Seifert Surfaces of Maximal Euler Characteristic

Given a link L ⊂ S 3 , a Seifert surface S for L is a compact, orientable surface with boundary L. The Euler characteristic χ(L) of the link L is dened to be the maximum over all Euler characteristics χ(S) of Seifert surfaces S for L. Seifert surfaces exist for all L, and this denition presents itself with the problem of calculating χ(L). An easily applicable method for producing Seifert surfaces is known as Seifert's algorithm. For a given link L, this algorithm takes as input a diagram D for L and returns a Seifert surface for L. The output of the algorithm generally depends on the choice of diagram, and as such, we dene the canonical Euler characteristic χ c (L) of a link L to be the maximum over all Euler characteristics χ(S), where S is now a Seifert surfaces for L produced by Seifert's algorithm. It is thus true that for all links L, we have χ(L) ≥ χ c (L). It will be seen to be straightforward to calculate χ(S), when S is a surface produced by Seifert's algorithm. Whenever the inequality χ(L) ≥ χ c (L) reduces to an equality, we thus have immediate access to nding χ(L). This thesis considers the question: For which links L do we have equality in χ(L) ≥ χ c (L)? We show rst that we do have equality for the class of links known as the alternating links. Following a completely dierent path, we prove then that we also have equality for a class of links known as the alternative links (of which alternating links are a special case), and also that the maximal Euler characteristic Seifert surface S is produced by applying Seifert's algorithm to an alternative diagram. I thank my supervisor Nathalie Wahl for her patience and resiliency in dealing with my many questions , and for her always helpful insights. Forord. En Seifertade S for et givet link L er en kompakt, orienterbar ade, som har L som rand. Eu-lerkarakteristikken χ(L) af linket L er pr. denition maksimum over alle Eulerkarakteristikker χ(S), hvor S er en Seifertade for L. Seifertader eksisterer for alle L, og denne denition giver anledning til spørgsmålet om, hvordan χ(L) beregnes. En letanvendelig metode til at producere Seifertader er Seiferts algorithme. For et givet link L tager algoritmen et diagram D for L som input, og returnerer en Seifertade for …