Calculating the first nontrivial 1-cocycle in the space of long knots (Q881062)

Knots invariants of finite order are well-known in knot theory. For spaces of knots in \(\mathbb R^3\) the Vassiliev theory defines the so-called cocycles of finite order. The zero-dimensional cocycles of this form are the finite-order invariants. In this paper the author considers the first nontrivial cocycle of finite order and positive dimension. This cocycle, denoted by \(v^1_3\), is one-dimensional and is of order 3. This cocycle was first found by the author of this paper and by D. Teiblyum as the result of computer calculations in 1995, and it is known as the Teiblyum-Turchin cocycle. This cocycle was then generalized by Vassiliev to the case of long knots in spaces \(\mathbb R^n\) of arbitrary \(n \geq 3\). Here the author describes three types of 1-cycles on which to compute the values of \(v_3^1\): (1) consider a knot \(k\) and the 1-cycle \(\hat k\) obtained by rotating \(k\) around a given line; (2) consider two knots \(k_1\) and \(k_2\) and the cycle \([k_1,k_2]\) obtained by taking their concatenation \(k_1 * k_2\) as starting point of the cycle and by dragging the knots one through the other in a certain way; (3) apply construction (2) to a pair of coinciding knots \(k\) and \(k\) and consider the resulting cycle (denoted by \(k \circ k\)). The main result of this paper is that the value of the cocycle \(v^1_3\) on the cycles \([k_1, k_2]\) and \(k \circ k\) is \(0\) and on the cycle \(\hat k\) reduced mod 2 is equal to the value of the first nontrivial invariant of second order \(v_2\) (that is, the Casson invariant) on \(k\), reduced mod~2.