Singularization of knots and closed braids

2014/05/21 by Thomas Fiedler, Fiedler, Thomas
Biochemistry, Genetics and Molecular Biology · Mathematics · #57M25 #Braid #Composite material #Connective tissue disorders research #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Materials science #Mathematics #Pure mathematics #math.GT #msc:57M25

paper · pdf · doi:10.48550/arxiv.1405.5562

The 1-cocycle is trivial

openalex publication_date 2014/05/21 · arxiv created 2019/01/16 · arxiv updated 2019/01/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We construct the first combinatorial 1-cocycle with values in the ℤ [x,x^-1]-module of isotopy classes of singular long knots in 3-space with a signed planar double point, and which represents a non trivial cohomology class in the topological moduli space of long knots. It can be interpreted as an invariant with values in a ℤ [x,x^-1]-module generated by 3-manifolds for each element of infinite order of the mapping class group of the complement of a satellite knot in S³. The 1-cocycle seems to be trivial on all loops for long knots which are not satellites but it is already non trivial on the Fox-Hatcher loop of any composite knot, on the loop which consists of \em dragging a trefoil through another trefoil and on the \em scan-arc for the 2-cable of the trefoil. The \em canonical resolution of the value of the 1-cocycle for dragging a knot through another knot leads to a symmetric bilinear form on the free ℤ [x,x^-1]-module of all unframed oriented knot types into itself. We conjecture that its radical contains only the trivial knot. Evaluating e.g. the Kauffman-Vogel HOMFLYPT polynomial for singular knots on the value of the 1-cocycle applied to the associated quadratic form, leads to a couple of new 3-variable knot polynomials. We construct also the first non trivial 1-cocycle for those closed positive 4-braids which contain a half-twist. It takes its values in a symmetric power of the ℤ-module of isotopy classes of closed positive 4-braids with a double point.

Citations

• Vassiliev’s knot invariants
• Isotopy invariants for closed braids and almost closed braids via loops in stratified spaces
• Knot polynomials via one parameter knot theory
• Quantum field theory and the Jones polynomial
• Invariants of 3-manifolds via link polynomials and quantum groups
• On the Vassiliev knot invariants
• Finite-type invariants of classical and virtual knots
• Virtual Knot Theory
• Finiteness of classifying spaces of relative diffeomorphism groups of 3–manifolds

Related

• Knot polynomials from 1-cocycles
• There are non homotopic framed homotopies of long knots
• The tangle-valued 1-cocycle for knots
• Polynomial invariants which can distinguish the orientations of knots
• Calculus of the first non-trivial 1-cocycle of the space of long knots