New perspectives on self-linking

Abstract

We initiate the study of classical knots through the homotopy class of the nth evaluation map of the knot, which is the induced map on the compactified n-point configuration space. Sending a knot to its nth evaluation map realizes the space of knots as a subspace of what we call the nth mapping space model for knots. We compute the homotopy types of the first three mapping space models, showing that the third model gives rise to an integer-valued invariant. We realize this invariant in two ways, in terms of collinearities of three or four points on the knot, and give some explicit computations. We show this invariant coincides with the second coefficient of the Conway polynomial, thus giving a new geometric definition of the simplest finite-type invariant. Finally, using this geometric definition, we give some new applications of this invariant relating to quadrisecants in the knot and to complexity of polygonal and polynomial realizations of a knot. r

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“…Any invariant given by Polyak-Viro formulas has a finite degree; by the Goussarov theorem [8] the converse also is true: any finite degree invariant can be expressed by such a formula. There exist also some other combinatorial formulas for some invariants, in particular the ones described in [12], [4], and in the present work (see also [24], [27]). …”

“…Namely, suppose that the ×-pair (a i 1 , a i 2 ) has, besides r footholds considered in the previous paragraph, a foothold a u such that |i 4 A geometrical explanation of the last condition can be seen in Figure 24 (ignoring for a while the crossing point with indices i 5 , i 6 which will be needed for the illustration of the next difficult case of three colliding ×-pairs described below). Finally, let us consider the collisions of three ×-pairs.…”

“…However these conditions cannot occur in our calculations related with knot invariants; see Proposition 6. 4, drawn at or under it; moreover, a condition 2 or2! can be supplied with a condition of type2a or2b; a condition of type3 can be supplied with a condition of type3a or3b; a condition4 or4!a or4!b can be completed by a condition of type4!!…”

“…If these numbers of some two pictures are equal, then we order these pictures by the type of the (unique) equality-type condition described in items1, 2,3,4,5, or6 of §2. 4 and participating in the definition of this block: the B-block with condition1 is older than that with condition2, etc. If these types also are equal for some two B-blocks with equally many active points, then we subordinate these blocks by the lexicographic order (in the Wilson line, counting from the left to the right) of the collection of active points participating in this condition.…”

An integral expression of the first nontrivial one-cocycle of the space of long knots in ℝ3

“…The invariant v 2 can also be interpreted as the linking number of colinearity manifolds [Budney et al 2005]. Notice that in each formulation (including the one in this paper) the value of v 2 is computed by counting some colinearity pairs on the knot.…”