## Tesi etd-07062011-061816 |

Thesis type

Tesi di laurea specialistica

Author

ACETO, PAOLO

email address

paoloaceto@gmail.com

URN

etd-07062011-061816

Title

Polynomial invariants for ribbon links and symmetric unions

Struttura

SCIENZE MATEMATICHE, FISICHE E NATURALI

Corso di studi

MATEMATICA

Supervisors

**relatore**Prof. Lisca, Paolo

Parole chiave

- Jones polynomial
- ribbon links
- symmetric unions

Data inizio appello

22/07/2011;

Consultabilità

Completa

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.

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.

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