On calculating the slice genera of 11- and 12-crossing knots

Lukas Lewark University of Bern, Mathematical Institute, Alpeneggstr. 22, 3012 Bern, Switzerland lukas.lewark@math.unibe.ch  and  Duncan McCoy Department of Mathematics, University of Texas at Austin, Austin, Texas, USA d.mccoy@math.utexas.edu
Abstract.

This paper contains the results of efforts to determine values of the smooth and the topological slice genus of 11- and 12-crossing 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 11-crossing 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 unknown111Unknown = ‘listed as unknown on KnotInfo [3] at the time of writing’. values of the smooth and topological slice genus for 11- and 12-crossing 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 11-crossing 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

Table 1. The remaining unknown values.

2. Taylor’s lower bound

The Seifert form of a knot yields several well-known lower bounds to the topological slice genus, namely the bounds coming from the Levine-Tristram signatures and the Fox-Milnor 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 higher-dimensional locally flat analogue of Seifert surfaces; more precisely, given a locally flat surface , the existence of a locally flat embedded compact oriented 3-manifold 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 3-manifold with equal to a push-off 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 3-manifold 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 well-known than the signatures and the Fox-Milnor 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 Levine-Tristram 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 straight-forward 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 :

  1. There is a non-negative integer and an integer such that , and the Hasse symbol of over is .

  2. The form induced by over is Witt-equivalent to an anisotropic four-dimensional 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 well-known fact that the Witt-class 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 Levine-Tristram signatures, the Fox-Milnor 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 2-bridge knots with differing smooth and topological slice genera [7].

Lemma 3.

Let be a knot with with a positive-definite Goeritz matrix . If and , then there is an integer matrix , such that .

Proof.

The double branched cover bounds a 4-manifold with intersection form given by the matrix [10]. It also bounds a smooth 4-manifold with and signature [10, 13]. Suppose that . This implies that the closed smooth 4-manifold 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 S-equivalence 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 Alexander-trivial if is a unit in . Let be the maximal rank of an Alexander-trivial 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 Alexander-trivial 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 .
Figure 1. A prime knot with , , (drawn with knotscape [12]).

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 non-trivial Alexander polynomial (and thus non-zero 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 Alexander-trivial 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 arXiv-version 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:

  1. changing a single crossing;

  2. changing a positive and a negative crossing; or

  3. 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 twice-punctured 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 12-crossing 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 10-crossing knots

The only 10-crossing 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 12-crossing diagram for . Since and , this implies that .
We can obtain by performing a single crossing change in a 12-crossing 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 4-genus of torus knots. Accepted for publication by Trans. Amer. Math. Soc., arXiv:1509.07634, 2015.
  • [2] S. Baader and L. Lewark. The stable 4-genus of alternating knots. Accepted for publication by Asian J. Math., arXiv:1505.03345, 2015.
  • [3] J.C. Cha and Livingston C. Knotinfo: Table of Knot Invariants. http://www.indiana.edu/~knotinfo, 2016.
  • [4] S. K. Donaldson. The orientation of Yang-Mills moduli spaces and -manifold topology. J. Differential Geom., 26(3):397–428, 1987.
  • [5] P. Feller. The degree of the Alexander polynomial is an upper bound for the topological slice genus. Geom. Topol., 20(3):1763–1771, 2016.
  • [6] P. Feller and L. Lewark. On classical upper bounds for slice genera. arXiv:1611.02679, 2016.
  • [7] P. Feller and D. McCoy. On 2-bridge knots with differing smooth and topological slice genera. Proc. Amer. Math. Soc., 144(12):5435–5442, 2016.
  • [8] M. H. Freedman and F. Quinn. Topology of 4-manifolds, volume 39 of Princeton Mathematical Series. Princeton University Press, Princeton, NJ, 1990.
  • [9] The GAP Group. GAP – Groups, Algorithms, and Programming, Version 4.7.8, 2015.
  • [10] C. McA. Gordon and R. A. Litherland. On the signature of a link. Invent. Math., 47(1):53–69, 1978.
  • [11] C. Herald, P. Kirk, and C. Livingston. Metabelian representations, twisted Alexander polynomials, knot slicing, and mutation. Math. Z., 265(4):925–949, 2010.
  • [12] J. Hoste and M. Thistlethwaite. Knotscape (version 1.01), 1999. Computer program.
  • [13] L. H. Kauffman and L. R. Taylor. Signature of links. Trans. Amer. Math. Soc., 216:351–365, 1976.
  • [14] R. C. Kirby and L. C. Siebenmann. Foundational essays on topological manifolds, smoothings, and triangulations. Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1977. With notes by John Milnor and Michael Atiyah, Annals of Mathematics Studies, No. 88.
  • [15] P. Koprowski and A. Czogała. Computing with quadratic forms over number fields. arXiv:1304.0708, 2013.
  • [16] T. Y. Lam. Introduction to quadratic forms over fields, volume 67 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2005.
  • [17] W. B. Raymond Lickorish. An introduction to knot theory, volume 175 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1997.
  • [18] H. Murakami. Algebraic unknotting operation. In Proceedings of the Second Soviet-Japan Joint Symposium of Topology (Khabarovsk, 1989), volume 8 of Questions Answers Gen. Topology, pages 283–292, 1990.
  • [19] The PARI Group, Bordeaux. PARI/GP, version 2.7.4, 2015. Programming library available from http://pari.math.u-bordeaux.fr/.
  • [20] A. Ranicki. High-dimensional knot theory. Springer Monographs in Mathematics. Springer-Verlag, New York, 1998. Algebraic surgery in codimension 2, With an appendix by Elmar Winkelnkemper.
  • [21] L. R. Taylor. On the genera of knots. In Topology of low-dimensional manifolds (Proc. Second Sussex Conf., Chelwood Gate, 1977), volume 722 of Lecture Notes in Math., pages 144–154. Springer, Berlin, 1979.
\enddoc@text

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 Alexander-trivial 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 Alexander-trivial subgroup
11n80 , , ,
12a187 , , ,
12a230 , , ,
12a317 , , ,
12a450 , , ,
12a542 ,
12a570 , , ,
12a908 , , ,
12a1118 , , ,
12a1185 , , ,