9 There are non homotopic framed homotopies of long knots
نویسنده
چکیده
Let M be the space of all, including singular, long knots in 3space and for which a fixed projection into the plane is an immersion. Let cl(Σ (1) iness) be the closure of the union of all singular knots in M with exactly one ordinary double point and such that the two resolutions represent the same (non singular) knot type. We call Σ (1) iness the inessential walls and we call Mess = M\ cl(Σ (1) iness) the essential diagram space. We construct a non trivial class in H(Mess;Z[A,A −1]) by an extension of the Kauffman bracket. This implies in particular that there are loops in Mess which consist of regular isotopies of knots together with crossing changings and which are not contractible in Mess (leading to the title of the paper). We conjecture that our construction gives rise to a new knot polynomial for knots of unknotting number one. 1
منابع مشابه
1 9 N ov 2 00 7 There are non homotopic framed homotopies of long knots
Let M be the space of all, including singular, long knots in 3space and for which a fixed projection into the plane is an immersion. Let Σ (1) iness be the union of all singular knots in M with exactly one ordinary double point and such that the two resolutions represent the same (non singular) knot type. We call Σ (1) iness the inessential walls and we call Mess = M\ Σ (1) iness the essential ...
متن کاملOn the Height of a Homotopy
Given 2 homotopic curves in a topological space, there are several ways to measure similarity between the curves, including Hausdorff distance and Fréchet distance. In this paper, we examine a different measure of similarity which considers the family of curves represented in the homotopy between the curves, and measures the longest such curve, known as the height of the homotopy. In other word...
متن کاملSolution of Bounded Nonlinear Systems of Equations Using Homotopies with Inexact Restoration
Nonlinear systems of equations often represent mathematical models of chemical production processes and other engineering problems. Homotopic techniques (in particular, the bounded homotopies introduced by Paloschi) are used for enhancing convergence to solutions, especially when a good initial estimate is not available. In this paper, the homotopy curve is considered as the feasible set of a m...
متن کاملAutomorphisms and Embeddings of Surfaces and Quadruple Points of Regular Homotopies
Let F be a closed surface. If i, i′ : F → R3 are two regularly homotopic generic immersions, then it has been shown in [5] that all generic regular homotopies between i and i′ have the same number mod 2 of quadruple points. We denote this number by Q(i, i′) ∈ Z/2. For F orientable we show that for any generic immersion i : F → R3 and any diffeomorphism h : F → F such that i and i ◦ h are regula...
متن کاملKnots, Operads and Double Loop Spaces
We show that the space of long knots in an euclidean space of dimension larger than three is a double loop space, proving a conjecture by Sinha. We also construct a double loop space structure on framed long knots, and show that the map forgetting the framing is not a double loop map in odd dimension. However there is always such a map in the reverse direction expressing the double loop space o...
متن کامل