Pairings of combinatorial 1-cocycles with loops in knot spaces

Abstract

We show that the evaluations of combinatorial 1-cocycles defined by Gauss diagrams with a triangle on the canonical loops in the space of long knots are finite type invariants. Using Gauss diagrams, we construct two $\mathbb{Z}$-valued combinatorial 1-cocycles $β_1, β_2$ and a $\mathbb{Z}/2\mathbb{Z}$-valued 1-cocycle $β_3$ on the space of long knots and prove their cocyclicity by verifying their invariance under higher Reidemeister moves coming from the codimension-two singularities of plane curves. We show that they represent genuinely new 1-cohomology classes and compute their pairings with the rotation, rolling, half rolling, bracket and half bracket loops. A key new feature is that $β_1$ and $β_2$ can pair nontrivially with bracket and half-bracket loops. We conjecture that $(α_3^1,β_1,β_2)$ over $\mathbb{Q}$, and their mod 2 reductions together with $β_3$ over $\mathbb{Z}/2\mathbb{Z}$, form bases of degree-one cohomology up to order 4 in the sense of Vassiliev. In addition, we show that the reparametrization loop is homotopic to the rolling loop concatenated with the rotation loop in $\operatorname{Emb}(S^1, S^3)$. Finally, we give the criteria for a 1-cohomology class in the long knot space to descend to 1-cohomology classes in $\operatorname{Emb}(S^1, S^3)$ and $\operatorname{Emb}(S^1, S^3)/\operatorname{Diff}^{+}(S^1)$. Using these criteria, we show that $β_1$ descends to a nontrivial 1-cohomology class in $\operatorname{Emb}(S^1,S^3)$, while $β_1 \bmod 2$ and $β_3$ descend to linearly independent nontrivial 1-cohomology classes in $\operatorname{Emb}(S^1, S^3)/\operatorname{Diff}^{+}(S^1)$.