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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.
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 :(More)
We study random knots and links in R using the Petaluma model, which is based on the petal projections developed in [2]. 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)
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)