Abstract
The trapezoidal conjecture is a classical famous conjecture posed by Fox, which states that the coefficient sequence of the Alexander polynomial of any alternating knot is trapezoidal. In this talk, we show this conjecture for knots and links of braid index 3.
Content
Previous Results
The alexander polynomial is symmetric. If is Alternierender Knoten, then the polynomial is alternating too (i.e. ).
Trapezoidal conjecture
It is conjectured that the alexander polynomial coeffiction alternating knots are trapezoidal (their absolute value increases, stays and decreases).
This holds for a wide class of objects.
Theorem (Stoimanow, ‘03)
An alternative link of braid index is
- as well as some other things.