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We show that the maximal number of singular moves required to pass between any two regularly homotopic planar or spherical curves with at most n crossings, grows quadratically with respect to n. Furthermore, this can be done with all curves along the way having at most n + 2 crossings.
Definition 1.1. Let F be a (finite) system of closed surfaces and M a 3-manifold. A regular homotopy Ht : F →M, t ∈ [0, 1] will be called closed if H0 = H1. We will denote a closed generic regular homotopy by CGRH. The number mod 2 of quadruple points of a generic regular homotopy Ht will be denoted by q(Ht) (∈ Z/2.) Max and Banchoff in [MB] proved that any… (More)
Let F be a closed surface. If i, i′ : F → R3 are two regularly homotopic generic immersions, then it has been shown in  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 :… (More)
We classify all order one invariants of immersions of a closed orientable surface F into R3, with values in an arbitrary Abelian group G. We show that for any F and G and any regular homotopy class A of immersions of F into R3, the group of all order one invariants on A is isomorphic to Gא0 ⊕B⊕B where Gא0 is the group of all functions from a set of… (More)
We study random knots and links in R using the Petaluma model, which is based on the petal projections developed in . In this model we obtain a formula for the distribution of the linking number of a random two-component link. We also obtain formulas for the expectations and the higher moments of the Casson invariant and the order-3 knot invariant v3.… (More)
We construct a new order 1 invariant for knot diagrams. We use it to determine the minimal number of Reidemeister moves needed to pass between certain pairs of knot diagrams.
We present a sequence of diagrams of the unknot for which the minimum number of Reidemeister moves required to pass to the trivial diagram is quadratic with respect to the number of crossings. These bounds apply both in S and in R.
Article history: Received 25 April 2011 Received in revised form 11 January 2012 Accepted 11 January 2012
Finite order invariants of stable immersions of a closed orientable surface into R have been defined in [N], where all order 1 invariants have been classified. In the present work we classify all finite order invariants of order n > 1, and show that they are all functions of the universal order 1 invariant constructed in [N]. The structure of the paper is… (More)
The universal order 1 invariant fU of immersions of a closed orientable surface into R3, whose existence has been established in [T. Nowik, Order one invariants of immersions of surfaces into 3-space, Math. Ann. 328 (2004) 261–283], is the direct sum