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 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 of 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 that others are either negative amphicheiral or concordant to negative amphicheiral knots, and thus are of order 2.

In the table, we await a complete calculation of concordance order for higher crossing knots. We have used the following criteria so far: A knot is 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, if there is a prime factor of the determinant that equals 3 mod 4 and has an odd exponent, then the algebraic order is 4.

Specific Knots

In the smooth category, Jabuka and Naik showed, using Heegaard Floer homology, that many knots of algebraic concordance order 4 are of order at least 6 in the smooth concordance group.

813, 914, 919, 1010, 1013, 1026, 1028, 1034, 11a13, 11a84, 11a91, 11a93, 11a110, 11a119, 11a175, 11a180, 11a185, 11a190, 11a195, 11a210, 11a333, 12a0169, 12a0197, 12a0204, 12a0243, 12a0303, 12a0307, 12a0380, 12a0385, 12a0437, 12a0482, 12a0497, 12a0518, 12a0532, 12a0550, 12a0579, 12a0583, 12a0585, 12a0596, 12a0644, 12a0650, 12a0690, 12a0691, 12a0728, 12a0736, 12a0744, 12a0758, 12a0774, 12a0792, 12a0803, 12a1024, 12a1129, 12a1130, 12a1139, 12a1140, 12a1146, 12a1148, 12a1166
   Using related methods, Grigsby, Ruberman and Strle showed that these two-bridge knots are of infinite order.

Lisca has completed the determination of the smooth concordance orders of 2-bridge knots.

930, 933, 944, 1058, 1060, 10102, 10119, 10135, 11a4, 11a8, 11a11, 11a24, 11a26, 11a30, 11a52, 11a56, 11a67, 11a76, 11a80, 11a88, 11a126, 11a160, 11a167, 11a170, 11a189, 11a233, 11a249, 11a257, 11a265, 11a270, 11a272, 11a287, 11a288, 11a289, 11a300, 11a303, 11a315, 11a350, 11n12, 11n48, 11n53, 11n55, 11n110, 11n114, 11n130, 11n165
   Adam Levine has shown that these knots have infinite smooth concordance order, building on the work of Grigsby, Ruberman, and Strle.

1091, 12a1199, 12a1222, 12a1231, and 12a1258 Paolo Lisca has determined that these knots have infinite concordance order.

Julia Collins, in her thesis [1], stated the following, which we quote in full. Since then, it was shown that 12a621 is slice.

Theorem. Of the prime knots of 12 or fewer crossings listed as having unknown concordance order, they are all of infinite order with the exception of the following: * 11n34 is slice because it has Alexander polynomial equal to 1. * 12a1288 is of order 2 because it is fully amphicheiral. * 11a5, 11a104, 11a112, 11a168, 11n85, 11n100, 12a309, 12a310, 12a387, 12a388, 12n286 and 12n388 are all of order 2, concordant to the Figure Eight knot 4a1. * 11a44, 11a47 and 11a109 are all of order 2, concordant to the knot 63. * 12a631 remains of unknown order, but is suspected to be finite order and possibly slice. * 12n846 remains of unknown order, and there are no suspicions as to whether it is of finite or infinite order.

References

[1] Collins, J., "On the concordance orders of knots." Arxiv preprint.

[2] Grigsby, E., Ruberman, D., and Strle, S., "Knot concordance and Heegaard Floer homology invariants in branched covers" Arxiv preprint.

[3] Jabuka, S. and Naik, S., "Order in the concordance group and Heegaard Floer homology." Arxiv preprint.

[4] Levine, A., "On knots with infinite smooth concordance order." Arxiv preprint.

[5] Lisca, P., "Sums of lens spaces bounding rational balls." Arxiv preprint.

[6] Lisca, P., "On 3-braid knots of finite concordance order." Arxiv preprint.

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

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

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

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