The Links–Gould Invariant as a Classical Generalization of the Alexander Polynomial?

ABSTRACT

In this article, we conjecture that the Links–Gould invariant of links, which we know is a generalization of the Alexander–Conway polynomial, shares some of its classical features. In particular, it seems to give a lower bound for the genus of links and to provide a criterion for fibredness of knots. We give some evidence for these two assumptions.

Keywords:

• Alexander–Conway polynomial
• fibredness
• genus
• knot
• link
• Links–Gould invariant

2000 AMS Subject Classification:

• 57M27

Acknowledgments

I owe many warm thanks to my advisor Emmanuel Wagner who taught me about Links–Gould invariants and has been of much help in designing and writing this article. I am most grateful to Hugh Morton for his help during my study of the Whitehead doubles of the trefoil, and for showing me a braid presentation for the untwisted double. I am indebted to David de Wit for the extended work he did implementing the LINKS-GOULD EXPLORER, and to Jon Links for telling me about this remarkable package. Finally, I would like to thank David Cimasoni and Anthony Conway for the fruitful and pleasant conversations, as well as for their comments and ideas.