## Knots

Three-page knot (or link) is a knot in a three-page book,
the union of three half-planes with common boundary
(the binding line) in the three-space. It is easy to prove (and
it was done more than 100 years ago in [1])
that any knot in the three-space is equivalent
to a three-page knot.
The most popular way for presenting knots is planar diagrams.
It is known that two planar diagrams present equivalent
knots if and only if they can be obtained from each other
by a finite sequence of Reidemeister moves [2]. Similar
results were obtained in [6], [7] for three-page knots.
It was shown that a (non-oriented) three-page knot can be
presented by a word in a 12-letter alphabet of length equal
to the number of vertices at the binding line. A set of
transforms of three-page knots was introduced such that
any two three-page knots which are equivalent, if regarded
as knots in the three-space, can be obtained from each
other by a finite number of transforms.

On the contrary to the planar diagram approach, in the case of
the three-page approach there exists a very simple geometric
rule for obtaining an actual three-page knot from its
combinatorial description, i.e. the corresponding word
in the 12-letter alphabet.

A similar approach was developed in [8] for n-page knots,
where n>3.

#### References

- H. Brunn:
*Über verknotete Kurven*, Mathematiker-Kongresses Zurich 1897, Leipzig (1898), 256--259.
- K Reidemeister:
*Knotentheorie*, Julius Springer, Berlin (1932).
- J. S. Birman:
*Braids, links, and mapping class groups*, Princeton University Press (1974).
- G. Burde and H. Zieschang:
*Knots*, de Gruyter (1986).
- D. Rolfsen:
*Knots and Links*, Publish or Perish, Houston (1990).
- I. Dynnikov:
*Three-page approach to knot theory. Encoding and local moves*, Functional Analysis and Its Appl. **33**, 4 (1999), 260--269.
- I. Dynnikov:
*Three-page approach to knot theory. Universal semigroup*, Functional Analysis and Its Appl. **34**, 1 (2000), 24--32.
- I. Dynnikov:
*Finitely presented groups and semigroups in knot theory*, Proc. of V.A.Steklov Inst. of Math. **231** (2000), to appear.