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 ℤ-valued combinatorial 1-cocycles β1, β2 and a ℤ/2ℤ-valued 1-cocycle β3 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 β1 and β2 can pair nontrivially with bracket and half-bracket loops. We conjecture that (α31,β1,β2) over ℚ, and their mod 2 reductions together with β3 over ℤ/2ℤ, 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 Emb(S1, S3). Finally, we give the criteria for a 1-cohomology class in the long knot space to descend to 1-cohomology classes in Emb(S1, S3) and Emb(S1, S3)/Diff+(S1). Using these criteria, we show that β1 descends to a nontrivial 1-cohomology class in Emb(S1,S3), while β1 bmod 2 and β3 descend to linearly independent nontrivial 1-cohomology classes in Emb(S1, S3)/Diff+(S1).
Related
BibTeX
@misc{zhang2026pairings,
title = {{Pairings of combinatorial 1-cocycles with loops in knot spaces}},
author = {Butian Zhang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.29946},
eprint = {2609.29946},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.29946},
}
RIS
TY - UNPB
TI - Pairings of combinatorial 1-cocycles with loops in knot spaces
AU - Butian Zhang
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2609.29946
UR - https://arxiv.org/abs/2609.29946
AB - 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 ℤ-valued combinatorial 1-cocycles β_1, β_2 and a ℤ/2ℤ-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 ℚ, and their mod 2 reductions together with β_3 over ℤ/2ℤ, 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 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 Emb(S^1, S^3) and Emb(S^1, S^3)/Diff^+(S^1). Using these criteria, we show that β_1 descends to a nontrivial 1-cohomology class in Emb(S^1,S^3), while β_1 bmod 2 and β_3 descend to linearly independent nontrivial 1-cohomology classes in Emb(S^1, S^3)/Diff^+(S^1).
ER -