USNA Mathematics Department seminar
The talks for the academic year 1998-99
are held in Chauvenet 116 at 3:45 pm unless
otherwise stated.
Speaker:
Mark Kidwell
USNA
Title:
A Knotty Solution to a Problem in Continued Fractions
Abstract: Let r be a rational number.
Write r (uniquely) as a continued fraction
[a0 ; a1, a2, ..., an ] with
a0 an integer,
a1, a2, ..., an
natural numbers and an > 1. Also
write r as the continued fraction
[b0 ; b1, b2, ..., bm]
where b0 is an arbitrary integer
and b1, b2, ..., bm
are arbitrary non-zero integers.
Must it be true that
|a0| + |a1| + |a2| + ... +
|am|
5
|b0| + |b1| +
|b2| + ... + |bm| ,
with strict inequality if at least one bi , i > 0,
is negative ?
We give an introduction to continued fractions
and then give an incredibly
round-about solution to the above problem via rational tangles,
alternating knots and
the Jones polynomial.
Time: Wednesday, Nov 18, 1998
Reception at 3:30 in the common room on the 3rd floor of Chauvenet
Hall.