# References: Brittenhams Letters

Excerpt from Letter 1 from Mark Brittenham.
I have gotten interested lately in computing/detecting unknotting number and four genus, and so have been running some calculations and searching the literature; but to start with I wanted to have a clearer picture of what the unknown values are. There are two sources I found/implemented which can thin the field considerably: Duncan McCoy's result on alternating knots, and Borodzik and Friedl's algebraic unknotting number calculations.
I implemented McCoy's result (arXiv:1312.1278), that an unknotting number 1 alternating knot has an unknotting crossing in any minimal diagram, in SnapPy, and can report that all of the 12-crossing alternating knots that list 1 as a possibility (but not a certainty) for their unknotting number actually have unknotting number greater than 1. I used SnapPy's DT_code() command for each knot (which provides 12-crossing diagrams) and had SnapPy identify every knot that resulted from changing one of the crossings (= changing the sign of one entry of the DT code). For the resulting knots K which were not hyperbolic (so that SnapPy's M.identify() command returned nothing), I used the simplify() and deconnect_sum() commands to identify their summands. This last step (really, the simplify command) is somewhat randomized, and so there were occasional failures, but repeating the experiment succeeded (and always yielded consistent results) in showing that all of the resulting knots were non-trivial. Attached is a file that represents a composite of these runs, and identifies the knots obtained by the crossing changes.
The other source of data that I want to mention is that Borodzik and Friedl's computations of algebraic unknotting number, on their "Knotorious" website, can identify both the unknotting number and topological four genus of a sizable number of knots - the alg unknotting number is an upper bound for topological slice genus, and many of their computations equal the lower bound reported on your site. In addition, many of their alg uknotting number values equal the upper bounds on unknotting number listed. In particular, their computations show that the following knots have unknotting number 3:
12n553, 12n554, 12n555, 12n556 .
These knots have unknotting number 2:
11n15, 11n29, 11n49, 11n58, 11n79, 11n83, 11n91, 11n92, 11n113, 11n117, 11n127, 11n132, 11n140, 11n146, 11n150, 11n155, 11n157, 11n163, 11n165, 11n167, 11n168, 11n170, 11n178, 12n10, 12n15, 12n17, 12n27, 12n33, 12n40, 12n52, 12n55, 12n56, 12n57, 12n60, 12n61, 12n62, 12n63, 12n66, 12n73, 12n78, 12n85, 12n95, 12n99, 12n109, 12n126, 12n130, 12n144, 12n145, 12n157, 12n161, 12n164, 12n171, 12n173, 12n174, 12n202(*delete), 12n206, 12n212(*delete), 12n219, 12n221, 12n223, 12n224, 12n225, 12n227, 12n247, 12n256, 12n257, 12n264, 12n266, 12n267, 12n268, 12n269, 12n278, 12n283, 12n297, 12n317, 12n333, 12n334, 12n335, 12n339, 12n355, 12n356, 12n357, 12n364, 12n365, 12n379, 12n380, 12n388, 12n389, 12n391, 12n393, 12n394, 12n397, 12n401, 12n409, 12n410, 12n413, 12n414, 12n420, 12n423, 12n440, 12n442, 12n451, 12n460, 12n462, 12n469, 12n480, 12n481, 12n486, 12n495, 12n497, 12n498, 12n505, 12n533, 12n540, 12n543, 12n546, 12n558, 12n561, 12n562, 12n567, 12n571, 12n580, 12n582, 12n583, 12n596, 12n598, 12n605, 12n611, 12n612, 12n615, 12n621, 12n622, 12n636, 12n637, 12n651, 12n652, 12n665, 12n669, 12n672, 12n706, 12n712, 12n713, 12n714, 12n717, 12n726, 12n737, 12n742, 12n743, 12n745, 12n746, 12n752, 12n755, 12n756, 12n757, 12n759, 12n760, 12n779, 12n781, 12n798, 12n810, 12n813, 12n817, 12n827, 12n837, 12n838, 12n840, 12n843, 12n845, 12n846, 12n847, 12n860, 12n869, 12n870, 12n874, 12n876, 12n877, 12n878, 12n879, 12n883, 12n886 ,
and these have unknotting number at least 2:
12n248, 12n253, 12n270, 12n601, 12n602, 12n630, 12n701,
while this has unknotting number at least 3:
12n642 .
Also, the following knots have topological slice genus equal to the reported lower bound in Knotinfo:
10_139, 10_145, 10_161, 11a211, 11n9, 11n77, 11n183, 12a153, 12a255, 12a414, 12a534, 12a542, 12a624, 12a636, 12a719, 12a1118, 12n59, 12n91, 12n105, 12n110, 12n120, 12n136, 12n148, 12n175, 12n187, 12n199, 12n207, 12n217, 12n220, 12n228, 12n239, 12n242, 12n328, 12n329, 12n366, 12n374, 12n402, 12n404, 12n417, 12n426, 12n472, 12n512, 12n518, 12n528, 12n574, 12n575, 12n591, 12n594, 12n640, 12n647, 12n660, 12n679, 12n680, 12n688, 12n689, 12n691, 12n692, 12n693, 12n694, 12n696, 12n725, 12n801, 12n850, 12n851, 12n888 .
Details from Brittenham Letter II.
Attached you should find the additional information that I had also mentioned, based on some random searches that I have been running over the past several months (although not so much the past month, I keep forgetting to restart the programs...). The basic idea is that the code builds a knot K at "random" (as a braid), and then checks to see what happens when you change each crossing, yielding L. If the known value of the unknotting number (or smooth 4-genus, or topological 4-genus, or algebraic unknotting number) for L is either one higher than the known upper bound for K, or one lower than the known lower bound, then this determines the value of the invariant for K. Thus far this has determined around 35 unknotting numbers and 1 smooth 4-genus, if I remember my count correctly. It also found a dozen or so knots where it could improve an upper or lower bound. I'm still on the hunt for what got me started with this, a new value for a knot with 10 or fewer crossings...

Using Snappy as a means to identify knots entered as braids, I have been conducting random searches for knots having projections for which a single crossing change can yield new information about unknotting numbers and other knot invariants. The basic idea is that, given a knot K, if a "crossing-adjacent" knot L has unknotting number n = one lower than the published (in Knotinfo) lower bound for the unknotting number of K, then u(K)=n+1; if L has unknotting number n = one higher than the published upper bound, then u(K)=n-1. The results of a random search - randomly building braid patterns, randomly introducing crossing information, identifying the resulting knot using SnapPy, and then changing each crossing in the braid diagram, identifying the knot, and comparing unknotting number information - are given below. A knot is represented by a pair of strings [a_1,..,a_n],[b_1,..,b_n]. The first string represents an unsigned braid word, with 0 = a crossing of the first and second strands, 1 = a crossing of the second and third, etc. The second string, consisting of 0's and 1's, is a choice of exponent 1/-1 for each braid generator. SnapPy was used to identify the resulting knots, and a Python script made comparisons of unknotting numbers (and other invariants) and flagged instances in which new information was gained from the adjacency. It is worth noting that some care must be taken with this approach to determining knot invariants. Knotinfo (which gives the data) and SnapPy (which identifies the knots) use different conventions for labeling the 10-crossing knots from 10_161 (the first of the Perko pair) to 10_165/10_166. Knotinfo does not include the Perko pair, and numbers to 165; SnapPy *does* include the Perko pair, and numbers to 166.

Unknotting numbers determined by this search:

K11a225 has u(K)=2

[3, 3, 1, 0, 3, 2, 3, 3, 3, 1, 1, 1, 1, 2, 3, 3, 2, 3, 1, 3, 0, 2]
[1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0]
Snappy: [11_523(0,0), K11a225(0,0)] Knotinfo: u = [2,3]
[3, 3, 1, 0, 3, 2, 3, 3, 3, 1, 1, 1, 1, 2, 3, 3, 2, 3, 1, 3, 0, 2]
[1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0]
Snappy: [t11291(0,0), 7_6(0,0), K8_241(0,0), K7a2(0,0)] Knotinfo: u = 1

K11a237 has u(K)=3
[5, 4, 1, 2, 0, 5, 3, 3, 3, 4, 1, 3, 3, 2, 1, 0, 5, 2, 4, 3, 5, 0, 3, 3, 2, 4]
[1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0]
Snappy: [11_165(0,0), K11a237(0,0)] Knotinfo: u = [2,3]
[5, 4, 1, 2, 0, 5, 3, 3, 3, 4, 1, 3, 3, 2, 1, 0, 5, 2, 4, 3, 5, 0, 3, 3, 2, 4]
[1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0]
Snappy: [11_513(0,0), K11a365(0,0)]
u = 4, per B. Owens, "Unknotting information from Heegaard Floer homology"

K11a337 has u(K)=3
[3, 1, 2, 3, 2, 2, 2, 1, 0, 4, 4, 0, 2, 0, 1, 4, 3, 0, 3, 3, 4, 4, 3, 3, 1]
[1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 1]
Snappy: [11_504(0,0), K11a337(0,0)] Knotinfo: u = [2,3]
[3, 1, 2, 3, 2, 2, 2, 1, 0, 4, 4, 0, 2, 0, 1, 4, 3, 0, 3, 3, 4, 4, 3, 3, 1]
[1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 1]
Snappy: ['11_513', 'K11a365']
u = 4, per B. Owens, "Unknotting information from Heegaard Floer homology"

K11a359 has u(K)=3
[3, 1, 2, 3, 2, 2, 2, 1, 0, 4, 4, 0, 2, 0, 1, 4, 3, 0, 3, 3, 4, 4, 3, 3, 1]
[1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 1]
Snappy: [11_541(0,0), K11a359(0,0)] Knotinfo: u = [2,3]
[3, 1, 2, 3, 2, 2, 2, 1, 0, 4, 4, 0, 2, 0, 1, 4, 3, 0, 3, 3, 4, 4, 3, 3, 1]
[1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 1]
Snappy: ['11_513', 'K11a365']
u = 4, per B. Owens, "Unknotting information from Heegaard Floer homology"

K12a39 has u(K)=2
[3, 4, 1, 0, 3, 2, 3, 1, 3, 2, 3, 1, 1, 1, 3, 0, 1, 2, 2, 0, 1, 4, 4, 3, 0]
[1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 1]
Snappy: [K12a39(0,0)] Knotinfo: u = [2,3]
[3, 4, 1, 0, 3, 2, 3, 1, 3, 2, 3, 1, 1, 1, 3, 0, 1, 2, 2, 0, 1, 4, 4, 3, 0]
[1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 1]
Snappy: [9_33(0,0), K9a11(0,0)] Knotinfo: u = 1

K12a76 has u(K)=2
[6, 3, 2, 4, 3, 2, 5, 0, 6, 0, 0, 6, 5, 2, 6, 1, 1, 2, 1, 4, 6, 5, 0, 3, 4, 5, 2, 7, 6, 4, 4, 3, 4, 4, 3, 5, 1, 5, 1, 3, 0, 7]
[0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 0, 1, 1, 0]
Snappy: [K12a76(0,0)] Knotinfo: u = [2,3]
[6, 3, 2, 4, 3, 2, 5, 0, 6, 0, 0, 6, 5, 2, 6, 1, 1, 2, 1, 4, 6, 5, 0, 3, 4, 5, 2, 7, 6, 4, 4, 3, 4, 4, 3, 5, 1, 5, 1, 3, 0, 7]
[0, 1, 0, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 0, 1, 1, 0]
Snappy: [9_21(0,0), K9a21(0,0)] Knotinfo: u = 1

K12a291 has u(K)=2
[0, 4, 1, 4, 3, 2, 1, 1, 1, 2, 3, 4, 2, 1, 0, 2, 3, 3, 0, 1, 0, 0, 3, 4, 1]
[0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1]
Snappy: [K12a291(0,0)] Knotinfo: u = [2,3]
[0, 4, 1, 4, 3, 2, 1, 1, 1, 2, 3, 4, 2, 1, 0, 2, 3, 3, 0, 1, 0, 0, 3, 4, 1]
[0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1]
Snappy: [9_30(0,0), K9a1(0,0)] Knotinfo: u =

K12a759 has u(K)=2
[0, 0, 0, 2, 3, 0, 0, 4, 4, 4, 1, 0, 1, 1, 2, 3, 2, 1, 4, 4, 4, 3, 3]
[1, 1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 1, 0, 1, 1, 0, 1]
Snappy: [K12a759(0,0)] Knotinfo: u = [2,3]
[0, 0, 0, 2, 3, 0, 0, 4, 4, 4, 1, 0, 1, 1, 2, 3, 2, 1, 4, 4, 4, 3, 3]
[1, 1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 0, 1]
Snappy: [t11034(0,0), 8_7(0,0), K8_236(0,0), K8a6(0,0)] Knotinfo: u = 1

K12a818 has u(K)=2
[1, 3, 2, 1, 1, 2, 4, 4, 3, 2, 2, 0, 5, 0, 0, 2, 3, 4, 2, 4, 0, 2, 3, 4, 1, 5, 2, 4, 0, 3]
[0, 0, 1, 1, 1, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0]
Snappy: [K12a818(0,0)] Knotinfo: u = [2,3]
[1, 3, 2, 1, 1, 2, 4, 4, 3, 2, 2, 0, 5, 0, 0, 2, 3, 4, 2, 4, 0, 2, 3, 4, 1, 5, 2, 4, 0, 3]
[0, 0, 1, 1, 1, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1]
Snappy: [8_17(0,0), K8a14(0,0)] Knotinfo: u = 1

K12a847 has u(K)=2
[3, 1, 3, 1, 2, 2, 2, 1, 1, 2, 2, 3, 2, 0, 0, 1, 4, 3, 4, 0, 2, 0, 0, 1, 2]
[0, 1, 0, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 1, 0, 1, 1, 1, 0]
Snappy: [K12a847(0,0)] Knotinfo: u = [2,3]
[3, 1, 3, 1, 2, 2, 2, 1, 1, 2, 2, 3, 2, 0, 0, 1, 4, 3, 4, 0, 2, 0, 0, 1, 2]
[0, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 1, 0, 1, 1, 1, 0]
Snappy: [t11291(0,0), 7_6(0,0), K8_241(0,0), K7a2(0,0)] Knotinfo: u = 1

K12a853 has u(K)=2
[4, 0, 0, 2, 2, 0, 1, 2, 1, 1, 2, 3, 4, 2, 1, 2, 3, 4, 0, 3, 2, 1, 4, 3, 0]
[1, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0]
Snappy: [K12a853(0,0)] Knotinfo: u = [2,3]
[4, 0, 0, 2, 2, 0, 1, 2, 1, 1, 2, 3, 4, 2, 1, 2, 3, 4, 0, 3, 2, 1, 4, 3, 0]
[1, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0]
Snappy: [K12a851(0,0)] Knotinfo: u = 1

K12a942 has u(K)=2
[1, 2, 4, 1, 4, 0, 1, 2, 3, 2, 0, 2, 4, 0, 1, 3, 1, 1, 2, 1, 2, 1, 1, 3, 3, 0, 3, 4, 0, 3, 3, 0, 2, 1, 2, 3, 2]
[0, 1, 0, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1]
Snappy: [K12a942(0,0)] Knotinfo: u = [2,3]
[1, 2, 4, 1, 4, 0, 1, 2, 3, 2, 0, 2, 4, 0, 1, 3, 1, 1, 2, 1, 2, 1, 1, 3, 3, 0, 3, 4, 0, 3, 3, 0, 2, 1, 2, 3, 2]
[0, 1, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1]
Snappy: [8_14(0,0), K8a1(0,0)] Knotinfo: u = 1

K12a970 has u(K)=2
[2, 3, 4, 2, 4, 1, 2, 2, 4, 4, 1, 2, 0, 0, 0, 2, 4, 1, 3, 0, 1, 0, 0]
[0, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0]
Snappy: [K12a970(0,0)] Knotinfo: u = [2,3]
[2, 3, 4, 2, 4, 1, 2, 2, 4, 4, 1, 2, 0, 0, 0, 2, 4, 1, 3, 0, 1, 0, 0]
[1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0]
Snappy: [K12a1219(0,0)] Knotinfo: u = 1

K12a1131 has u(K)=2
[4, 0, 2, 1, 3, 2, 1, 3, 0, 2, 4, 0, 0, 1, 4, 0, 0, 4, 4, 2, 4, 2, 3, 2, 0]
[1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 1, 0, 0]
Snappy: [K12a1131(0,0)] Knotinfo: u = [2,3]
[4, 0, 2, 1, 3, 2, 1, 3, 0, 2, 4, 0, 0, 1, 4, 0, 0, 4, 4, 2, 4, 2, 3, 2, 0]
[1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 1, 0, 0]
Snappy: [m289(0,0), 6_2(0,0), K5_19(0,0), K6a2(0,0)] Knotinfo: u = 1

K12a1142 has u(K)=2
[1, 3, 4, 2, 4, 0, 1, 1, 0, 1, 0, 4, 3, 0, 0, 4, 1, 4, 2, 1, 3, 4, 3, 1, 2]
[0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 1]
Snappy: [K12a1142(0,0)] Knotinfo: u = [2,3]
[1, 3, 4, 2, 4, 0, 1, 1, 0, 1, 0, 4, 3, 0, 0, 4, 1, 4, 2, 1, 3, 4, 3, 1, 2]
[0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 1]
Snappy: [t12656(0,0), 7_7(0,0), K8_294(0,0), K7a1(0,0)] Knotinfo: u = 1

K12a1153 has u(K)=2
[2, 0, 4, 3, 2, 1, 0, 1, 2, 4, 0, 3, 0, 1, 4, 2, 3, 0, 1, 2, 3, 2, 3, 0, 0, 4, 2, 3, 4, 2, 4, 3, 3]
[1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1]
Snappy: [K12a1153(0,0)] Knotinfo: u = [2,3]
[2, 0, 4, 3, 2, 1, 0, 1, 2, 4, 0, 3, 0, 1, 4, 2, 3, 0, 1, 2, 3, 2, 3, 0, 0, 4, 2, 3, 4, 2, 4, 3, 3]
[1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1]
Snappy: [9_26(0,0), K9a15(0,0)] Knotinfo: u = 1

K12a1157 has u(K)=3
[1, 3, 3, 0, 1, 4, 3, 4, 0, 3, 4, 4, 1, 1, 3, 4, 2, 4, 3, 1, 2, 3, 3, 0, 0]
[0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 1]
Snappy: [t07658(0,0), K8_146(0,0), K12a1157(0,0)] Knotinfo: u = [3,4]
[1, 3, 3, 0, 1, 4, 3, 4, 0, 3, 4, 4, 1, 1, 3, 4, 2, 4, 3, 1, 2, 3, 3, 0, 0]
[0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 1]
Snappy: [v2858(0,0), 10_8(0,0), K7_103(0,0), K10a114(0,0)] Knotinfo: u = 2

K12a1158 has u(K)=2
[3, 3, 3, 1, 1, 2, 3, 2, 3, 3, 5, 4, 2, 1, 2, 3, 4, 5, 4]
[1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1]
Snappy: [K12a1158(0,0)] Knotinfo: u = [2,3]
[3, 3, 3, 1, 1, 2, 3, 2, 3, 3, 5, 4, 2, 1, 2, 3, 4, 5, 4]
[1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1]
Snappy: [o9_42258(0,0), 8_11(0,0), K8a9(0,0)] Knotinfo: u = 1

K12a1171 has u(K)=2
[2, 3, 0, 0, 2, 1, 2, 2, 3, 1, 1, 3, 3, 0, 3, 1, 2, 1, 3, 0, 0, 3, 2, 1]
[1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1]
Snappy: [K12a1171(0,0)] Knotinfo: u = [2,3]
[2, 3, 0, 0, 2, 1, 2, 2, 3, 1, 1, 3, 3, 0, 3, 1, 2, 1, 3, 0, 0, 3, 2, 1]
[1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 0, 1]
Snappy: [K12a1169(0,0)] Knotinfo: u =

K12a1200 has u(K)=2
[3, 2, 1, 4, 3, 2, 4, 3, 4, 4, 3, 4, 4, 1, 3, 0, 3]
[0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 0]
Snappy: [K12a1200(0,0)] Knotinfo: u = [2,3]
[3, 2, 1, 4, 3, 2, 4, 3, 4, 4, 3, 4, 4, 1, 3, 0, 3]
[0, 1, 0, 1, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 1, 0, 0]
Snappy: [9_30(0,0), K9a1(0,0)] Knotinfo: u = 1

K12a1242 has u(K)=3
[3, 2, 2, 1, 2, 0, 2, 0, 4, 0, 2, 1, 0, 2, 2, 0, 2, 0, 3, 2, 0, 1, 1]
[0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1]
Snappy: [K12a1242(0,0)] Knotinfo: u = [3,4]
[3, 2, 2, 1, 2, 0, 2, 0, 4, 0, 2, 1, 0, 2, 2, 0, 2, 0, 3, 2, 0, 1, 1]
[0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1]
Snappy: [v2858(0,0), 10_8(0,0), K7_103(0,0), K10a114(0,0)] Knotinfo: u = 2

K12a1259 has u(K)=2
[4, 0, 1, 3, 2, 4, 3, 0, 0, 0, 0, 4, 4, 1, 3, 4, 4, 0, 3, 1, 4, 2, 4]
[0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 0]
Snappy: [K12a1259(0,0)] Knotinfo: u = [2,3]
[4, 0, 1, 3, 2, 4, 3, 0, 0, 0, 0, 4, 4, 1, 3, 4, 4, 0, 3, 1, 4, 2, 4]
[0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 0]
Snappy: [K12a1258(0,0)] Knotinfo: u = 1

K12n116 has u(K)=2
[4, 2, 0, 1, 3, 0, 1, 2, 1, 0, 4, 4, 2, 3, 1, 2, 1, 3, 2]
[0, 1, 1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 1]
Snappy: [K12n116(0,0)] Knotinfo: u = [1,2]
[4, 2, 0, 1, 3, 0, 1, 2, 1, 0, 4, 4, 2, 3, 1, 2, 1, 3, 2]
[0, 1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1]
Snappy: ['o9_44057', '9_10', 'K9a39'] Knotinfo: u = 3

K12n200 has u(K)=2
[2, 3, 0, 0, 0, 3, 2, 0, 2, 2, 1, 0, 3, 2, 1, 0, 3, 2, 3, 2, 2, 1, 2, 3]
[1, 0, 0, 1, 1, 1, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 1, 1, 0, 1]
Snappy: [t09735(0,0), K8_199(0,0), K12n200(0,0)] Knotinfo: u = [1,2]
[2, 3, 0, 0, 0, 3, 2, 0, 2, 2, 1, 0, 3, 2, 1, 0, 3, 2, 3, 2, 2, 1, 2, 3]
[1, 0, 0, 1, 1, 1, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 1, 1, 0, 1]
Snappy: ['o9_39339', '9_35', 'K9a40'] Knotinfo: u = 3

K12n455 has u(K)=2
[0, 2, 0, 3, 2, 0, 0, 2, 1, 2, 2, 2, 3, 0, 0, 0, 1, 1]
[1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 0, 1, 1]
Snappy: [K12n455(0,0)] Knotinfo: u = [2,3]
[0, 2, 0, 3, 2, 0, 0, 2, 1, 2, 2, 2, 3, 0, 0, 0, 1, 1]
[1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1]
Snappy: [t11291(0,0), 7_6(0,0), K8_241(0,0), K7a2(0,0)] Knotinfo: u = 1

K12n476 has u(K)=2
[1, 3, 2, 0, 3, 2, 2, 1, 0, 2, 1, 2, 0, 2, 1, 1, 3, 2, 1, 0]
[0, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1]
Snappy: [K12n476(0,0)] Knotinfo: u = [2,3]
[1, 3, 2, 0, 3, 2, 2, 1, 0, 2, 1, 2, 0, 2, 1, 1, 3, 2, 1, 0]
[0, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 1, 1]
Snappy: [K12n467(0,0)] Knotinfo: u = 1

K12n512 has u(K)=2
[1, 2, 3, 3, 0, 1, 2, 2, 4, 2, 0, 3, 1, 0, 4, 3, 2, 2, 0, 0, 1]
[0, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0]
Snappy: [K12n512(0,0)] Knotinfo: u = [1,2]
[1, 2, 3, 3, 0, 1, 2, 2, 4, 2, 0, 3, 1, 0, 4, 3, 2, 2, 0, 0, 1]
[0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0]
Snappy: ['9_38', 'K9a30'] Knotinfo: u = 3

K12n601 has u(K)=2
[1, 3, 1, 2, 1, 3, 2, 1, 1, 3, 2, 0, 3, 1, 3, 3, 0, 3, 0, 2, 1, 0, 2, 1]
[1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0]
Snappy: [K12n601(0,0)] Knotinfo: u = [2,3]
[1, 3, 1, 2, 1, 3, 2, 1, 1, 3, 2, 0, 3, 1, 3, 3, 0, 3, 0, 2, 1, 0, 2, 1]
[1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0]
Snappy: [t12656(0,0), 7_7(0,0), K8_294(0,0), K7a1(0,0)] Knotinfo: u = 1

K12n602 has u(K)=2
[3, 1, 2, 3, 0, 1, 2, 3, 1, 2, 1, 1, 0, 2, 0, 1, 3, 1]
[0, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0]
Snappy: [K12n602(0,0)] Knotinfo: u = [2,3]
[3, 1, 2, 3, 0, 1, 2, 3, 1, 2, 1, 1, 0, 2, 0, 1, 3, 1]
[0, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0]
Snappy: [t12656(0,0), 7_7(0,0), K8_294(0,0), K7a1(0,0)] Knotinfo: u = 1

K12n606 has u(K)=2
[1, 2, 3, 1, 1, 4, 0, 4, 1, 1, 2, 1, 0, 2, 4, 0, 2, 2, 3]
[1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1]
Snappy: [K12n606(0,0)] Knotinfo: u = [1,2]
[1, 2, 3, 1, 1, 4, 0, 4, 1, 1, 2, 1, 0, 2, 4, 0, 2, 2, 3]
[1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 1, 0, 0, 1, 1, 0, 1]
Snappy: ['9_38', 'K9a30'] Knotinfo: u = 3

K12n685 has u(K)=2
[0, 1, 3, 0, 2, 0, 3, 2, 1, 0, 1, 2, 1, 0, 1, 2, 2, 1, 1, 1, 0, 1, 1, 2]
[0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0]
Snappy: [K12n685(0,0)] Knotinfo: u = [1,2]
[0, 1, 3, 0, 2, 0, 3, 2, 1, 0, 1, 2, 1, 0, 1, 2, 2, 1, 1, 1, 0, 1, 1, 2]
[0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0]
Snappy: ['9_13', 'K9a34'] Knotinfo: u = 3

K12n699 has u(K)=2
[1, 3, 0, 1, 4, 2, 1, 0, 3, 2, 0, 1, 2, 0, 0, 1, 2, 3, 3, 0, 2, 2, 2, 0, 0]
[1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0]
Snappy: [K12n699(0,0)] Knotinfo: u = [1,2]
[1, 3, 0, 1, 4, 2, 1, 0, 3, 2, 0, 1, 2, 0, 0, 1, 2, 3, 3, 0, 2, 2, 2, 0, 0]
[1, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0]
Snappy: ['o9_44057', '9_10', 'K9a39'] Knotinfo: u = 3

K12n773 has u(K)=2
[3, 2, 2, 0, 0, 2, 0, 2, 3, 0, 1, 0, 3, 2, 0, 3, 3, 3, 0, 3, 1, 2, 1, 1]
[1, 1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1]
Snappy: [K12n773(0,0)] Knotinfo: u = [2,3]
[3, 2, 2, 0, 0, 2, 0, 2, 3, 0, 1, 0, 3, 2, 0, 3, 3, 3, 0, 3, 1, 2, 1, 1]
[0, 1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1]
Snappy: [K12n767(0,0)] Knotinfo: u = 1

K12n797 has u(K)=2
[0, 1, 3, 4, 2, 3, 2, 0, 3, 1, 2, 3, 0, 4, 3, 2, 2, 4, 2]
[1, 1, 1, 0, 1, 0, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1]
Snappy: [K12n797(0,0)] Knotinfo: u = [1,2]
[0, 1, 3, 4, 2, 3, 2, 0, 3, 1, 2, 3, 0, 4, 3, 2, 2, 4, 2]
[1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1]
Snappy: ['9_13', 'K9a34'] Knotinfo: u = 3

success: K12n862 has u(K)=2
[1, 1, 2, 3, 1, 4, 1, 1, 4, 0, 1, 2, 4, 1, 3, 3, 2, 1, 4, 0, 0, 3, 4]
[0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
Snappy: [K12n862(0,0)] Knotinfo: u = [1,2]
[1, 1, 2, 3, 1, 4, 1, 1, 4, 0, 1, 2, 4, 1, 3, 3, 2, 1, 4, 0, 0, 3, 4]
[0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
Snappy: ['9_38', 'K9a30'] Knotinfo: u = 3

-------------------------------------------------------

Smooth four genus determined by this search:

K12n542 has smooth four genus 1
[0, 3, 1, 0, 0, 2, 3, 2, 3, 3, 1, 2, 3, 1, 2, 1, 2, 1]
[0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 1, 0]
Snappy: [K12n542(0,0)] Knotinfo: g_4 = [1,2]
[0, 3, 1, 0, 0, 2, 3, 2, 3, 3, 1, 2, 3, 1, 2, 1, 2, 1]
[0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0]
Snappy: [K12n312(0,0)] Knotinfo: g_4 = 0

--------------------------------------------------------

Improved unknotting number inequalities:

(the target unknotting number must be within one of the known

K11a361 has u(K) = [3,4]
[5, 4, 1, 2, 0, 5, 3, 3, 3, 4, 1, 3, 3, 2, 1, 0, 5, 2, 4, 3, 5, 0, 3, 3, 2, 4]
[1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0]
Snappy: ['11_242', 'K11a361'] Knotinfo: u = [2,3,4]
[5, 4, 1, 2, 0, 5, 3, 3, 3, 4, 1, 3, 3, 2, 1, 0, 5, 2, 4, 3, 5, 0, 3, 3, 2, 4]
[1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0]
Snappy: ['11_513', 'K11a365']
u = 4, per B. Owens, "Unknotting information from Heegaard Floer homology"

K11a366 has u(K) = [3,4]
[5, 4, 1, 2, 0, 5, 3, 3, 3, 4, 1, 3, 3, 2, 1, 0, 5, 2, 4, 3, 5, 0, 3, 3, 2, 4]
[1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 0]
Snappy: ['11_202', 'K11a366'] Knotinfo: u = [2,3,4]
[5, 4, 1, 2, 0, 5, 3, 3, 3, 4, 1, 3, 3, 2, 1, 0, 5, 2, 4, 3, 5, 0, 3, 3, 2, 4]
[1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0]
Snappy: ['11_513', 'K11a365']
u = 4, per B. Owens, "Unknotting information from Heegaard Floer homology"

K12a1162 has u(K) = [2,3]
[4, 4, 1, 1, 4, 0, 4, 0, 0, 4, 4, 0, 2, 1, 0, 0, 4, 3, 0, 4, 2, 3, 4, 1, 4]
[1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0]
Snappy: [K12a1162(0,0)] Knotinfo: u = [2,3,4]
[4, 4, 1, 1, 4, 0, 4, 0, 0, 4, 4, 0, 2, 1, 0, 0, 4, 3, 0, 4, 2, 3, 4, 1, 4]
[1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0]
Snappy: [v2858(0,0), 10_8(0,0), K7_103(0,0), K10a114(0,0)] Knotinfo: u = 2

K12n47 has u(K) = [1,2]
[3, 4, 0, 2, 1, 2, 3, 5, 0, 0, 4, 5, 0, 3, 5, 0, 3, 1, 3, 2, 5, 0, 2, 1, 2, 1]
[0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0]
Snappy: [K12n47(0,0)] Knotinfo: u = [1,2,3]
[3, 4, 0, 2, 1, 2, 3, 5, 0, 0, 4, 5, 0, 3, 5, 0, 3, 1, 3, 2, 5, 0, 2, 1, 2, 1]
[0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1]
Snappy: [9_19(0,0), K9a3(0,0)] Knotinfo: u = 1

K12n193 has u(K) = [2,3]
[4, 0, 3, 0, 3, 2, 3, 0, 3, 3, 4, 2, 3, 2, 1, 2, 4, 1, 1]
[0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 0]
Snappy: [K12n193(0,0)] Knotinfo: u = [1,2,3]
[4, 0, 3, 0, 3, 2, 3, 0, 3, 3, 4, 2, 3, 2, 1, 2, 4, 1, 1]
[0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 0]
Snappy: ['9_13', 'K9a34'] Knotinfo: u = 3

K12n208 has u(K) = [2,3]
[2, 3, 1, 1, 4, 0, 0, 1, 3, 0, 1, 2, 3, 2, 4, 3, 1, 1, 3, 3, 2, 0, 2, 1, 3]
[1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0]
Snappy: [K12n208(0,0)] Knotinfo: u = [1,2,3]
[2, 3, 1, 1, 4, 0, 0, 1, 3, 0, 1, 2, 3, 2, 4, 3, 1, 1, 3, 3, 2, 0, 2, 1, 3]
[1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0]
Snappy: ['9_38', 'K9a30'] Knotinfo: u = 3

K12n236 has u(K) = [2,3]
[1, 2, 1, 1, 3, 3, 0, 0, 3, 1, 0, 1, 1, 2, 0, 0, 3, 3, 2, 1, 0, 0, 3, 3]
[0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, 0, 1]
Snappy: [K12n236(0,0)] Knotinfo: u = [1,2,3]
[1, 2, 1, 1, 3, 3, 0, 0, 3, 1, 0, 1, 1, 2, 0, 0, 3, 3, 2, 1, 0, 0, 3, 3]
[0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1]
Snappy: ['o9_44057', '9_10', 'K9a39'] Knotinfo: u = 3

K12n239 has u(K) = [2,3]
[3, 3, 1, 3, 0, 4, 1, 0, 0, 3, 0, 2, 1, 3, 2, 1, 4]
[0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1]
Snappy: [K12n239(0,0)] Knotinfo: u = [1,2,3]
[3, 3, 1, 3, 0, 4, 1, 0, 0, 3, 0, 2, 1, 3, 2, 1, 4]
[0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1]
Snappy: ['9_13', 'K9a34'] Knotinfo: u = 3

K12n258 has u(K)<=2 = [1,2]
[1, 2, 2, 0, 1, 1, 2, 1, 1, 3, 0, 1, 3, 0, 1, 2, 1, 1, 2, 2, 0, 3, 1, 1]
[1, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1]
Snappy: [K12n258(0,0)] Knotinfo: u = [1,2,3]
[1, 2, 2, 0, 1, 1, 2, 1, 1, 3, 0, 1, 3, 0, 1, 2, 1, 1, 2, 2, 0, 3, 1, 1]
[1, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1]
Snappy: [11_135(0,0), K11n55(0,0)] Knotinfo: u = 1

K12n260 has u(K) = [2,3]
[2, 3, 0, 0, 0, 3, 2, 0, 2, 2, 1, 0, 3, 2, 1, 0, 3, 2, 3, 2, 2, 1, 2, 3]
[1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 1, 1, 0, 1]
Snappy: [K12n260(0,0)] Knotinfo: u = [1,2,3]
[2, 3, 0, 0, 0, 3, 2, 0, 2, 2, 1, 0, 3, 2, 1, 0, 3, 2, 3, 2, 2, 1, 2, 3]
[1, 0, 0, 1, 1, 1, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 1, 1, 0, 1]
Snappy: ['o9_39339', '9_35', 'K9a40'] Knotinfo: u = 3

K12n456 has u(K) = [1,2]
[2, 0, 5, 2, 3, 2, 1, 0, 1, 2, 2, 2, 4, 0, 5, 3, 3, 3, 4, 3, 1, 0]
[0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 0]
Snappy: [K12n456(0,0)] Knotinfo: u = [1,2,3]
[2, 0, 5, 2, 3, 2, 1, 0, 1, 2, 2, 2, 4, 0, 5, 3, 3, 3, 4, 3, 1, 0]
[0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 1, 1, 1, 0]
Snappy: [t11291(0,0), 7_6(0,0), K8_241(0,0), K7a2(0,0)] Knotinfo: u = 1

K12n475 has u(K) = [1,2]
[2, 2, 3, 2, 1, 2, 4, 3, 4, 4, 2, 2, 2, 0, 3, 2, 4, 0, 0, 4, 2, 0, 0, 1, 0]
[0, 0, 1, 1, 1, 1, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0]
Snappy: [v3320(0,0), K7_121(0,0), K12n475(0,0)] Knotinfo: u = [1,2,3]
[2, 2, 3, 2, 1, 2, 4, 3, 4, 4, 2, 2, 2, 0, 3, 2, 4, 0, 0, 4, 2, 0, 0, 1, 0]
[0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0]
Snappy: [m289(0,0), 6_2(0,0), K5_19(0,0), K6a2(0,0)] Knotinfo: u = 1

K12n769 has u(K) = [2,3]
[1, 2, 3, 0, 1, 0, 2, 3, 3, 3, 1, 4, 3, 2, 1, 3, 2, 2, 0, 2, 4, 0, 2, 4, 3]
[1, 1, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 0, 0, 1]
Snappy: [K12n769(0,0)] Knotinfo: u = [1,2,3]
[1, 2, 3, 0, 1, 0, 2, 3, 3, 3, 1, 4, 3, 2, 1, 3, 2, 2, 0, 2, 4, 0, 2, 4, 3]
[1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 0, 0, 1]
Snappy: ['9_13', 'K9a34'] Knotinfo: u = 3