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,

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.

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.