding 2-cable polynomials and colored Jones polynomials, which are not mutants.

math.GT↗

arXiv1997

The Fundamental Theorem of Vassiliev Invariants

The "fundamental theorem of Vassiliev invariants" says that every weight system can be integrated to a knot invariant. We discuss four different approaches to the proof of this theorem: a topological/combinatorial approach following M. Hutchings, a geometrical approach following Kontsevich, an algebraic approach following Drinfel'd's theory of associators, and a physical approach coming from the Chern-Simons quantum field theory. Each of these approaches is unsatisfactory in one way or another, and hence we argue that we still don't really understand the fundamental theorem of Vassiliev invariants.

q-alg↗