I would like to know if there are any examples where the HOMFLY polynomial fails. Like two knots having the same polynomial when they shouldn't, or having different ones when they should be the same (this one is not possible I think, since then HOMFLY shouldn't be called invariant any more), or maybe chirality issues. Both Alexander and Jones have their breakpoints, so I assume HOMFLY does too.
What I've found until now is that we can not prove that HOMFLY polynomial is not perfect, but it is not likely to be perfect. I just wonder if anyone found a point where it fails, that I couldn't, an example that proves it wrong.
$5_1$ and $10_{132}$ are different knots with same Homfly polynomial. You can look at [Yılmaz Altun, "On the Homfly Polynomial", International Mathematical Forum, 2, 2007, no. 56, 2753 - 2757] and references therein.
The knots 88 and 10129 have the same HOMFLY polynomial but distinct Kauffman polynomials.
Reference: An Introduction to Knot Theory (1997) - Lickorish
Chapter 16 Exploring the HOMFLY and Kauffman Polynomials
- Badges
- Science topic
- Similar topics
- Engineering
- Petroleum Engineering
- Gas