Chord Diagrams, d-Diagrams, and Knots

On hyperbolic once-punctured-torus bundles IV: Automata for lightning curves

Finite-type invariants of classical and virtual knots

In the present paper, we suggest a new combinatorial approach to knot theory based on embeddings of knots and links into a union of three half-planes with the same boundary. The idea to embed knots into a “book” is quite natural and was considered already in [1]. Among recent papers on embeddings of knots into a book with infinitely many pages, we mention [2] and [3] (see also references therein). The restriction of the number of pages to three (or any other number ≥3) provides a convenient way toencode links by words in a finite alphabet. For those words, we give a finite set of local changes that realizes the equivalence of links by analogy with the Reidemeister moves for planar link diagrams.

Abstract. For an equivalence relation E on the words in some finite alphabet, we consider the recognition problem (decide whether two words are equivalent), the invariant problem (calculate a function constant on precisely the equivalence classes), the normal form problem (calculate a particular member of an equivalence class, given an arbitrary member) and the first member problem (calculate the first member of an equivalence class, given an arbitrary member). A solution for any of these problems yields solutions for all earlier ones in the list. We show that, for polynomial time recognizable E, the first member problem is always in the class Az (solvable in polynomial time with an oracle for an NP set) and can be complete for this class even when the normal form problem is solvable in polynomial time. To distinguish between the other problems in the list, we construct an E whose invariant problem is not solvable in polynomial time with an oracle for E (although the first member problem is in NPz fqco-NPZ), and we construct an E whose normal form problem is not solvable in polynomial time with an oracle for a certain solution of its invariant problem. Key words, polynomial time, Turing reducible, NP-complete, equivalence relation, certificate, normal form Introduction. This paper was stimulated by [3], where finite structures (e.g. graphs)

Introduced in S. Nelson and M. Sakurai [Biquandle arrow weight enhancements, Internat. J. Math. 34(8) (2023) 2350046], biquandle arrow weight invariants are enhancements of the biquandle counting invariant for oriented virtual and classical knots defined from biquandle-colored Gauss diagrams using a tensor over an abelian group satisfying certain properties. In this paper, we categorify the biquandle arrow weight polynomial invariant using biquandle coloring quivers, obtaining new infinite families of polynomial invariants of oriented virtual and classical knots.

Gauss diagrams, or more generally chord diagrams are a well-established tool in the study of topology of knots and of planar curves. In this paper we present a system description of Gauss-lintel, our implementation in SWI-Prolog of a suite of algorithms for exploring chord diagrams. Gauss-lintel employs a datatype which we call “lintel”, which is a list representation of an odd-even matching for the set of integers [0,...,2n–1], for efficiently generating Gauss diagrams and testing their properties, including one important property called realizability. We report on extensive experiments in generation and enumeration of various classes of Gauss diagrams, as well as on experimental testing of several published descriptions of realizability.

We prove that the algebra $ \cal A $ of chord diagrams, the dual to the associated graded algebra of Vassiliev knot invariants, is isomorphic to the universal enveloping algebra of a Casimir Lie algebra in a certain tensor category (the PROP for Casimir Lie algebras). This puts on a firm ground a known statement that the algebra $ \cal A $ looks and behaves like a universal enveloping algebra. An immediate corollary of our result is the conjecture of [BGRT] on the Kirillov-Duflo isomorphism for algebras of chord diagrams.¶ Our main tool is a general construction of a functor from the category $ \tt CycOp $ of cyclic operads to the category $ \tt ModOp $ of modular operads which is left adjoint to the tree part functor $ {\tt ModOp} \to {\tt CycOp} $ . The algebra of chord diagrams arises when this construction is applied to the operad $ {\tt LIE} $ . Another example of this construction is Kontsevich's graph complex which corresponds to the operad $ {\tt LIE}_\infty $ for homotopy Lie algebras.