Figure-eight knot ($4_1$)

The figure-eight knot is the unique knot of crossing number of a knot four. It is prime, alternating, and has Alexander polynomial of a knot $-t+3-t^{-1}$ up to a unit.

The fundamental group of the figure-eight knot complement has the displayed two-generator one-relator presentation. It follows from the Artin automorphism of a braid for $({\sigma }_1{\sigma }_2^{-1}{)}^2$ by eliminating the third meridian from its fixed-generator presentation.