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"
Adv. Math 217 (2008) 2353-2376
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"
Adv. Math 217 (2008) 2353-2376
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"
Adv. Math 217 (2008) 2353-2376
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
value for a crossing-adjacent knot)
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"
Adv. Math 217 (2008) 2353-2376
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"
Adv. Math 217 (2008) 2353-2376
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