Journal of Knot Theory and Its Ramifications

We determine all 2- and 3-cocycles for Laver tables, an infinite sequence of finite structures obeying the left-self-distributivity law; in particular, we describe simple explicit bases. This provides a number of new positive braid invariants and paves the way for further potential topological applications. An important tool for constructing a combinatorially meaningful basis of 2-cocycles is the right-divisibility relation on Laver tables, which turns out to be a partial ordering.

Arone and Turchin defined graph-complexes computing the rational homotopy of the spaces of long embeddings. The graph-complexes split into a direct sum by the number of loops in graphs. In this paper, we compute the homology of its two-loop part.

A local move called a C_n-move is closely related to Vassiliev invariants. The C_n-distance between two knots K and L, denoted by d_Cn(K, L), is the minimal number of C_n-moves needed to transform K into L. In the case of n ≥ 3, we show that for any pair of knots K₁ and K₂ with d_Cn(K₁, K₂) = 1 and for any given natural number m, there exist infinitely many knots J_j (j = 1, 2, …) such that d_Cn(K₁, J_j) = d_Cn(J_j, K₂) = 1 and they have the same Vassiliev invariants of order less than or equal to m.

Three disjoint rays in ℝ³ form Borromean rays provided their union is knotted, but the union of any two components is unknotted. We construct infinitely many Borromean rays, uncountably many of which are pairwise inequivalent. We obtain uncountably many Borromean hyperplanes.

The flat plumbing basket presentation of a link is introduced by Furihata, Hirasawa and Kobayashi. In this paper, we show that the pass-equivalence and an equivalence introduced by using the flat plumbing basket presentation are the same relation. Furthermore, we obtain an evaluation of the minimal number of bands used for the flat pluming basket presentation for a knot coming from the degree of the Alexander polynomial and the three genus of the knot.