Arf Invariant
A Link invariant which always has the value 0 or 1. A Knot has Arf Invariant 0 if the Knot is``pass equivalent'' to the Unknot and 1 if it is pass equivalent to the Trefoil Knot. If 
, 
, and
are projections which are identical outside the region of the crossing diagram, and and are Knots while is a 2-component Link with a nonintersecting crossing diagram where the two left and right strandsbelong to the different Links, then
 | (1) |
is the Linking Number of 
and 
. The Arf invariant can be determined from the AlexanderPolynomial or Jones Polynomial for a Knot. For 
the Alexander Polynomial of 
, the Arfinvariant is given by | (2) |
of a Knot , | (3) |
References
Adams, C. C. The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. New York: W. H. Freeman, pp. 223-231, 1994. Jones, V. ``A Polynomial Invariant for Knots via von Neumann Algebras.'' Bull. Amer. Math. Soc. 12, 103-111, 1985.
Weisstein, E. W. ``Knots.'' Mathematica notebook Knots.m.
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