knots are equivalent, then the polynomials associated to
them turn out to be the same. This thus provided an
algebraic means to distinguish knots: if two knots have
different Alexander polynomials, then they are not
equivalent. In the 1960s British mathematician John
Horton Conway discovered an explicit method for
implementing Alexander’s polynomials, and in 1984
Vaughn Jones developed an alternative (simpler) choice
of polynomial to associate with each knot. (It was with
the “Jones polynomial” that Jones was able to establish
Tait’s claim.) The same year, eight mathematicians inde-
pendently discovered another polynomial that general-
izes Jones’s approach. (This polynomial is called the
HOMFLYPT polynomial—an acronym of their names.)
Unfortunately, it is not yet known whether Jones’s tech-
nique provides a perfect correspondence between dis-
tinct knots and distinct polynomials—it is feasible that
two nonequivalent knots could produce the same Jones’s
polynomial. (No one has yet seen an example of this.)
Knot theory is an active area of current research. It
is used in the practical study of complicated molecules
such as DNA, in the analysis of electrical circuits, and
in the planning of street and highway networks.
Although the mirror image of a knot is deemed
equivalent to the original knot, mathematicians have
proved that no knot (except the unknot) can be physi-
cally deformed into its mirror image. It is also interest-
ing to note that no knots exist in two-dimensional
space or four-dimensional space (meaning that any
knot in these spaces can be deformed into the unknot).
See also
BRAID
.
Koch curve See
FRACTAL
.
Kolmogorov, Andrey Nikolaevich (1903–1987) Rus-
sian Probability theory, Analysis, Dynamical systems
Born on April 25, 1903, in Tambov, Russia, scholar
Andrey Kolmogorov is remembered for his significant
contributions to the field of
PROBABILITY
theory and his
groundbreaking work in the study of
DYNAMICAL SYS
-
TEM
s. He also contributed to the study of F
OURIER
SERIES
,
ANALYSIS
, and
TOPOLOGY
. Applying his work to
planetary studies, Kolmogorov was also able to establish
the mathematical stability of our solar system.
Kolmogorov began his working career as a railway
conductor. In his spare time, however, he studied math-
ematics and physics, and managed to write a treatise
on S
IR
I
SAAC
N
EWTON
’s laws of mechanics. In 1920 he
entered Moscow State University. By the time he com-
pleted his undergraduate degree he had published eight
influential papers on the topics of
SET THEORY
and
analysis. Kolmogorov published another 10 papers
before receiving his doctoral degree in 1929.
Two years later Kolmogorov was appointed a pro-
fessor of mathematics at Moscow State University. At
this time, he began work on writing Grundbegriffe der
Wahrscheinlichkeitsrechnung (Foundations on the the-
ory of probability), in which he attempted, successfully,
to build up the entire theory of probability from a
finite set of axioms using nothing but logical rigor. Kol-
mogorov received national fame when the work was
published in 1933. The piece was translated into the
English language by Nathan Morrison in 1950.
In 1938 Kolmogorov was appointed department
head of probability and statistics at the newly estab-
lished Steklov Mathematical Institute in Russia. Soon
afterward Kolmogorov’s attention turned to the study
of turbulence, and in 1941 he wrote two important
papers on the nature of turbulent airflow from jet
engines, laying down the founding principles of the the-
ory of dynamical systems. He later applied this work to
the study of planetary motion.
Kolmogorov received many awards for his out-
standing work. In 1939 he was elected to the U.S.S.R.
Academy of Science, and over the following years, he
received eight prizes from the state and the academy.
He was also elected to a number of foreign academies,
including the Royal Statistical Society of London in
1956, the American Academy of Arts and Sciences in
1959, the Netherlands Academy of Sciences in 1963,
the R
OYAL
S
OCIETY
of London in 1964, and the French
Academy of Sciences in 1968. Kolmogorov was also
awarded the Balzan International Prize in 1962.
Kolmogorov died in Moscow, Russia, on October
20, 1987. His work paved the way for continued
research in the fields of Fourier analysis, “Markov
chains” in probability theory, and topological analysis.
Kronecker, Leopold (1823–1891) German Number
theory Born on December 7, 1823, in Liegitz, Prussia
(now Legnica, Poland), mathematician and philosopher
Leopold Kronecker is noted as a scholar for his consid-
erable contributions to the field of
NUMBER THEORY
. As
296 Koch curve