## 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

Commissione

**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. <br>Slice and ribbon links <br>were first introduced by Fox in 1962. A knot is said to be <br>(smoothly) slice if it bounds a properly embedded disc in the four dimensional ball.<br>A knot is ribbon if the radial function may be chosen to be a Morse function<br>with no local maxima when restricted to the disc. Ribbon knots can be described<br>in a purely 3-dimensional way as knots bounding immersed discs with only "nice singularities".<br>The main problem which motivates our work is the so called slice-ribbon conjecture:<br> Is every slice knot a ribbon knot?<br>This question was first asked by Ralph Fox in 1962 and it is still open.<br>The notions of sliceness and of being ribbon naturally extends from knots to links<br>and the same conjecture stands in this more general context.<br><br>The 3-dimensional characterization of ribbon links allows one to study<br>them in a combinatorial setting, which makes possible the use of <br>polynomial invariants. These are the main tools of our work.<br>Following Eisermann's paper we will look at the multiplicity of a certain<br>factor in the Jones polynomial of links, showing that for ribbon links this multiplicity<br>is completely determined. We will also discuss some possible generalizations of this result.<br><br>We then introduce symmetric unions. A symmetric union presentation for a knot<br>is a diagram obtained by starting with a connected sum of a knot with its mirror image<br>and then adding some extra crossings on the axis. This construction was first introduced by<br>Kinoshita and Terasaka. Every symmetric union knot diagram represents<br>a ribbon knot, therefore the following question arise naturally:<br>Does every ribbon knot admit a symmetric union presentation?<br>This is an open question and no obstructions are known. Again a similar notion<br>can be defined for links and the same open problem stands.<br>Following Eisermann and Lamm we will consider<br>the problem of uniqueness of symmetric presentations of a given knot.<br>In order for this problem to make sense, one should specify when two<br>symmetric diagrams are considered to be \emph{symmetrically equivalent}.<br>We will list a set of symmetric Reidemeister moves for this purpose.<br>This combinatorial approach allows one to introduce polynomial invariants <br>in the symmetric context. In fact, a 2-variable refinement of the classical<br>Jones polynomial is defined, and it is shown that it can distinguish between<br>different symmetric union presentations of the same knot.<br>

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