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. Using related methods, Grigsby, Ruberman and Strle showed many two-bridge knots are of infinite order.

The 2-bridge knots shown to have infinite order by GRS are the following:

8_13, 9_14, 9_19, 10_10, 10_13, 10_26, 10_28, 10_34, 11a_13, 11a_84, 11a_91, 11a_93, 11a_110, 11a_119, 11a_175, 11a_180, 11a_185, 11a_190, 11a_195, 11a_210, 11a_333, 12a_0169, 12a_0197, 12a_0204, 12a_0243, 12a_0303, 12a_0307, 12a_0380, 12a_0385, 12a_0437, 12a_0482, 12a_0497, 12a_0518, 12a_0532, 12a_0550, 12a_0579, 12a_0583, 12a_0585, 12a_0596, 12a_0644, 12a_0650, 12a_0690, 12a_0691, 12a_0728, 12a_0736, 12a_0744, 12a_0758, 12a_0774, 12a_0792, 12a_0803, 12a_1024, 12a_1129, 12a_1130, 12a_1139, 12a_1140, 12a_1146, 12a_1148, 12a_1166

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

Adam Levine has shown the following knots have infinite smooth concordance order, building on the work of Grigsby, Ruberman, and Strle:

9_30,
9_33,
9_44,
10_58,
10_60,
10_102,
10_119,
10_135,
11a_4,
11a_8,
11a_11,
11a_24,
11a_26,
11a_30,
11a_52,
11a_56,
11a_67,
11a_76,
11a_80,
11a_88,
11a_126,
11a_160,
11a_167,
11a_170,
11a_189,
11a_233,
11a_249,
11a_257,
11a_265,
11a_270,
11a_272,
11a_287,
11a_288,
11a_289,
11a_300,
11a_303,
11a_315,
11a_350,
11n_12,
11n_48,
11n_53,
11n_55,
11n_110,
11n_114,
11n_130,
11n_165,

Paolo Lisca has determined that the 10_91, 12_a1199, 12_a1222, 12_a1231, and 12_a1258, have infinite concordance order.

Julia Collins, in her thesis "On the concordance orders of knots," arxiv.org:1206.0669, stated the following, which we quote in full. Since then, it was shown that 12a_621 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.

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.

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

P. Lisca, "On 3-braid knots of finite concordance order," to appear.

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

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

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