0.5The Cayley graph of the group 
Fig. 0.6Flips on a pentagon
This leads to the relation:
Note that unlike the case of
Definition 0.2. The group
Fig. 0.7Voronoï tiling change
(1)
(2)
(3)
(4)
Just like we formulated the
It turns out that groups
says that the product of diagonals of an inscribed quadrilateral equals the sum of products of its opposite faces, see Fig. 0.8.
Fig. 0.8The Ptolemy relation
We can use it when considering triangulations of a given surface: when performing a flip, we replace one diagonal (x) with the other diagonal (y) by using this relation. It is known that if we consider all five triangulations of the pentagon and perform five flips all the way around, we return to the initial triangulation with the same label, see Fig. 0.6.
This well known fact gives rise to presentations of
Thus, by analysing the groups
(1)Invariants of braids on 2-surfaces valued in polytopes;
(2)Invariants of knots;
(3)Relations to groups
(4)Braids on
Going slightly beyond, we can investigate braids in
We will not say much about the groups
(1)Generators (codimension 1) correspond to simplicial (k − 2)-polytopes with k vertices;
(2)The most interesting relations (codimension 2) correspond to (k − 2)-polytopes with k + 1 vertices.
It would be extremely interesting to establish the connection between
It is also worth mentioning, that the relations in the group
Finally, our invariants may not be just group-valued: some variations of
Note that the present book is very much open-ended. On one hand, the invariants of manifolds are calculated in some explicit cases. On the other hand, one can vary “the
The present book has the following structure. In Part 1 we review basic notions of knot theory and combinatorial group theory: groups and their presentations, van Kampen diagrams, braid theory, knot theories and the theory of 2-dimensional knots. Part 2 is devoted to the parity theory and its applications to cobordisms of knots and free knots. In Part 3 we present the theory of
