  Exceptional points (EPs) are peculiar band singularities and play a vital
role in a rich array of unusual optical phenomena and non-Hermitian band
theory. In this paper, we provide a topological classification of isolated EPs
based on homotopy theory. In particular, the classification indicates that an
$n$-th order EP in two dimensions is fully characterized by the braid group
B$_n$, with its eigenenergies tied up into a geometric knot along a closed path
enclosing the EP. The quantized discriminant invariant of the EP is the writhe
of the knot. The knot crossing number gives the number of bulk Fermi arcs
emanating from each EP. Furthermore, we put forward a non-Hermitian no-go
theorem, which governs the possible configurations of EPs and their splitting
rules on a two-dimensional lattice and goes beyond the previous fermion
doubling theorem. We present a simple algorithm generating the non-Hermitian
Hamiltonian with a prescribed knot. Our framework constitutes a systematic
topological classification of the EPs and paves the way towards exploring the
intriguing phenomena related to the enigmatic non-Hermitian band degeneracy.
