Concordance Order Calculator

This website will calculate both the algebraic and geometric concordance orders of any linear combination of prime knots with 9 or fewer crossings.

Input the coefficients of the knots you want in your linear combination.

Result

Details of calculation

Consider the free group generated by the 9-crossing prime knots. This has 87 generators and we use the knots listed in the table below as a basis. These knots have been grouped by their algebraic (A) and geometric (G) concordance orders, and each row of the table corresponds to generators which map to a complementary summand of the concordance group. For example, the knots in the first row correspond to a summand Z46 (in both the algebraic and geometric concordance groups), whilst the knots in the second row correspond to a summand (Z/4Z)2 in the algebraic concordance group.

There only possible relations that are not known among the listed generators could be among the five knots (92 - 74), (821 - 818 - 31), (923 - 92 - 31),( 940 - 818 - 41 - 31), and (932 - 9r32). It is known that the subgroup they generate contains and infinite cyclic subgroup and is of rank at most five.

Your knot Your knot written in terms of this new basis is:

These terms are highlighted in the table below, and the meaning of the different colours of highlighting is explained beneath the table.

To find the (algebraic or geometric) concordance order of your knot, take the maximum of the orders of the individual highlighted knots.

A = ∞, G = ∞ 31, 51, 52, 62, 71, 72, 73, 74, 75, 76, 82, 84, 85, 86, 87, 814, 816, 819, 91, 93, 94, 95, 96, 97, 99, 910,
911, 913, 915, 917, 918, 920, 921, 922, 925, 926, 931, 932, 935, 936, 938, 943, 945, 947, 948, 949
A = 4, G = ∞ 77, 934
A = 2, G = ∞ 81, 813, (815 - 72 - 31), (92 - 74), (912 - 52), 914, (916 - 73 - 31), 919, (928 - 31), 930, 933, (942 + 85 - 31), (944 - 41)
A = 1, G = ∞ (821 - 818 - 31), (98 - 814), (923 - 92 - 31), (929 - 928 +2(31)), (932r - 932), (933r - 933), (939 + 72 - 41), (940 - 818 - 41 - 31)
A = 2, G = 2 41, 63, 83, 812, 817, 818
A = 1, G = 2 (817r - 817)
A = 1, G = 1 61, 88, 89, (810 + 31), (811 - 31), 820, (924 - 41), 927, (937 - 41), 941, 946

Key: Yellow means read the order as given in the table. Green means divide the order by 4. Blue means divide the order by 2.
For example, if 77 is highlighted in green then it is of algebraic order 1; if it is highlighted in blue then it is of algebraic order 2; if it is highlighted in yellow then it is of algebraic order 4.