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)
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)
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 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)
Let F be a closed non-orientable surface. We classify all finite order invariants of immersions of F into R 3 , with values in any Abelian group. We show they are all functions of the universal order 1 invariant that we construct as T ⊕ P ⊕ Q where T is a Z valued invariant reflecting the number of triple points of the immersion, and P, Q are Z/2 valued… (More)
We give a complete description of all order 1 invariants of planar curves.
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T he theories as developed by European mathematicians prior to 1870 differed from the modern ones in that none of them used the modern theory of limits. Fermat develops what is sometimes called a " precalculus " theory, where the optimal value is determined by some special condition such as equality of roots of some equation. The same can be said for his… (More)