Oriented Knot

An oriented knot is an oriented link of one component, or equivalently, it is a knot which has been given an orientation. Given an oriented knot , reversing the orientation of  may give rise to an inequivalent knot.

Giving knots orientations are important to many applications of knot theory. Most importantly, providing orientations for knots allows for defining the sum of oriented knots simply by taking the connected sum of the knots regarded as oriented manifolds. Attempting to define a similar sum operation on non-oriented knots turns out not to be well-defined.

As another example, knot orientations are necessary for producing Seifert surfaces for knots via the Seifert algorithm, which quite explicitly uses the orientation.

REFERENCES:

Cerf, C. "Atlas of Oriented Knots and Links." Topology Atlas Invited Contributions 3, No. 2, 1-32, 1998. https://at.yorku.ca/t/a/i/c/31.htm.