Homogeneous links, Seifert surfaces, digraphs and the reduced Alexander polynomial
Abstract
We give a geometric proof of the following result of Juhasz. \emph{Let $a_g$ be the leading coefficient of the Alexander polynomial of an alternating knot $K$. If $|a_g|<4$ then $K$ has a unique minimal genus Seifert surface.} In doing so, we are able to generalise the result, replacing `minimal genus' with `incompressible' and `alternating' with `homogeneous'. We also examine the implications of our proof for alternating links in general.
- Publication:
-
arXiv e-prints
- Pub Date:
- January 2011
- DOI:
- 10.48550/arXiv.1101.1412
- arXiv:
- arXiv:1101.1412
- Bibcode:
- 2011arXiv1101.1412B
- Keywords:
-
- Mathematics - Geometric Topology;
- 57M25;
- 57M27;
- 57M15
- E-Print:
- 37 pages, 28 figures