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The arf invariant is a link invariant that 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.
Arf invariants are implemented in the Wolfram Language as KnotData[knot, "ArfInvariant"].
The numbers of prime knots on , 2, ... crossings having Arf invariants 0 and 1 are summarized in the table below.
OEIS | counts of prime knots with , 2, ... crossings | |
0 | A131433 | 0, 0, 0, 0, 1, 1, 3, 10, 25, 82, ... |
1 | A131434 | 0, 0, 1, 1, 1, 2, 4, 11, 24, 83, ... |
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 strands belong to the different links, then
(1) |
where is the linking number of and .
The Arf invariant can be determined from the Alexander polynomial or Jones polynomial for a knot. For the Alexander polynomial of , the Arf invariant is given by
(2) |
(Jones 1985). Here, the factor takes care of the ambiguity introduced by the fact that the Alexander polynomial is defined only up to multiples of . (Alternately, this indeterminacy is also taken care of by the Conway definition of the polynomial.)
For the Jones polynomial of a knot ,
(3) |
(Jones 1985), where i is the imaginary number.
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.
Sloane, N. J. A. Sequences A131433 and A131434 in "The On-Line Encyclopedia of Integer Sequences."
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