Abstract. The operations of conjugation and multiplication in a group
satisfy four properties that can be interpreted as moves on knotted
trivalent graphs that represent handle-body knots. These moves are
codimension 1 singularities. The codimension 2 singularities give rise to
movie moves for knotted foams. These movie moves are chains in a higher
dimensional homology theory. The pictures then can be dualized to produce
polytopes and analyzed from the point of view of abstract tensors. A system
of equations arrises that has a nice solution from the above considerations.

The motivation for using a kind of knot-diagrams for representing
maps from the three-sphere into other spaces came from a long-term
research project that aims at getting hands on homotopy
and homology groups of shrinking wedges. The most-cited
shrinking wedges are the shrinking wedge of circles
(also know as „the Hawaiian Earrings”) and the shrinking wedge
of two-spheres (also knows as the „two-dimensional Hawaiian Earrings”
or the „Milnor-Barratt-space”, [MB]), which is an example of a
two-dimensional
space having a non-zero three-dimensional singular homology group.
It is known that there cannot be a simple
universal formula for the homology groups of a shrinking wedge ([EdaK] vs.
[EKRZ, Thm.1.6]) while, conversely, the results of [EdaK] & [EKRZ] could
only be achieved by analyzing the possible maps generating the elements of
the homology groups by the methods of geometric topology. The classical
methods to determine the algebraic structure of homology and homotopy
groups give only little information on the geometric structure of the
underlying maps. Therefore, as a basis for proceeding further into the
direction of shrinking wedges and of other spaces under consideration in
wild algebraic topology, the concept of visualizing by diagrams of knotted
surfaces was developed.
The research project to be reported about is still just in its initial
phase. The algebraic structure of the third homotopy group of the
suspension of the torus can be determined by classical methods. The talk
will have to restrict to explaining the concept
of theses diagrams of knotted surfaces, to showing the
diagrams of the generators of our group, and to explaining what parts of the
proof that the corresponding maps are the free generators of the third
homotopy group of the suspension of the torus can already been given by
according diagrammatic methods.



[EdaK] Eda, Katsuya; Kawamura, Kazuhiro: „Homotopy and homology groups of
the $n$-dimensional Hawaiian earring”, Fund. Math.,
Vol. 165 (2000), no. 1, 17–28.
[MB] Barratt, M. G.; Milnor, John: „An example of anomalous singular
homology”, Proc. Amer. Math. Soc., Vol. 13 (1962) 293–297.
[EKRZ] Eda, Katsuya; Karimov, Umed H.; Repovs, Dusan; Zastrow, Andreas:
„On snake cones, alternating cones and related constructions”,
Glas. Mat. Ser. III, Vol. 48(68) (2013), no. 1, 115–135.