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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 2 and in R 2 .
We classify all order one invariants of immersions of a closed orientable surface F into R 3 , 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 R 3 , the group of all order one invariants on A is isomorphic to G ℵ 0 ⊕ B ⊕ B where G ℵ 0 is the group of all functions from a set… (More)
Let GI denote the space of all generic immersions of a surface F into a 3-manifold M. Let q(H R) denote the number mod 2 of quadruple points of a generic regular homotopy H R : FPM. We are interested in de"ning an invariant Q : GIP9/2 such that q(H R)"Q(H)!Q(H) for any generic regular homotopy H R : FPM. Such an invariant exists i! q"0 for any closed… (More)
We classify all finite order invariants of immersions of a closed orientable surface into R 3 , with values in any Abelian group. We show that they are all functions of order one invariants.
Let F be a closed surface. If i, i : F → R 3 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 → R 3 and any diffeomorphism h : F… (More)
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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.
We study random knots and links in R 3 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)
The universal order 1 invariant f U of immersions of a closed orientable surface into R 3 , 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 f U = n∈Z f H n ⊕ n∈Z f T n ⊕ M ⊕ Q where each f H n , f T n is a Z valued invariant and M, Q are Z/2… (More)