On calculating the slice genera of 11 and 12crossing knots
Abstract.
This paper contains the results of efforts to determine values of the smooth and the topological slice genus of 11 and 12crossing knots. Upper bounds for these genera were produced by using a computer to search for genus one concordances between knots. For the topological slice genus further upper bounds were produced using the algebraic genus. Lower bounds were obtained using a new obstruction from the Seifert form and by use of Donaldson’s diagonalization theorem. These results complete the computation of the topological slice genera for all knots with at most 11 crossings and leaves the smooth genera unknown for only two 11crossing knots. For 12 crossings there remain merely 25 knots whose smooth or topological slice genus is unknown.
1. Introduction
This paper contains the results of efforts to determine unknown^{1}^{1}1Unknown = ‘listed as unknown on KnotInfo [3] at the time of writing’. values of the smooth and topological slice genus for 11 and 12crossing knots. In order to determine the slice genus of a knot one needs to produce an upper bound, typically by exhibiting a surface cobounding the knot, and then establishing a lower bound which shows that surface has optimal genus. We apply a variety of methods to produce new upper and lower bounds.
Our lower bounds come from two sources. The first is an invariant due to Taylor, which is a lower bound for the topological slice genus. Though is determined by the Seifert form, it is generally difficult to compute. In order to apply it, we deduce from a new, computable obstruction to by applying the theory of quadratic forms over the adic numbers to the symmetrization of the Seifert form. Our other source of lower bounds is Donaldson’s diagonalization theorem, from which we obtain an obstruction to the smooth slice genus being equal to the signature bound, i.e. for alternating knots.
The majority of our upper bounds arise from a computer search to find knots related by crossing changes and crossing resolutions and using that the slice genus (smooth or topological) of knots related by a concordance of genus one differs by at most one. In other cases, we obtained upper bounds for by computing the recently introduced algebraic genus. Similar to Taylor’s invariant, the algebraic genus is an invariant depending only on the Seifert form. That it is an upper bound is a consequence of Freedman’s disk theorem.
Altogether these methods allow us to complete the calculation of for 11crossing knots. In total, for knots with up to 12 crossings, there remain 22 unknown values for the smooth slice genus, and 7 unknown values for the topological slice genus.
Table 1 summarizes the knots with crossing number at most 12 for which the topological and smooth slice genera are not known and their possible values. We hope that this paper will be helpful in drawing attention to these remaining unknown values, which may well require new and more interesting techniques to determine.
Each of the four following sections is devoted to one of the slice genus bounds we used. First, we looked for genus one concordances (Section 5); then, we applied the more sophisticated tools to the remaining unknown genera: the obstruction from Taylor’s bound (Section 2), the obstruction from Donaldson’s theorem (Section 3), and the upper bound from the algebraic genus (Section 4). Details about calculations are contained in the appendices.
Knot  
0 or 1  0  
1 or 2  1  
1 or 2  1  
1 or 2  1  
1 or 2  1  
2  1 or 2  
1 or 2  1  
1 or 2  1  
1 or 2  1  
1 or 2  1  
1 or 2  1  
2  1 or 2  
1 or 2  1 or 2  
2  1 or 2 
Knot  
1 or 2  1  
1 or 2  1  
1 or 2  1  
1 or 2  1  
1 or 2  1  
1 or 2  1  
1 or 2  1  
2  1 or 2  
1 or 2  1 or 2  
1 or 2  1  
2  1 or 2  
1 or 2  1  
1 or 2  1 
2. Taylor’s lower bound
The Seifert form of a knot yields several wellknown lower bounds to the topological slice genus, namely the bounds coming from the LevineTristram signatures and the FoxMilnor condition. All of these lower bounds are subsumed by Taylor’s lower bound [21]: let be the maximal rank of an isotropic subgroup of , i.e. a subgroup on which is identically zero. Taylor’s bound is then the following:
Since this bound has previously only been explicitly stated in the literature as a bound on the smooth slice genus, we briefly indicate why this is true. The key ingredient is the existence of the higherdimensional locally flat analogue of Seifert surfaces; more precisely, given a locally flat surface , the existence of a locally flat embedded compact oriented 3manifold with boundary . Following [20, p. XXI], such an may be constructed as follows: let denote an open tubular neighborhood of . We want to extend the projection onto the first factor to a function . Then, because topological transversality holds (cf. [8, Ch. 9] and also [14, Essay III, §1]), there is a homotopy making topologically transverse to . Thus is a locally flat 3manifold with equal to a pushoff of . To obtain , note that by Alexander duality and being a ), we have
So one may take to be a generator for .
Given this key ingredient, Taylor’s bound can be obtained by a minor generalization to the proof that a topologically slice knot is algebraically slice (see e.g. [17, Ch. 8]). Indeed, given a Seifert surface with Seifert form and a properly embedded locally flat surface cobounding we can form the closed embedded locally flat surface . Take to be an embedded 3manifold with boundary , and let be the inclusion. Standard homological and linear algebra arguments show that the kernel of is an isotropic subgroup of of required rank.
Taylor’s bound is much less wellknown than the signatures and the FoxMilnor condition, most likely because of two reasons: no algorithm to calculate has been produced so far, and it seems unlikely that will be much stronger than all the LevineTristram signatures taken together. Still, we exhibit the following computable lower bound to the slice genus coming from the Seifert form, which has to the best knowledge of the authors not been stated in the literature before, although it was implicitly used by the first author in [2]. This bound arises from Taylor’s bound by a straightforward application of the theory of quadratic forms, for which we refer to any of the standard textbooks such as the one by Lam [16].
Theorem 1.
Let be a knot with a –dimensional Seifert form . Denote by the integral quadratic form given by . Denote by the discriminant of (note ). If there is an odd prime such that the two following, equivalent conditions are satisfied, then :

There is a nonnegative integer and an integer such that , and the Hasse symbol of over is .

The form induced by over is Wittequivalent to an anisotropic fourdimensional form.
Proof.
An isotropic subgroup of rank of gives rise to an isotropic subspace of dimension of . If (ii) is satisfied, such a subspace has dimension at most . Thus . The equivalence of the conditions (i) and (ii) follows from the wellknown fact that the Wittclass of a quadratic form over a local field is determined by its discriminant and Hasse symbol. ∎
We used scripts [15] written for PARI/GP [19] to test for which knots this lower bound would be applicable, finding the six knots:
, , , , , . 
For these knots, the topological slice genus is at least , and, by the upper bounds already known, in fact equal to . For all of them, one takes the odd prime in Theorem 1 to be . The first two knots have discriminant , the last four discriminant .
Taylor’s invariant is not just a lower bound for the slice genus, but may be considered as knot invariant in its own right, with further noteworthy properties; e.g., is an algebraic concordance invariant and equals the minimal slice genus among knots with Seifert form . As a side effect of our efforts to compute slice genera, we have also computed for all but five knots with up to 12 crossings. Note that is bounded below by the LevineTristram signatures, the FoxMilnor condition and the bound from Theorem 1.
Proposition 2.
For all prime knots with up to 10 crossings, one has
(2.1) 
For 11 and 12 crossings, there are 16 exceptions, and 7 potential exceptions to Equation 2.1:

The following 16 knots are algebraically slice, i.e. , but their topological sliceness has been obstructed by twisted Alexander polynomials [11] (in fact, they all have topological slice genus equal to ):
, , , , , , , ,
, , , , , , , .

We were unable to determine the Taylor invariant of the two knots
For both of them, . For these knots, it is not known whether the topological slice genus is or .

The Taylor invariant of the five knots
, , , ,
can be shown to be by explicitly exhibiting an isotropic subgroup of the appropriate rank (cf. Appendix B). For these knots, it is not known whether the topological slice genus is or .
3. An obstruction from Donaldson’s theorem
Now we state our obstruction from Donaldson’s theorem. This obstruction has previously been applied to help find 2bridge knots with differing smooth and topological slice genera [7].
Lemma 3.
Let be a knot with with a positivedefinite Goeritz matrix . If and , then there is an integer matrix , such that .
Proof.
The double branched cover bounds a 4manifold with intersection form given by the matrix [10]. It also bounds a smooth 4manifold with and signature [10, 13]. Suppose that . This implies that the closed smooth 4manifold is positive definite. Therefore, has intersection form isomorphic to the diagonal lattice [4]. The inclusion induces an injection and hence an embedding of intersection forms. This gives the desired matrix factorization. ∎
Using this lemma, we find that the following alternating knots cannot have smooth slice genus one, and thus all have smooth slice genus two. In each case, GAP’s command OrthogonalEmbeddings was used to find minimal dimension matrix factorizations [9].
11a211, 12a244, 12a255, 12a414, 12a534, 12a542, 12a719, 12a810, 12a908, 12a1118, 12a1142, 12a1185 
4. The algebraic genus
The algebraic genus is a knot invariant determined by the Sequivalence class of Seifert forms of a knot , giving an upper bound for . It was recently introduced by Feller and the first author [6]; we refer to that paper for a detailed treatment, and only briefly state a definition and some properties of here. A subgroup is called Alexandertrivial if is a unit in . Let be the maximal rank of an Alexandertrivial subgroup. Then the algebraic genus of is defined as
and the algebraic genus of is defined as the minimum algebraic genus of a Seifert form of . At the moment, no way is known to compute for a general knot. However, a randomized algorithm as in [1], implemented in PARI/GP [19], gives good upper bounds for , and thus for . The upper bound for given by subsumes the bound coming from the algebraic unknotting number [18]:
We found 19 knots for which the bound given by
is not strong enough, but the algebraic genus determines the previously unknown topological slice genus.
Appendix C lists a Seifert matrix and a basis for an Alexandertrivial subgroup for each of those knots.
for .  

for {187, 230, 317, 450, 542, 570, 908, 1118, 1185, 1189, 1208}.  
for {52, 63, 225, 558, 665, 886}.  
for . 
Note that the minimum of the algebraic genus and the smooth slice genus is a very efficient upper bound for the topological slice genus of small knots:
Proposition 4.
For all prime knots with up to 11 crossings, one has
(4.1) 
Indeed, Equation 4.1 even holds for all prime knots with up to 12 crossings with the potential exceptions of the 7 knots for which is unknown (see Table 1). Note that Equation 4.1 need not hold for higher crossing numbers. For example, the knot in Figure 1 is topologically but not smoothly slice (it is concordant to the –pretzel knot) and has nontrivial Alexander polynomial (and thus nonzero algebraic genus).
Just as the Taylor invariant, the algebraic genus may be considered as a knot invariant of independent interest. We were able to determine the algebraic
genus for all knots with up to 12 crossings, except for the following 54 knots, all of which
have Alexander polynomial not equal to , but algebraic unknotting number equal to , which implies :
, , , , , , , , ,  

for  {164, 166, 177, 244, 265, 298, 396, 413, 493, 503, 735, 769, 810, 873, 895, 905, 1013, 1047, 1142, 1168, 1203, 1211, 1221, 1222, 1225, 1226, 1229, 1230, 1248, 1260, 1283, 1288} 
for  {334, 379, 388, 460, 480, 495, 549, 583, 737, 813, 846, 869} 
For all other 2923 prime knots with up 12 crossings, the algebraic genus equals the maximum of two of its lower bounds: the Taylor invariant and . A sharp upper bound for the algebraic genus is given in 2341 cases by the algebraic unknotting number, i.e. ; in the other 582 cases, we explicitly found an Alexandertrivial subgroup of sufficient rank (for those knots, ). To avoid overly bloating the appendix, bases for those subgroups are included in a separate text file with the arXivversion of this article.
5. Genus one concordances
Lemma 5.
Let and be knots in . If can be obtained from by one of the following operations:

changing a single crossing;

changing a positive and a negative crossing; or

taking oriented resolutions of two crossings,
then and .
Proof (sketch)..
In all three cases can be obtained from by adjoining two oriented bands. This shows that in each case, there is a smoothly embedded twicepunctured torus with . Since we can glue to any properly embedded (smooth or locally flat) surface with boundary or , we see that the desired inequalities hold. ∎
The above lemma was used to generate upper bounds for the slice genus as follows. For a given knot , we take a diagram and obtain a new diagram by performing one of the three given operations. By considering the Jones polynomial and the crossing number of , we obtain a small set of possibilities, , for the knot represented by . This gives the following upper bound on the smooth slice genus:
(5.1) 
Remark.
In practice, we found that the combination of the Jones polynomials and the bound on crossing number was often sufficient to identify exactly. However, even when the set contains more than one knot, we often found that all the knots in had the same smooth slice genus.
Using a computer, we applied the above method to each knot where the smooth slice genus was not listed on KnotInfo [3]. Comparing the resulting upper bounds with previously known lower bounds allowed to deduce the exact value for the knots listed below. In Appendix A, we state the operation performed and the resulting set which gave the desired upper bound for each knot.
5.1. The smooth slice genus of 11 and 12crossing knots
Using the methods outlined above, the value of the smooth genus can be determined for the knots listed in LABEL:table:g1. In all cases, this means that the smooth genus is equal to the previously known lower bound. Although it does not determine the value completely, we also obtain a new upper bound for . This is done by observing can be transformed into , which has , by a crossing change.
5.2. The topological slice genus of 10crossing knots
The only 10crossing knots for which the topological genera are not known are and . We can determine for these by obtaining them by a crossing change from knots for which the topological genus is already known.
We can obtain by performing a single crossing change in a 12crossing diagram for . Since and , this implies that .  
We can obtain by performing a single crossing change in a 12crossing diagram for . Since and , this implies that . 
Note that this uses the topological genera of and which were recently determined by Feller [5].
Acknowledgments
The second author would like to thank his supervisor, Brendan Owens, for his continued guidance. Both authors would like to thank Peter Feller for the conversations that led to the undertaking of this project.
References
 [1] S. Baader, P. Feller, L. Lewark, and L. Liechti. On the topological 4genus of torus knots. Accepted for publication by Trans. Amer. Math. Soc., arXiv:1509.07634, 2015.
 [2] S. Baader and L. Lewark. The stable 4genus of alternating knots. Accepted for publication by Asian J. Math., arXiv:1505.03345, 2015.
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 [7] P. Feller and D. McCoy. On 2bridge knots with differing smooth and topological slice genera. Proc. Amer. Math. Soc., 144(12):5435–5442, 2016.
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 [10] C. McA. Gordon and R. A. Litherland. On the signature of a link. Invent. Math., 47(1):53–69, 1978.
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 [18] H. Murakami. Algebraic unknotting operation. In Proceedings of the Second SovietJapan Joint Symposium of Topology (Khabarovsk, 1989), volume 8 of Questions Answers Gen. Topology, pages 283–292, 1990.
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Appendix A
For each knot in LABEL:table:g1, we state an operation from Lemma 5 along with the resulting family , which can be used to give the required bound on . In all cases we used only the diagram corresponding to the PD notation listed on KnotInfo [3]. In this table,

‘cc’=‘crossing change’,

‘+cc’=‘a positive and a negative crossing change’

and ‘res’=‘oriented resolutions oftwo crossings’.
Knot  Operation  

11a1  +cc  [U] 
11a3  +cc  [U] 
11a4  +cc  [U] 
11a6  +cc  [U] 
11a7  +cc  [U] 
11a13  +cc  [6_1] 
11a14  +cc  [3_1#(3_1)] 
11a15  +cc  [3_1#(3_1)] 
11a16  +cc  [6_1] 
11a17  +cc  [6_1] 
11a18  cc  [8_8] 
11a19  +cc  [U] 
11a20  +cc  [3_1] 
11a21  +cc  [U] 
11a23  +cc  [U] 
11a24  +cc  [6_1] 
11a25  +cc  [U] 
11a26  +cc  [6_1] 
11a27  +cc  [U] 
11a29  +cc  [6_1] 
11a30  +cc  [U] 
11a32  +cc  [U] 
11a33  +cc  [6_1] 
11a37  +cc  [6_1] 
11a38  +cc  [6_1] 
11a39  +cc  [6_1] 
11a42  +cc  [U] 
11a44  +cc  [U] 
11a45  +cc  [6_1] 
11a46  +cc  [U] 
11a47  +cc  [U] 
11a49  +cc  [5_2] 
11a50  +cc  [U] 
11a51  cc  [8_9, 4_1#4_1] 
11a52  +cc  [U] 
11a53  +cc  [3_1] 
11a54  +cc  [U] 
11a55  +cc  [U] 
11a57  +cc  [U] 
11a59  +cc  [6_1] 
11a60  +cc  [5_2] 
11a61  res  [6_1] 
11a63  +cc  [5_2] 
11a64  +cc  [3_1] 
11a65  +cc  [U] 
11a66  +cc  [U] 
11a67  +cc  [U] 
11a68  +cc  [U] 
11a72  +cc  [3_1#(3_1)] 
11a75  +cc  [6_1] 
11a76  +cc  [U] 
11a79  +cc  [U] 
11a81  +cc  [U] 
11a83  +cc  [3_1] 
11a84  +cc  [6_1] 
11a85  +cc  [U] 
11a89  +cc  [6_1] 
11a90  +cc  [U] 
11a92  +cc  [U] 
11a93  +cc  [U] 
11a97  +cc  [6_1] 
11a99  +cc  [6_1] 
11a102  +cc  [6_1] 
11a105  +cc  [3_1] 
11a107  +cc  [U] 
11a108  +cc  [U] 
11a109  +cc  [3_1#(3_1)] 
11a110  +cc  [6_1] 
11a111  +cc  [U] 
11a118  +cc  [U] 
11a119  +cc  [6_1] 
11a125  +cc  [U] 
11a126  +cc  [3_1#(3_1)] 
11a128  +cc  [6_1] 
11a130  +cc  [U] 
11a131  +cc  [U] 
11a132  +cc  [U] 
11a133  +cc  [U] 
11a134  cc  [8_8] 
11a135  +cc  [6_1] 
11a137  +cc  [6_1] 
11a141  +cc  [6_1] 
11a144  +cc  [7_2] 
11a145  res  [6_1] 
11a147  +cc  [U] 
11a148  res  [6_1] 
11a151  +cc  [8_20] 
11a152  +cc  [U] 
11a153  +cc  [U] 
11a154  cc  [6_1] 
11a155  cc  [8_20] 
11a156  +cc  [8_20] 
11a157  +cc  [U] 
11a158  +cc  [U] 
11a159  +cc  [U] 
11a161  +cc  [7_6] 
11a162  +cc  [U] 
11a163  +cc  [3_1#(3_1)] 
11a166  +cc  [U] 
11a170  +cc  [U] 
11a171  +cc  [U] 
11a172  +cc  [U] 
11a173  cc  [8_20] 
11a174  +cc  [U] 
11a175  +cc  [U] 
11a176  +cc  [U] 
11a178  +cc  [U] 
11a181  +cc  [6_1] 
11a183  +cc  [U] 
11a185  +cc  [U] 
11a188  +cc  [6_1] 
11a193  +cc  [U] 
11a197  cc  [8_8] 
11a199  +cc  [6_1] 
11a202  +cc  [6_1] 
11a205  +cc  [U] 
11a214  +cc  [6_1] 
11a217  +cc  [U] 
11a218  +cc  [U] 
11a219  res  [6_1] 
11a221  +cc  [8_20] 
11a228  +cc  [U] 
11a229  +cc  [U] 
11a231  +cc  [U] 
11a232  +cc  [U] 
11a239  +cc  [U] 
11a248  +cc  [U] 
11a249  +cc  [U] 
11a251  +cc  [3_1#(3_1)] 
11a252  +cc  [3_1#(3_1)] 
11a253  +cc  [6_1] 
11a254  +cc  [3_1#(3_1)] 
11a258  +cc  [6_1] 
11a262  +cc  [U] 
11a265  +cc  [U] 
11a268  +cc  [U] 
11a269  +cc  [U] 
11a270  +cc  [U] 
11a271  +cc  [U] 
11a273  +cc  [U] 
11a274  +cc  [U] 
11a277  +cc  [U] 
11a278  +cc  [6_1] 
11a279  +cc  [U] 
11a281  +cc  [6_1] 
11a284  +cc  [U] 
11a285  +cc  [U] 
11a288  +cc  [U] 
11a293  +cc  [5_2] 
11a294  +cc  [U] 
11a296  res  [6_1] 
11a297  +cc  [U] 
11a301  +cc  [U] 
11a303  +cc  [U] 
11a304  +cc  [5_2] 
11a305  +cc  [U] 
11a312  res  [3_1#(3_1)] 
11a313  +cc  [U] 
11a314  +cc  [U] 
11a315  +cc  [U] 
11a317  +cc  [U] 
11a322  +cc  [U] 
11a323  +cc  [6_1] 
11a324  res  [6_1] 
11a325  +cc  [U] 
11a327  cc  [8_20] 
11a331  +cc  [U] 
11a332  +cc  [U] 
11a333  +cc  [U] 
11a346  +cc  [3_1] 
11a347  +cc  [U] 
11a349  +cc  [U] 
11a350  +cc  [6_1] 
11a352  cc  [6_1] 
11n3  +cc  [U] 
11n5  +cc  [U] 
11n6  +cc  [U] 
11n7  +cc  [U] 
11n11  +cc  [U] 
11n15  +cc  [U] 
11n17  res  [U] 
11n23  +cc  [5_2] 
11n24  +cc  [U] 
11n29  +cc  [U] 
11n30  +cc  [5_2] 
11n32  +cc  [6_1] 
11n33  +cc  [6_1] 
11n36  +cc  [U] 
11n40  res  [U] 
11n44  +cc  [U] 
11n46  res  [U] 
11n51  +cc  [U] 
11n54  res  [U] 
11n58  +cc  [3_1#(3_1)] 
11n60  +cc  [U] 
11n65  cc  [3_1#(3_1)] 
11n66  +cc  [U] 
11n79  +cc  [6_1] 
11n91  res  [U] 
11n92  +cc  [U] 
11n94  +cc  [U] 
11n98  cc  [3_1#(3_1)] 
11n99  res  [U] 
11n102  +cc  [U] 
11n112  +cc  [U] 
11n113  res  [U] 
11n115  +cc  [U] 
11n117  +cc  [6_1] 
11n119  +cc  [6_1] 
11n120  +cc  [U] 
11n127  res  [U] 
11n128  +cc  [6_1] 
11n129  +cc  [U] 
11n133  +cc  [3_1] 
11n137  res  [6_2] 
11n138  +cc  [6_1] 
11n140  res  [6_1] 
11n142  +cc  [6_1] 
11n146  res  [U] 
11n148  res  [8_9, 4_1#4_1] 
11n150  +cc  [U] 
11n155  +cc  [6_1] 
11n157  +cc  [U] 
11n160  +cc  [U] 
11n161  +cc  [U] 
11n162  res  [U] 
11n163  +cc  [U] 
11n165  +cc  [U] 
11n166  +cc  [U] 
11n167  res  [6_1] 
11n168  +cc  [U] 
11n170  res  [6_1] 
11n173  +cc  [5_2] 
11n177  +cc  [U] 
11n178  res  [U] 
11n179  +cc  [U] 
11n182  +cc  [U] 
12a4  +cc  [6_1] 
12a10  +cc  [6_1] 
12a39  +cc  [8_20] 
12a45  +cc  [3_1#(3_1)] 
12a49  res  [8_8] 
12a50  +cc  [8_20] 
12a65  +cc  [3_1#(3_1)] 
12a66  +cc  [3_1#(3_1)] 
12a75  +cc  [7_4] 
12a76  res  [6_1] 
12a86  res  [8_8] 
12a89  cc  [8_8] 
12a103  +cc  [6_1] 
12a104  cc  [8_8] 
12a108  +cc  [8_20] 
12a120  +cc  [8_20] 
12a125  +cc  [6_1] 
12a127  +cc  [6_1] 
12a128  +cc  [6_1] 
12a129  +cc  [3_1#(3_1)] 
12a135  +cc  [6_1] 
12a147  +cc  [7_4] 
12a148  +cc  [7_4] 
12a150  res  [8_8] 
12a160  cc  [10_77] 
12a161  res  [8_8] 
12a163  res  [8_8] 
12a164  +cc  [8_20] 
12a166  +cc  [8_20] 
12a167  cc  [3_1#(3_1)#3_1] 
12a168  cc  [3_1#(3_1)] 
12a175  +cc  [6_1] 
12a177  +cc  [6_1] 
12a178  +cc  [6_1] 
12a181  +cc  [6_1] 
12a193  +cc  [8_6] 
12a194  res  [10_22, 10_35] 
12a195  +cc  [8_6] 
12a196  cc  [10_22, 10_35] 
12a200  +cc  [9_46] 
12a204  +cc  [6_1] 
12a212  +cc  [8_20] 
12a231  +cc  [7_4] 
12a247  cc  [10_129, 8_8] 
12a259  +cc  [6_1] 
12a260  +cc  [3_1#(3_1)] 
12a265  +cc  [6_1] 
12a289  +cc  [7_4] 
12a291  +cc  [6_1] 
12a292  +cc  [8_20] 
12a296  +cc  [6_1] 
12a298  +cc  [8_20] 
12a302  cc  [8_8] 
12a311  +cc  [7_4] 
12a312  res  [6_1] 
12a327  cc  [8_8] 
12a338  +cc  [6_1] 
12a339  res  [6_1] 
12a342  +cc  [6_1] 
12a353  cc  [8_8] 
12a354  res  [8_8] 
12a357  +cc  [6_1] 
12a364  +cc  [8_20] 
12a370  +cc  [7_4] 
12a372  res  [8_8] 
12a375  res  [3_1#(3_1)#3_1] 
12a376  res  [3_1#(3_1)] 
12a379  res  [3_1#(3_1)] 
12a380  cc  [10_129, 8_8] 
12a381  +cc  [6_1] 
12a395  res  [8_8] 
12a396  +cc  [8_20] 
12a399  res  [8_8] 
12a400  cc  [9_41] 
12a413  +cc  [8_20] 
12a423  res  [8_8] 
12a424  +cc  [3_1#(3_1)] 
12a434  +cc  [8_20] 
12a436  +cc  [3_1#(3_1)] 
12a438  +cc  [6_1] 
12a448  +cc  [6_1] 
12a449  +cc  [6_1] 
12a454  +cc  [6_1] 
12a459  cc  [9_46] 
12a462  +cc  [3_1#(3_1)] 
12a463  +cc  [6_1] 
12a465  +cc  [6_1] 
12a468  +cc  [8_20] 
12a481  cc  [11n49] 
12a482  +cc  [6_1] 
12a489  +cc  [6_1] 
12a493  +cc  [6_1] 
12a494  +cc  [6_1] 
12a496  cc  [10_129, 8_8] 
12a503  cc  [10_75] 
12a505  res  [8_9, 4_1#4_1] 
12a544  +cc  [3_1#(3_1)] 
12a545  cc  [8_8] 
12a549  +cc  [6_1] 
12a554  +cc  [6_1] 
12a564  res  [6_1] 
12a580  cc  [10_12] 
12a581  res  [3_1#(3_1)] 
12a582  res  [3_1#(3_1)] 
12a597  +cc  [6_1] 
12a598  +cc  [8_20] 
12a601  res  [6_1] 
12a609  res  [8_8] 
12a621  +cc  [8_20] 
12a634  res  [3_1#(3_1)] 
12a639  cc  [10_87] 
12a642  +cc  [8_20] 
12a643  +cc  [10_129, 8_8] 
12a644  +cc  [10_129, 8_8] 
12a649  +cc  [10_129, 8_8] 
12a665  +cc  [6_1] 
12a668  +cc  [3_1#(3_1)] 
12a669  +cc  [10_129, 8_8] 
12a677  cc  [8_8] 
12a680  cc  [10_87] 
12a684  +cc  [6_1] 
12a687  +cc  [6_1] 
12a689  +cc  [6_1] 
12a690  +cc  [6_1] 
12a691  +cc  [6_1] 
12a692  cc  [3_1#(3_1)#3_1] 
12a693  +cc  [8_6] 
12a704  +cc  [10_129, 8_8] 
12a706  +cc  [3_1#(3_1)] 
12a725  res  [6_2] 
12a730  +cc  [7_4] 
12a735  res  [6_1] 
12a741  +cc  [7_4] 
12a749  res  [3_1#(3_1)] 
12a750  res  [6_1] 
12a752  +cc  [6_1] 
12a757  cc  [8_20] 
12a767  res  [6_1] 
12a769  +cc  [6_1] 
12a771  res  [8_8] 
12a783  +cc  [6_1] 
12a784  +cc  [3_1#(3_1)] 
12a789  +cc  [3_1#(3_1)] 
12a791  +cc  [10_129, 8_8] 
12a812  +cc  [7_4] 
12a815  cc  [5_1#(5_1)] 
12a816  +cc  [10_129, 8_8] 
12a818  +cc  [3_1#(3_1)] 
12a824  cc  [10_48, 5_2#(5_2)] 
12a825  +cc  [8_20] 
12a826  +cc  [10_129, 8_8] 
12a827  +cc  [3_1#(3_1)] 
12a833  cc  [10_87] 
12a835  cc  [10_48, 5_2#(5_2)] 
12a841  res  [6_2] 
12a842  +cc  [6_1] 
12a845  +cc  [3_1#(3_1)] 
12a852  cc  [3_1#(3_1)] 
12a853  cc  [3_1#(3_1)] 
12a862  res  [8_8] 
12a870  cc  [8_20] 
12a871  cc  [8_20] 
12a873  +cc  [8_20] 
12a878  +cc  [8_20] 
12a886  res  [8_9, 4_1#4_1] 
12a895  cc  [10_87] 
12a896  +cc  [3_1#(3_1)] 
12a898  +cc  [8_20] 
12a899  +cc  [10_129, 8_8] 
12a901  +cc  [8_20] 
12a911  cc  [10_22, 10_35] 
12a912  +cc  [6_1] 
12a914  res  [8_8] 
12a916  cc  [3_1#(3_1)] 
12a921  res  [8_9, 4_1#4_1] 
12a939  res  [10_3] 
12a940  +cc  [6_1] 
12a941  res  [10_22, 10_35] 
12a942  +cc  [10_137, 10_155, 11n37] 
12a957  res  [8_8] 
12a967  +cc  [7_4] 
12a971  +cc  [6_1] 
12a981  +cc  [8_20] 
12a983  +cc  [7_4] 
12a988  +cc  [7_4] 
12a989  res  [8_8] 
12a999  +cc  [8_20] 
12a1000  +cc  [8_20] 
12a1012  +cc  [8_9, 4_1#4_1] 
12a1014  +cc  [6_1] 
12a1016  +cc  [6_1] 
12a1025  res  [10_22, 10_35] 
12a1028  +cc  [8_20] 
12a1039  +cc  [6_1] 
12a1040  +cc  [6_1] 
12a1050  +cc  [6_1] 
12a1061  res  [9_27] 
12a1066  cc  [10_22, 10_35] 
12a1085  +cc  [6_1] 
12a1095  +cc  [6_1] 
12a1103  res  [8_8] 
12a1109  res  [9_27] 
12a1110  res  [9_41] 
12a1115  +cc  [7_4] 
12a1116  +cc  [7_4] 
12a1124  +cc  [8_9, 4_1#4_1] 
12a1127  res  [6_1] 
12a1138  +cc  [6_1] 
12a1145  cc  [10_22, 10_35] 
12a1147  cc  [10_22, 10_35] 
12a1148  +cc  [6_1] 
12a1149  +cc  [6_1] 
12a1150  +cc  [6_1] 
12a1151  +cc  [6_1] 
12a1160  res  [6_1] 
12a1163  cc  [10_3] 
12a1165  cc  [10_3] 
12a1171  +cc  [6_1] 
12a1174  +cc  [8_20] 
12a1175  res  [9_27] 
12a1179  +cc  [6_1] 
12a1194  cc  [10_129, 8_8] 
12a1200  +cc  [6_1] 
12a1201  +cc  [8_20] 
12a1205  +cc  [10_129, 8_8] 
12a1226  +cc  [8_20] 
12a1254  +cc  [8_9, 4_1#4_1] 
12a1256  +cc  [6_1] 
12a1259  +cc  [6_1] 
12a1275  +cc  [6_1] 
12a1278  +cc  [6_2] 
12a1279  +cc  [6_1] 
12a1281  res  [3_1#(3_1)] 
12a1282  cc  [10_3] 
12a1284  +cc  [8_9, 4_1#4_1] 
12a1285  +cc  [8_9, 4_1#4_1] 
12a1286  +cc  [8_4] 
12a1288  +cc  [8_9, 4_1#4_1] 
12n47  +cc  [6_1] 
12n60  +cc  [3_1#(3_1)] 
12n61  +cc  [3_1#(3_1)] 
12n75  +cc  [3_1#(3_1)] 
12n80  cc  [8_20] 
12n84  cc  [3_1#(3_1)] 
12n92  +cc  [3_1#(3_1)] 
12n101  +cc  [3_1#(3_1)] 
12n109  cc  [8_20] 
12n113  res  [3_1] 
12n115  cc  [10_153] 
12n116  res  [U] 
12n118  res  [U] 
12n137  +cc  [3_1#(3_1)] 
12n140  res  [8_20] 
12n147  cc  [8_8] 
12n157  res  [U] 
12n159  res  [U] 
12n167  +cc  [10_129, 8_8] 
12n171  res  [U] 
12n176  res  [U] 
12n190  res  [8_21] 
12n192  cc  [10_153] 
12n193  res  [U] 
12n197  res  [8_8] 
12n200  res  [6_1] 
12n202  res  [8_8] 
12n204  +cc  [7_4] 
12n206  +cc  [6_1] 
12n208  res  [U] 
12n211  +cc  [6_1] 
12n212  res  [U] 
12n216  cc  [8_8] 
12n219  +cc  [3_1#(3_1)] 
12n227  +cc  [6_1] 
12n233  res  [3_1] 
12n236  res  [U] 
12n247  res  [U] 
12n248  res  [U] 
12n253  res  [U] 
12n258  +cc  [6_1] 
12n260  res  [U] 
12n267  cc  [8_20] 
12n270  res  [U] 
12n291  cc  [3_1#(3_1)] 
12n304  res  [3_1#(3_1)] 
12n307  res  [6_1] 
12n324  +cc  [6_1] 
12n334  cc  [6_1] 
12n345  cc  [8_20] 
12n351  +cc  [6_1] 
12n359  +cc  [6_1] 
12n376  cc  [8_9, 4_1#4_1] 
12n379  cc  [8_20] 
12n383  res  [U] 
12n388  cc  [6_1] 
12n391  res  [8_8] 
12n396  +cc  [6_1] 
12n409  res  [U] 
12n410  res  [8_8] 
12n411  res  [6_1] 
12n439  cc  [3_1#(3_1)] 
12n441  res  [5_2] 
12n442  +cc  [6_1] 
12n443  cc  [3_1#(3_1)] 
12n451  res  [U] 
12n454  res  [U] 
12n456  res  [U] 
12n460  +cc  [6_1] 
12n469  res  [U] 
12n475  res  [U] 
12n480  +cc  [6_1] 
12n489  res  [8_8] 
12n495  +cc  [8_20] 
12n496  cc  [7_4] 
12n500  res  [U] 
12n514  res  [U] 
12n519  res  [6_1] 
12n520  res  [U] 
12n522  res  [U] 
12n524  +cc  [6_1] 
12n525  +cc  [6_1] 
12n531  +cc  [3_1#(3_1)] 
12n532  +cc  [6_1] 
12n537  +cc  [6_1] 
12n543  res  [U] 
12n554  res  [6_1] 
12n564  res  [U] 
12n569  res  [3_1#(3_1)] 
12n577  cc  [10_140] 
12n583  cc  [6_1] 
12n596  res  [3_1#(3_1)] 
12n601  res  [3_1#(3_1)] 
12n606  res  [U] 
12n608  res  [6_1] 
12n621  res  [U] 
12n626  res  [7_2] 
12n630  +cc  [6_1] 
12n631  +cc  [6_1] 
12n672  res  [10_129, 8_8] 
12n673  res  [U] 
12n675  cc  [3_1#(3_1)] 
12n678  +cc  [8_20] 
12n681  +cc  [3_1#(3_1)] 
12n685  res  [U] 
12n698  res  [5_2] 
12n699  res  [U] 
12n700  res  [3_1] 
12n701  res  [U] 
12n707  res  [3_1] 
12n717  res  [U] 
12n726  res  [U] 
12n730  res  [U] 
12n734  res  [3_1] 
12n735  res  [U] 
12n737  res  [6_1] 
12n742  res  [U] 
12n759  res  [6_1] 
12n769  res  [U] 
12n777  res  [6_1] 
12n783  res  [6_1] 
12n794  +cc  [6_1] 
12n796  res  [5_2] 
12n797  res  [U] 
12n804  res  [8_9, 4_1#4_1] 
12n805  +cc  [6_1] 
12n808  cc  [8_20] 
12n809  cc  [11n116] 
12n811  res  [8_20] 
12n813  +cc  [6_1] 
12n814  res  [U] 
12n815  +cc  [6_1] 
12n818  +cc  [6_1] 
12n822  res  [U] 
12n824  +cc  [6_1] 
12n829  cc  [8_20] 
12n833  +cc  [8_20] 
12n844  +cc  [6_1] 
12n846  +cc  [6_1] 
12n854  res  [8_8] 
12n855  +cc  [6_1] 
12n856  +cc  [6_1] 
12n859  +cc  [6_1] 
12n861  res  [U] 
12n862  res  [U] 
12n863  res  [5_2] 
12n867  cc  [7_4] 
12n869  cc  [8_20] 
12n873  +cc  [6_1] 
12n875  +cc  [9_46] 
Appendix B
This appendix contains Seifert matrices (from KnotInfo [3]) and bases of isotropic subspaces (computed with PARI/GP [19]) of the five knots referred to at the end of Section 2. Each of them has Taylor invariant equal to .
Knot  Seifert matrix  Basis of an isotropic subgroup 

12a244  ,  
12a810  ,  
12a905  ,  
12n555  ,  
12n642 
Appendix C
For each knot in the second table of Section 4, we give a Seifert matrix (from KnotInfo [3]) and a basis of an Alexandertrivial subgroup of maximal rank (computed with PARI/GP [19] using a randomized algorithm as in [1]). The basis is chosen such that the matrix of the restriction of the Seifert form to the subgroup with respect to the basis has the following form:
where is a quadratic matrix of dimension two less, of the same form.
Knot  Seifert matrix  Basis of an Alexandertrivial subgroup 

11n80  , , ,  
12a187  , , ,  
12a230  , , ,  
12a317  , , ,  
12a450  , , ,  
12a542  ,  
12a570  , , ,  
12a908  , , ,  
12a1118  , , ,  
12a1185  , , , 