  Historically originated as a sub-field of topology, knot theory is an active
area of mathematical investigation that has strong connections with a diverse
set of scientific fields such as algebra, biology, and statistical mechanics. A
popular and important concept in linear algebra, Gram determinants enjoy a
connection with the mathematical theory of knots. In this article, we expose
this concept and present several types of Gram determinants in what can be
considered as a survey of the current Gram determinants of interest to knot
theorists; examples are included to illustrate the definitions. In particular,
we pay special attention to a recently defined determinant from a M\"obius band
and we further study its structure. At the end, some speculation is presented
regarding the closed formula for the Gram determinant of type $(Mb)_1$, a
problem that arouses serious interest among knot theorists.
