self-linking number

{{Short description|Invariant of framed knots}}

In knot theory, the self-linking number is an invariant of framed knots. It is related to the linking number of curves.

A framing of a knot is a choice of a non-zero non-tangent vector at each point of the knot. More precisely, a framing is a choice of a non-zero section in the normal bundle of the knot, i.e. a (non-zero) normal vector field. Given a framed knot C, the self-linking number is defined to be the linking number of C with a new curve obtained by pushing points of C along the framing vectors.

Given a Seifert surface for a knot, the associated Seifert framing is obtained by taking a tangent vector to the surface pointing inwards and perpendicular to the knot. The self-linking number obtained from a Seifert framing is always zero.{{cite journal |last1=Sumners |first1=De Witt L. |last2=Cruz-White |first2=Irma I. |last3=Ricca |first3=Renzo L. |year=2021 |title=Zero helicity of Seifert framed defects |journal=J. Phys. A |volume=54 |issue=29 |page=295203 |doi=10.1088/1751-8121/abf45c |bibcode=2021JPhA...54C5203S |s2cid=233533506 }}

The blackboard framing of a knot is the framing where each of the vectors points in the vertical (z) direction. The self-linking number obtained from the blackboard framing is called the Kauffman self-linking number of the knot. This is not a knot invariant because it is only well-defined up to regular isotopy.