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Tesi etd-07062011-061816

Thesis type
Tesi di laurea specialistica
email address
Polynomial invariants for ribbon links and symmetric unions
Corso di studi
relatore Prof. Lisca, Paolo
Parole chiave
  • Jones polynomial
  • ribbon links
  • symmetric unions
Data inizio appello
Riassunto analitico
This thesis is about ribbon links, symmetric unions and boundary links.
Slice and ribbon links
were first introduced by Fox in 1962. A knot is said to be
(smoothly) slice if it bounds a properly embedded disc in the four dimensional ball.
A knot is ribbon if the radial function may be chosen to be a Morse function
with no local maxima when restricted to the disc. Ribbon knots can be described
in a purely 3-dimensional way as knots bounding immersed discs with only "nice singularities".
The main problem which motivates our work is the so called slice-ribbon conjecture:
Is every slice knot a ribbon knot?
This question was first asked by Ralph Fox in 1962 and it is still open.
The notions of sliceness and of being ribbon naturally extends from knots to links
and the same conjecture stands in this more general context.

The 3-dimensional characterization of ribbon links allows one to study
them in a combinatorial setting, which makes possible the use of
polynomial invariants. These are the main tools of our work.
Following Eisermann's paper we will look at the multiplicity of a certain
factor in the Jones polynomial of links, showing that for ribbon links this multiplicity
is completely determined. We will also discuss some possible generalizations of this result.

We then introduce symmetric unions. A symmetric union presentation for a knot
is a diagram obtained by starting with a connected sum of a knot with its mirror image
and then adding some extra crossings on the axis. This construction was first introduced by
Kinoshita and Terasaka. Every symmetric union knot diagram represents
a ribbon knot, therefore the following question arise naturally:
Does every ribbon knot admit a symmetric union presentation?
This is an open question and no obstructions are known. Again a similar notion
can be defined for links and the same open problem stands.
Following Eisermann and Lamm we will consider
the problem of uniqueness of symmetric presentations of a given knot.
In order for this problem to make sense, one should specify when two
symmetric diagrams are considered to be \emph{symmetrically equivalent}.
We will list a set of symmetric Reidemeister moves for this purpose.
This combinatorial approach allows one to introduce polynomial invariants
in the symmetric context. In fact, a 2-variable refinement of the classical
Jones polynomial is defined, and it is shown that it can distinguish between
different symmetric union presentations of the same knot.