Strong and weak (1, 2, 3) homotopies on knot projections

A knot projection is an image of a generic immersion from a circle into a two-dimensional sphere. We can find homotopies between any two knot projections by local replacements of knot projections of three types, called Reidemeister moves. This paper defines an equivalence relation for knot projections called weak (1, 2, 3) homotopy, which consists of Reidemeister moves of type 1, weak type 2, and weak type 3. This paper defines the first nontrivial invariant under weak (1, 2, 3) homotopy. We use this invariant to show that there exist an infinite number of weak (1, 2, 3) homotopy equivalence classes of knot projections. By contrast, all equivalence classes of knot projections consisting of the other variants of a triple type, i.e. Reidemeister moves of (1, strong type 2, strong type 3), (1, weak type 2, strong type 3), and (1, strong type 2, weak type 3), are contractible.

PLANE CURVES IN AN IMMERSED GRAPH IN ℝ2

For any chord diagram on a circle there exists a complete graph on sufficiently many vertices such that any generic immersion of it to the plane contains a plane-closed curve whose chord diagram contains the given chord diagram as a sub-chord diagram. For any generic immersion of the complete graph on six vertices to the plane, the sum of averaged invariants of all Hamiltonian plane curves in it is congruent to one quarter modulo one-half.

REGULAR PROJECTIONS OF GRAPHS WITH AT MOST THREE DOUBLE POINTS

A generic immersion of a planar graph into the 2-space is said to be knotted if there does not exist a trivial embedding of the graph into the 3-space obtained by lifting the immersion with respect to the natural projection from the 3-space to the 2-space. In this paper, we show that if a generic immersion of a planar graph is knotted then the number of double points of the immersion is more than or equal to three. To prove this, we also show that an embedding of a graph obtained from a generic immersion of the graph (does not need to be planar) with at most three double points is totally free if it contains neither a Hopf link nor a trefoil knot.

COMPLETELY DISTINGUISHABLE PROJECTIONS OF SPATIAL GRAPHS

A generic immersion of a finite graph into the 2-space with p double points is said to be completely distinguishable if any two of the 2p embeddings of the graph into the 3-space obtained from the immersion by giving over/under information to each double point are not ambient isotopic in the 3-space. We show that only non-trivializable graphs and non-planar graphs have a non-trivial completely distinguishable immersion. We give examples of non-trivial completely distinguishable immersions of several non-trivializable graphs, the complete graph on n vertices and the complete bipartite graph on m + n vertices.

A MATRIX INVARIANT OF CURVES IN S2

Let k: S1 → S2 be a generic immersion with n double points. We present an algorithm that assigns to k a partitioned n × n matrix over Z/2Z, and show that k gives rise to an orthogonal decomposition of (Z/2Z)n. We discuss a connection between this decomposition and the trip matrix of an alternating knot diagram produced from k.

REPRESENTATIVE BRAIDS FOR LINKS ASSOCIATED TO PLANE IMMERSED CURVES

In [ AC 2], A'Campo associates a link in S3 to any proper generic immersion of a disjoint union of arcs into a 2-disc. We give a sample algorithmic way to produce, from the immersion, a representative braid for such links. As a by-product we get a minimal representative braid for any algebraic link, from a divide associated to a real deformation of the polynomial defining the link.

REGULAR HOMOTOPY AND VASSILIEV INVARIANTS OF GENERIC IMMERSIONS Sk → ℝ2k-1, k ≥ 4

The regular homotopy class of a generic immersion Sk → ℝ2k-1 is calculated in terms of its self intersection manifold with natural additional structures. There is a natural notion of Vassiliev invariants of generic immersions. These may take values in any Abelian group G. It is proved that, for any m, the group of mth order G-valued invariants modulo invariants of lower order is isomorphic to G and that the Vassiliev invariants are not sufficient to separate generic immersions, which can not be obtained from each other by a regular homotopy through generic immersions.

Multiple points of codimension one immersions of oriented manifolds

Work by L.S.Pontrjagin(18) and M.W.Hirsch(7) allows us to identify the stablen-stemwith the bordism group of oriented compact closed smoothn-manifolds immersed in ℝn+1. In a recent paper (11), U. Koschorke discusses invariants thereby defined onby analysing the self-intersections of immersed manifolds. In particular he discusses the homomorphismdefined by assigning to a generic immersionMn→ ℝn+1number (modulo 2) of its (n+ 1)-fold points.