(Algebraic) Concordance Order


Every knot represents an element in the concordance group, a countably generated abelian group. The order of that element is called the concordance order of the knot.

Levine defined a homomorphism of the concordance group onto an algebraically defined group, isomorphic to the countably infinite direct sum of (an infinite number of) copies of Z_2, Z_4, and Z. The algebraic order of algebraic concordance order of a knot is the order of the image in Levine's alebraic concordance group.

Techniques for the computation of the orders of elements in the algebraic concordance group appear in a paper by Toshiyuki Morita. Livingston and Naik have shown that many knots of algebraic order 4 are infinite order in the concordance group. Andrius Tamulis proved that may knots of algebraic order 2 are of higher order in the concordance group, and proved that others are either negative amphicheiral, or concordant to negative amphicheiral knots, and thus are of order 2.

In the table, for higher crossing knots we await a complete calculation of concordance order. We have used the following criteria so far. A knot if of infinite order algebraically if and only if its signature function is nonzero. If the signature function is 0 and the determinant is a square, the order is 2 or 4. If, in addition, there is a prime factor of the determinant that equals 3 mod 4 and has odd exponent exponent , then the algebraic order 4.
References

T. Morita, "Orders of knots in the algebraic knot cobordism group," Osaka J. Math. 25 (1988), 859-864.

A. Tamulis, " Knots of ten or fewer crossings of algebraic order 2," J. Knot Theory Ramifications 11 (2002), no. 2, 211--222.

C.  Livingston and S. Naik, "Knot concordance and torsion," Asian J. Math. 5 (2001), no. 1, 161--167.

C. Livingstonand S. Naik, "Obstructing four-torsion in the classical knot concordance group," J. Differential Geom. 51 (1999), no. 1, 1--12.