(Smooth and Topological) 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 Z2, Z4, 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 many 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. Additional references to more recent work by Jabuka and Naik, by Grigsby, Ruberman, and Strle, by Lisca, and by Levine, are given below.

References

E. Grigsby, D. Ruberman, S. Strle, Knot concordance and Heegaard Floer homology invariants in branched covers, math.GT/0701460.

S. Jabuka and S. Naik, Order in the concordance group and Heegaard Floer homology, math.GT/0611023.

A. Levine, On knots of infinite smooth concordance order, arxiv.org/abs/0805.2410.

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

P. Lisca, Sums of lens spaces bounding rational balls, arXiv:0705.1950.

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

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.

Further information on particular knots.