The Alternating Knot 10₆₆

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Acknowledgement

KnotPlot

PD Presentation: X₁₄₂₅ X_3,10,4,11 X_5,14,6,15 X_7,16,8,17 X_15,6,16,7 X_17,20,18,1 X_11,18,12,19 X_19,12,20,13 X_13,8,14,9 X_9,2,10,3

Gauss Code: {-1, 10, -2, 1, -3, 5, -4, 9, -10, 2, -7, 8, -9, 3, -5, 4, -6, 7, -8, 6}

DT (Dowker-Thistlethwaite) Code: 4 10 14 16 2 18 8 6 20 12

Minimum Braid Representative:

Length is 11, width is 4
Braid index is 4

A Morse Link Presentation:

3D Invariants:

Symmetry Type Unknotting Number 3-Genus Bridge/Super Bridge Index Nakanishi Index

Reversible 3 3 3 / NotAvailable 1

Alexander Polynomial: 3t^-3 - 9t^-2 + 16t^-1 - 19 + 16t - 9t² + 3t³

Conway Polynomial: 1 + 7z² + 9z⁴ + 3z⁶

Other knots with the same Alexander/Conway Polynomial: {K11a245, ...}

Determinant and Signature: {75, -6}

Jones Polynomial: q^-13 - 4q^-12 + 7q^-11 - 10q^-10 + 12q^-9 - 13q^-8 + 11q^-7 - 8q^-6 + 6q^-5 - 2q^-4 + q^-3

Other knots (up to mirrors) with the same Jones Polynomial: {...}

A2 (sl(3)) Invariant: q^-40 - 2q^-36 + q^-34 - 2q^-32 - 3q^-26 + 2q^-24 - 2q^-22 + 3q^-20 + 2q^-18 + 3q^-14 - q^-12 + q^-10

HOMFLY-PT Polynomial: 2a⁶ + 5a⁶z² + 4a⁶z⁴ + a⁶z⁶ + 2a⁸ + 9a⁸z² + 8a⁸z⁴ + 2a⁸z⁶ - 4a¹⁰ - 8a¹⁰z² - 3a¹⁰z⁴ + a¹² + a¹²z²

Kauffman Polynomial: - 2a⁶ + 5a⁶z² - 4a⁶z⁴ + a⁶z⁶ + a⁷z + 2a⁷z³ - 5a⁷z⁵ + 2a⁷z⁷ + 2a⁸ - 6a⁸z² + 8a⁸z⁴ - 8a⁸z⁶ + 3a⁸z⁸ - 5a⁹z + 20a⁹z³ - 22a⁹z⁵ + 6a⁹z⁷ + a⁹z⁹ + 4a¹⁰ - 8a¹⁰z² + 8a¹⁰z⁴ - 13a¹⁰z⁶ + 7a¹⁰z⁸ - 6a¹¹z + 22a¹¹z³ - 28a¹¹z⁵ + 11a¹¹z⁷ + a¹¹z⁹ + a¹² + 5a¹²z² - 13a¹²z⁴ + 3a¹²z⁶ + 4a¹²z⁸ + a¹³z³ - 7a¹³z⁵ + 7a¹³z⁷ + 2a¹⁴z² - 8a¹⁴z⁴ + 7a¹⁴z⁶ - 3a¹⁵z³ + 4a¹⁵z⁵ + a¹⁶z⁴

V₂ and V₃, the type 2 and 3 Vassiliev invariants: {7, -17}

Khovanov Homology:
(The squares with yellow highlighting are those on the "critical diagonals", where j-2r=s+1 or j-2r=s+1, where s=-6 is the signature of 10₆₆. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

t^rq^j r = -10 r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0

j = -5 1

j = -7 2 1

j = -9 4

j = -11 4 2

j = -13 7 4

j = -15 6 4

j = -17 6 7

j = -19 4 6

j = -21 3 6

j = -23 1 4

j = -25 3

j = -27 1

n Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2 q^-36 - 4q^-35 + 3q^-34 + 10q^-33 - 24q^-32 + 10q^-31 + 36q^-30 - 62q^-29 + 15q^-28 + 77q^-27 - 104q^-26 + 8q^-25 + 116q^-24 - 122q^-23 - 10q^-22 + 131q^-21 - 105q^-20 - 28q^-19 + 113q^-18 - 67q^-17 - 36q^-16 + 73q^-15 - 27q^-14 - 28q^-13 + 33q^-12 - 4q^-11 - 12q^-10 + 8q^-9 + q^-8 - 2q^-7 + q^-6

3 q^-69 - 4q^-68 + 3q^-67 + 6q^-66 - 4q^-65 - 16q^-64 + 7q^-63 + 39q^-62 - 20q^-61 - 65q^-60 + 24q^-59 + 118q^-58 - 38q^-57 - 183q^-56 + 39q^-55 + 277q^-54 - 37q^-53 - 378q^-52 + 11q^-51 + 490q^-50 + 28q^-49 - 584q^-48 - 92q^-47 + 661q^-46 + 160q^-45 - 696q^-44 - 237q^-43 + 701q^-42 + 301q^-41 - 660q^-40 - 365q^-39 + 601q^-38 + 403q^-37 - 514q^-36 - 426q^-35 + 407q^-34 + 436q^-33 - 307q^-32 - 405q^-31 + 187q^-30 + 378q^-29 - 107q^-28 - 301q^-27 + 16q^-26 + 243q^-25 + 18q^-24 - 155q^-23 - 56q^-22 + 105q^-21 + 43q^-20 - 45q^-19 - 41q^-18 + 23q^-17 + 21q^-16 - 4q^-15 - 12q^-14 + 3q^-13 + 3q^-12 + q^-11 - 2q^-10 + q^-9

4 q^-112 - 4q^-111 + 3q^-110 + 6q^-109 - 8q^-108 + 4q^-107 - 19q^-106 + 20q^-105 + 29q^-104 - 42q^-103 + 9q^-102 - 66q^-101 + 80q^-100 + 110q^-99 - 135q^-98 - 30q^-97 - 175q^-96 + 255q^-95 + 333q^-94 - 312q^-93 - 230q^-92 - 441q^-91 + 616q^-90 + 878q^-89 - 479q^-88 - 731q^-87 - 1070q^-86 + 1081q^-85 + 1881q^-84 - 368q^-83 - 1430q^-82 - 2200q^-81 + 1318q^-80 + 3138q^-79 + 237q^-78 - 1907q^-77 - 3575q^-76 + 1037q^-75 + 4094q^-74 + 1172q^-73 - 1809q^-72 - 4647q^-71 + 331q^-70 + 4344q^-69 + 2008q^-68 - 1196q^-67 - 5038q^-66 - 478q^-65 + 3893q^-64 + 2500q^-63 - 330q^-62 - 4771q^-61 - 1198q^-60 + 2979q^-59 + 2647q^-58 + 578q^-57 - 3999q^-56 - 1738q^-55 + 1786q^-54 + 2444q^-53 + 1378q^-52 - 2839q^-51 - 1946q^-50 + 539q^-49 + 1837q^-48 + 1821q^-47 - 1497q^-46 - 1656q^-45 - 401q^-44 + 940q^-43 + 1683q^-42 - 378q^-41 - 962q^-40 - 726q^-39 + 138q^-38 + 1072q^-37 + 171q^-36 - 279q^-35 - 523q^-34 - 218q^-33 + 433q^-32 + 197q^-31 + 49q^-30 - 194q^-29 - 186q^-28 + 92q^-27 + 64q^-26 + 73q^-25 - 28q^-24 - 67q^-23 + 10q^-22 + 2q^-21 + 23q^-20 + 2q^-19 - 13q^-18 + 3q^-17 - 2q^-16 + 3q^-15 + q^-14 - 2q^-13 + q^-12

5 q^-165 - 4q^-164 + 3q^-163 + 6q^-162 - 8q^-161 + q^-159 - 6q^-158 + 10q^-157 + 17q^-156 - 20q^-155 - 23q^-154 + 5q^-153 + 19q^-152 + 39q^-151 + 12q^-150 - 61q^-149 - 101q^-148 - 5q^-147 + 170q^-146 + 184q^-145 - 18q^-144 - 300q^-143 - 410q^-142 - 31q^-141 + 657q^-140 + 817q^-139 + 46q^-138 - 1078q^-137 - 1525q^-136 - 368q^-135 + 1791q^-134 + 2713q^-133 + 892q^-132 - 2537q^-131 - 4411q^-130 - 2072q^-129 + 3315q^-128 + 6721q^-127 + 3899q^-126 - 3764q^-125 - 9456q^-124 - 6640q^-123 + 3698q^-122 + 12370q^-121 + 10117q^-120 - 2753q^-119 - 15075q^-118 - 14217q^-117 + 965q^-116 + 17153q^-115 + 18385q^-114 + 1738q^-113 - 18285q^-112 - 22316q^-111 - 4942q^-110 + 18387q^-109 + 25406q^-108 + 8386q^-107 - 17462q^-106 - 27561q^-105 - 11583q^-104 + 15773q^-103 + 28564q^-102 + 14359q^-101 - 13581q^-100 - 28639q^-99 - 16437q^-98 + 11104q^-97 + 27810q^-96 + 18043q^-95 - 8553q^-94 - 26444q^-93 - 19023q^-92 + 5915q^-91 + 24507q^-90 + 19703q^-89 - 3206q^-88 - 22235q^-87 - 19991q^-86 + 446q^-85 + 19501q^-84 + 19912q^-83 + 2389q^-82 - 16321q^-81 - 19456q^-80 - 5102q^-79 + 12832q^-78 + 18254q^-77 + 7555q^-76 - 8880q^-75 - 16569q^-74 - 9441q^-73 + 5094q^-72 + 13934q^-71 + 10504q^-70 - 1276q^-69 - 10992q^-68 - 10611q^-67 - 1645q^-66 + 7475q^-65 + 9713q^-64 + 3966q^-63 - 4336q^-62 - 8010q^-61 - 4932q^-60 + 1377q^-59 + 5833q^-58 + 5178q^-57 + 510q^-56 - 3616q^-55 - 4307q^-54 - 1807q^-53 + 1689q^-52 + 3311q^-51 + 2028q^-50 - 368q^-49 - 1953q^-48 - 1881q^-47 - 422q^-46 + 1055q^-45 + 1298q^-44 + 631q^-43 - 294q^-42 - 807q^-41 - 589q^-40 + 8q^-39 + 358q^-38 + 388q^-37 + 145q^-36 - 146q^-35 - 217q^-34 - 95q^-33 + 6q^-32 + 93q^-31 + 81q^-30 + 2q^-29 - 40q^-28 - 17q^-27 - 17q^-26 + 5q^-25 + 20q^-24 + 3q^-23 - 7q^-22 + 2q^-21 - 2q^-20 - 2q^-19 + 3q^-18 + q^-17 - 2q^-16 + q^-15

6 q^-228 - 4q^-227 + 3q^-226 + 6q^-225 - 8q^-224 - 3q^-222 + 14q^-221 - 16q^-220 - 2q^-219 + 39q^-218 - 42q^-217 - 4q^-216 + 2q^-215 + 57q^-214 - 41q^-213 - 35q^-212 + 100q^-211 - 140q^-210 - 20q^-209 + 64q^-208 + 248q^-207 - 79q^-206 - 169q^-205 + 89q^-204 - 504q^-203 - 74q^-202 + 376q^-201 + 1014q^-200 + 158q^-199 - 527q^-198 - 506q^-197 - 1941q^-196 - 613q^-195 + 1353q^-194 + 3625q^-193 + 2006q^-192 - 656q^-191 - 2762q^-190 - 6736q^-189 - 3705q^-188 + 2700q^-187 + 10343q^-186 + 9133q^-185 + 2273q^-184 - 6914q^-183 - 18678q^-182 - 14669q^-181 + 838q^-180 + 22137q^-179 + 27114q^-178 + 15411q^-177 - 8703q^-176 - 39411q^-175 - 40753q^-174 - 13203q^-173 + 33653q^-172 + 57844q^-171 + 47459q^-170 + 2693q^-169 - 61888q^-168 - 83258q^-167 - 49065q^-166 + 31675q^-165 + 91783q^-164 + 98405q^-163 + 37579q^-162 - 70153q^-161 - 129424q^-160 - 104830q^-159 + 5020q^-158 + 110215q^-157 + 152282q^-156 + 92208q^-155 - 52386q^-154 - 158190q^-153 - 161908q^-152 - 41454q^-151 + 101207q^-150 + 186918q^-149 + 146792q^-148 - 14279q^-147 - 158247q^-146 - 198592q^-145 - 88400q^-144 + 71194q^-143 + 192656q^-142 + 181743q^-141 + 26139q^-140 - 136562q^-139 - 207712q^-138 - 119619q^-137 + 36403q^-136 + 177073q^-135 + 192865q^-134 + 56166q^-133 - 107215q^-132 - 197392q^-131 - 133783q^-130 + 6344q^-129 + 152178q^-128 + 188492q^-127 + 76464q^-126 - 76882q^-125 - 177616q^-124 - 138757q^-123 - 20779q^-122 + 122336q^-121 + 176291q^-120 + 93488q^-119 - 43178q^-118 - 150494q^-117 - 139215q^-116 - 49577q^-115 + 84670q^-114 + 155711q^-113 + 108395q^-112 - 3359q^-111 - 112182q^-110 - 131152q^-109 - 77814q^-108 + 37707q^-107 + 121253q^-106 + 113524q^-105 + 37275q^-104 - 61777q^-103 - 106650q^-102 - 94527q^-101 - 10940q^-100 + 72032q^-99 + 98738q^-98 + 64897q^-97 - 8764q^-96 - 64499q^-95 - 87954q^-94 - 45268q^-93 + 19104q^-92 + 63094q^-91 + 66699q^-90 + 29079q^-89 - 17154q^-88 - 58073q^-87 - 52186q^-86 - 18491q^-85 + 20800q^-84 + 44006q^-83 + 38976q^-82 + 15622q^-81 - 20922q^-80 - 34705q^-79 - 28920q^-78 - 8018q^-77 + 14209q^-76 + 25770q^-75 + 23531q^-74 + 3647q^-73 - 10973q^-72 - 18731q^-71 - 14862q^-70 - 4181q^-69 + 7614q^-68 + 14519q^-67 + 9264q^-66 + 2619q^-65 - 5186q^-64 - 8412q^-63 - 7223q^-62 - 1819q^-61 + 4099q^-60 + 4838q^-59 + 4331q^-58 + 1087q^-57 - 1710q^-56 - 3504q^-55 - 2577q^-54 - 232q^-53 + 759q^-52 + 1806q^-51 + 1363q^-50 + 517q^-49 - 718q^-48 - 945q^-47 - 484q^-46 - 312q^-45 + 267q^-44 + 417q^-43 + 399q^-42 - 8q^-41 - 145q^-40 - 92q^-39 - 169q^-38 - 30q^-37 + 45q^-36 + 109q^-35 + 12q^-34 - 11q^-33 + 12q^-32 - 33q^-31 - 15q^-30 - 4q^-29 + 22q^-28 - 6q^-26 + 8q^-25 - 3q^-24 - 2q^-23 - 2q^-22 + 3q^-21 + q^-20 - 2q^-19 + q^-18

Computer Talk. The data above can be recomputed by Mathematica using the package KnotTheory`. Following setup, the sample Mathematica session below reproduces most of the above data (Mathematica system prompts in blue, human input in red, Mathematica output in black):

In[1]:=

<< KnotTheory`

Loading KnotTheory` (version of August 30, 2005, 10:15:35)...

In[2]:=
PD[Knot[10, 66]]

Out[2]=
PD[X[1, 4, 2, 5], X[3, 10, 4, 11], X[5, 14, 6, 15], X[7, 16, 8, 17], > X[15, 6, 16, 7], X[17, 20, 18, 1], X[11, 18, 12, 19], X[19, 12, 20, 13], > X[13, 8, 14, 9], X[9, 2, 10, 3]]

In[3]:=
GaussCode[Knot[10, 66]]

Out[3]=
GaussCode[-1, 10, -2, 1, -3, 5, -4, 9, -10, 2, -7, 8, -9, 3, -5, 4, -6, 7, -8, > 6]

In[4]:=
DTCode[Knot[10, 66]]

Out[4]=
DTCode[4, 10, 14, 16, 2, 18, 8, 6, 20, 12]

In[5]:=
br = BR[Knot[10, 66]]

Out[5]=
BR[4, {-1, -1, -1, 2, -1, -3, -2, -2, -2, -3, -3}]

In[6]:=
{First[br], Crossings[br]}

Out[6]=
{4, 11}

In[7]:=
BraidIndex[Knot[10, 66]]

Out[7]=
4

In[8]:=
Show[DrawMorseLink[Knot[10, 66]]]

Out[8]=
-Graphics-

In[9]:=
#[Knot[10, 66]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}

Out[9]=
{Reversible, 3, 3, 3, NotAvailable, 1}

In[10]:=
alex = Alexander[Knot[10, 66]][t]

Out[10]=
3 9 16 2 3 -19 + -- - -- + -- + 16 t - 9 t + 3 t 3 2 t t t

In[11]:=
Conway[Knot[10, 66]][z]

Out[11]=
2 4 6 1 + 7 z + 9 z + 3 z

In[12]:=
Select[AllKnots[], (alex === Alexander[#][t])&]

Out[12]=
{Knot[10, 66], Knot[11, Alternating, 245]}

In[13]:=
{KnotDet[Knot[10, 66]], KnotSignature[Knot[10, 66]]}

Out[13]=
{75, -6}

In[14]:=
Jones[Knot[10, 66]][q]

Out[14]=
-13 4 7 10 12 13 11 8 6 2 -3 q - --- + --- - --- + -- - -- + -- - -- + -- - -- + q 12 11 10 9 8 7 6 5 4 q q q q q q q q q

In[15]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]

Out[15]=
{Knot[10, 66]}

In[16]:=
A2Invariant[Knot[10, 66]][q]

Out[16]=
-40 2 -34 2 3 2 2 3 2 3 -12 -10 q - --- + q - --- - --- + --- - --- + --- + --- + --- - q + q 36 32 26 24 22 20 18 14 q q q q q q q q

In[17]:=
HOMFLYPT[Knot[10, 66]][a, z]

Out[17]=
6 8 10 12 6 2 8 2 10 2 12 2 6 4 2 a + 2 a - 4 a + a + 5 a z + 9 a z - 8 a z + a z + 4 a z + 8 4 10 4 6 6 8 6 > 8 a z - 3 a z + a z + 2 a z

In[18]:=
Kauffman[Knot[10, 66]][a, z]

Out[18]=
6 8 10 12 7 9 11 6 2 8 2 -2 a + 2 a + 4 a + a + a z - 5 a z - 6 a z + 5 a z - 6 a z - 10 2 12 2 14 2 7 3 9 3 11 3 13 3 > 8 a z + 5 a z + 2 a z + 2 a z + 20 a z + 22 a z + a z - 15 3 6 4 8 4 10 4 12 4 14 4 16 4 > 3 a z - 4 a z + 8 a z + 8 a z - 13 a z - 8 a z + a z - 7 5 9 5 11 5 13 5 15 5 6 6 8 6 > 5 a z - 22 a z - 28 a z - 7 a z + 4 a z + a z - 8 a z - 10 6 12 6 14 6 7 7 9 7 11 7 > 13 a z + 3 a z + 7 a z + 2 a z + 6 a z + 11 a z + 13 7 8 8 10 8 12 8 9 9 11 9 > 7 a z + 3 a z + 7 a z + 4 a z + a z + a z

In[19]:=
{Vassiliev[2][Knot[10, 66]], Vassiliev[3][Knot[10, 66]]}

Out[19]=
{7, -17}

In[20]:=
Kh[Knot[10, 66]][q, t]

Out[20]=
-7 -5 1 3 1 4 3 6 4 q + q + ------- + ------ + ------ + ------ + ------ + ------ + ------ + 27 10 25 9 23 9 23 8 21 8 21 7 19 7 q t q t q t q t q t q t q t 6 6 7 6 4 7 4 4 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 19 6 17 6 17 5 15 5 15 4 13 4 13 3 11 3 q t q t q t q t q t q t q t q t 2 4 2 > ------ + ----- + ---- 11 2 9 2 7 q t q t q t

In[21]:=
ColouredJones[Knot[10, 66], 2][q]

Out[21]=
-36 4 3 10 24 10 36 62 15 77 104 8 116 q - --- + --- + --- - --- + --- + --- - --- + --- + --- - --- + --- + --- - 35 34 33 32 31 30 29 28 27 26 25 24 q q q q q q q q q q q q 122 10 131 105 28 113 67 36 73 27 28 33 > --- - --- + --- - --- - --- + --- - --- - --- + --- - --- - --- + --- - 23 22 21 20 19 18 17 16 15 14 13 12 q q q q q q q q q q q q 4 12 8 -8 2 -6 > --- - --- + -- + q - -- + q 11 10 9 7 q q q q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 10₆₆

10₆₅
10₆₇

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q^-36 - 4q^-35 + 3q^-34 + 10q^-33 - 24q^-32 + 10q^-31 + 36q^-30 - 62q^-29 + 15q^-28 + 77q^-27 - 104q^-26 + 8q^-25 + 116q^-24 - 122q^-23 - 10q^-22 + 131q^-21 - 105q^-20 - 28q^-19 + 113q^-18 - 67q^-17 - 36q^-16 + 73q^-15 - 27q^-14 - 28q^-13 + 33q^-12 - 4q^-11 - 12q^-10 + 8q^-9 + q^-8 - 2q^-7 + q^-6
3	q^-69 - 4q^-68 + 3q^-67 + 6q^-66 - 4q^-65 - 16q^-64 + 7q^-63 + 39q^-62 - 20q^-61 - 65q^-60 + 24q^-59 + 118q^-58 - 38q^-57 - 183q^-56 + 39q^-55 + 277q^-54 - 37q^-53 - 378q^-52 + 11q^-51 + 490q^-50 + 28q^-49 - 584q^-48 - 92q^-47 + 661q^-46 + 160q^-45 - 696q^-44 - 237q^-43 + 701q^-42 + 301q^-41 - 660q^-40 - 365q^-39 + 601q^-38 + 403q^-37 - 514q^-36 - 426q^-35 + 407q^-34 + 436q^-33 - 307q^-32 - 405q^-31 + 187q^-30 + 378q^-29 - 107q^-28 - 301q^-27 + 16q^-26 + 243q^-25 + 18q^-24 - 155q^-23 - 56q^-22 + 105q^-21 + 43q^-20 - 45q^-19 - 41q^-18 + 23q^-17 + 21q^-16 - 4q^-15 - 12q^-14 + 3q^-13 + 3q^-12 + q^-11 - 2q^-10 + q^-9
4	q^-112 - 4q^-111 + 3q^-110 + 6q^-109 - 8q^-108 + 4q^-107 - 19q^-106 + 20q^-105 + 29q^-104 - 42q^-103 + 9q^-102 - 66q^-101 + 80q^-100 + 110q^-99 - 135q^-98 - 30q^-97 - 175q^-96 + 255q^-95 + 333q^-94 - 312q^-93 - 230q^-92 - 441q^-91 + 616q^-90 + 878q^-89 - 479q^-88 - 731q^-87 - 1070q^-86 + 1081q^-85 + 1881q^-84 - 368q^-83 - 1430q^-82 - 2200q^-81 + 1318q^-80 + 3138q^-79 + 237q^-78 - 1907q^-77 - 3575q^-76 + 1037q^-75 + 4094q^-74 + 1172q^-73 - 1809q^-72 - 4647q^-71 + 331q^-70 + 4344q^-69 + 2008q^-68 - 1196q^-67 - 5038q^-66 - 478q^-65 + 3893q^-64 + 2500q^-63 - 330q^-62 - 4771q^-61 - 1198q^-60 + 2979q^-59 + 2647q^-58 + 578q^-57 - 3999q^-56 - 1738q^-55 + 1786q^-54 + 2444q^-53 + 1378q^-52 - 2839q^-51 - 1946q^-50 + 539q^-49 + 1837q^-48 + 1821q^-47 - 1497q^-46 - 1656q^-45 - 401q^-44 + 940q^-43 + 1683q^-42 - 378q^-41 - 962q^-40 - 726q^-39 + 138q^-38 + 1072q^-37 + 171q^-36 - 279q^-35 - 523q^-34 - 218q^-33 + 433q^-32 + 197q^-31 + 49q^-30 - 194q^-29 - 186q^-28 + 92q^-27 + 64q^-26 + 73q^-25 - 28q^-24 - 67q^-23 + 10q^-22 + 2q^-21 + 23q^-20 + 2q^-19 - 13q^-18 + 3q^-17 - 2q^-16 + 3q^-15 + q^-14 - 2q^-13 + q^-12
5	q^-165 - 4q^-164 + 3q^-163 + 6q^-162 - 8q^-161 + q^-159 - 6q^-158 + 10q^-157 + 17q^-156 - 20q^-155 - 23q^-154 + 5q^-153 + 19q^-152 + 39q^-151 + 12q^-150 - 61q^-149 - 101q^-148 - 5q^-147 + 170q^-146 + 184q^-145 - 18q^-144 - 300q^-143 - 410q^-142 - 31q^-141 + 657q^-140 + 817q^-139 + 46q^-138 - 1078q^-137 - 1525q^-136 - 368q^-135 + 1791q^-134 + 2713q^-133 + 892q^-132 - 2537q^-131 - 4411q^-130 - 2072q^-129 + 3315q^-128 + 6721q^-127 + 3899q^-126 - 3764q^-125 - 9456q^-124 - 6640q^-123 + 3698q^-122 + 12370q^-121 + 10117q^-120 - 2753q^-119 - 15075q^-118 - 14217q^-117 + 965q^-116 + 17153q^-115 + 18385q^-114 + 1738q^-113 - 18285q^-112 - 22316q^-111 - 4942q^-110 + 18387q^-109 + 25406q^-108 + 8386q^-107 - 17462q^-106 - 27561q^-105 - 11583q^-104 + 15773q^-103 + 28564q^-102 + 14359q^-101 - 13581q^-100 - 28639q^-99 - 16437q^-98 + 11104q^-97 + 27810q^-96 + 18043q^-95 - 8553q^-94 - 26444q^-93 - 19023q^-92 + 5915q^-91 + 24507q^-90 + 19703q^-89 - 3206q^-88 - 22235q^-87 - 19991q^-86 + 446q^-85 + 19501q^-84 + 19912q^-83 + 2389q^-82 - 16321q^-81 - 19456q^-80 - 5102q^-79 + 12832q^-78 + 18254q^-77 + 7555q^-76 - 8880q^-75 - 16569q^-74 - 9441q^-73 + 5094q^-72 + 13934q^-71 + 10504q^-70 - 1276q^-69 - 10992q^-68 - 10611q^-67 - 1645q^-66 + 7475q^-65 + 9713q^-64 + 3966q^-63 - 4336q^-62 - 8010q^-61 - 4932q^-60 + 1377q^-59 + 5833q^-58 + 5178q^-57 + 510q^-56 - 3616q^-55 - 4307q^-54 - 1807q^-53 + 1689q^-52 + 3311q^-51 + 2028q^-50 - 368q^-49 - 1953q^-48 - 1881q^-47 - 422q^-46 + 1055q^-45 + 1298q^-44 + 631q^-43 - 294q^-42 - 807q^-41 - 589q^-40 + 8q^-39 + 358q^-38 + 388q^-37 + 145q^-36 - 146q^-35 - 217q^-34 - 95q^-33 + 6q^-32 + 93q^-31 + 81q^-30 + 2q^-29 - 40q^-28 - 17q^-27 - 17q^-26 + 5q^-25 + 20q^-24 + 3q^-23 - 7q^-22 + 2q^-21 - 2q^-20 - 2q^-19 + 3q^-18 + q^-17 - 2q^-16 + q^-15
6	q^-228 - 4q^-227 + 3q^-226 + 6q^-225 - 8q^-224 - 3q^-222 + 14q^-221 - 16q^-220 - 2q^-219 + 39q^-218 - 42q^-217 - 4q^-216 + 2q^-215 + 57q^-214 - 41q^-213 - 35q^-212 + 100q^-211 - 140q^-210 - 20q^-209 + 64q^-208 + 248q^-207 - 79q^-206 - 169q^-205 + 89q^-204 - 504q^-203 - 74q^-202 + 376q^-201 + 1014q^-200 + 158q^-199 - 527q^-198 - 506q^-197 - 1941q^-196 - 613q^-195 + 1353q^-194 + 3625q^-193 + 2006q^-192 - 656q^-191 - 2762q^-190 - 6736q^-189 - 3705q^-188 + 2700q^-187 + 10343q^-186 + 9133q^-185 + 2273q^-184 - 6914q^-183 - 18678q^-182 - 14669q^-181 + 838q^-180 + 22137q^-179 + 27114q^-178 + 15411q^-177 - 8703q^-176 - 39411q^-175 - 40753q^-174 - 13203q^-173 + 33653q^-172 + 57844q^-171 + 47459q^-170 + 2693q^-169 - 61888q^-168 - 83258q^-167 - 49065q^-166 + 31675q^-165 + 91783q^-164 + 98405q^-163 + 37579q^-162 - 70153q^-161 - 129424q^-160 - 104830q^-159 + 5020q^-158 + 110215q^-157 + 152282q^-156 + 92208q^-155 - 52386q^-154 - 158190q^-153 - 161908q^-152 - 41454q^-151 + 101207q^-150 + 186918q^-149 + 146792q^-148 - 14279q^-147 - 158247q^-146 - 198592q^-145 - 88400q^-144 + 71194q^-143 + 192656q^-142 + 181743q^-141 + 26139q^-140 - 136562q^-139 - 207712q^-138 - 119619q^-137 + 36403q^-136 + 177073q^-135 + 192865q^-134 + 56166q^-133 - 107215q^-132 - 197392q^-131 - 133783q^-130 + 6344q^-129 + 152178q^-128 + 188492q^-127 + 76464q^-126 - 76882q^-125 - 177616q^-124 - 138757q^-123 - 20779q^-122 + 122336q^-121 + 176291q^-120 + 93488q^-119 - 43178q^-118 - 150494q^-117 - 139215q^-116 - 49577q^-115 + 84670q^-114 + 155711q^-113 + 108395q^-112 - 3359q^-111 - 112182q^-110 - 131152q^-109 - 77814q^-108 + 37707q^-107 + 121253q^-106 + 113524q^-105 + 37275q^-104 - 61777q^-103 - 106650q^-102 - 94527q^-101 - 10940q^-100 + 72032q^-99 + 98738q^-98 + 64897q^-97 - 8764q^-96 - 64499q^-95 - 87954q^-94 - 45268q^-93 + 19104q^-92 + 63094q^-91 + 66699q^-90 + 29079q^-89 - 17154q^-88 - 58073q^-87 - 52186q^-86 - 18491q^-85 + 20800q^-84 + 44006q^-83 + 38976q^-82 + 15622q^-81 - 20922q^-80 - 34705q^-79 - 28920q^-78 - 8018q^-77 + 14209q^-76 + 25770q^-75 + 23531q^-74 + 3647q^-73 - 10973q^-72 - 18731q^-71 - 14862q^-70 - 4181q^-69 + 7614q^-68 + 14519q^-67 + 9264q^-66 + 2619q^-65 - 5186q^-64 - 8412q^-63 - 7223q^-62 - 1819q^-61 + 4099q^-60 + 4838q^-59 + 4331q^-58 + 1087q^-57 - 1710q^-56 - 3504q^-55 - 2577q^-54 - 232q^-53 + 759q^-52 + 1806q^-51 + 1363q^-50 + 517q^-49 - 718q^-48 - 945q^-47 - 484q^-46 - 312q^-45 + 267q^-44 + 417q^-43 + 399q^-42 - 8q^-41 - 145q^-40 - 92q^-39 - 169q^-38 - 30q^-37 + 45q^-36 + 109q^-35 + 12q^-34 - 11q^-33 + 12q^-32 - 33q^-31 - 15q^-30 - 4q^-29 + 22q^-28 - 6q^-26 + 8q^-25 - 3q^-24 - 2q^-23 - 2q^-22 + 3q^-21 + q^-20 - 2q^-19 + q^-18

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	PD[Knot[10, 66]]
Out[2]=	PD[X[1, 4, 2, 5], X[3, 10, 4, 11], X[5, 14, 6, 15], X[7, 16, 8, 17], > X[15, 6, 16, 7], X[17, 20, 18, 1], X[11, 18, 12, 19], X[19, 12, 20, 13], > X[13, 8, 14, 9], X[9, 2, 10, 3]]
In[3]:=	GaussCode[Knot[10, 66]]
Out[3]=	GaussCode[-1, 10, -2, 1, -3, 5, -4, 9, -10, 2, -7, 8, -9, 3, -5, 4, -6, 7, -8, > 6]
In[4]:=	DTCode[Knot[10, 66]]
Out[4]=	DTCode[4, 10, 14, 16, 2, 18, 8, 6, 20, 12]
In[5]:=	br = BR[Knot[10, 66]]
Out[5]=	BR[4, {-1, -1, -1, 2, -1, -3, -2, -2, -2, -3, -3}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{4, 11}
In[7]:=	BraidIndex[Knot[10, 66]]
Out[7]=	4
In[8]:=	Show[DrawMorseLink[Knot[10, 66]]]

Out[8]=	-Graphics-
In[9]:=	#[Knot[10, 66]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 3, 3, 3, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 66]][t]
Out[10]=	3 9 16 2 3 -19 + -- - -- + -- + 16 t - 9 t + 3 t 3 2 t t t
In[11]:=	Conway[Knot[10, 66]][z]
Out[11]=	2 4 6 1 + 7 z + 9 z + 3 z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 66], Knot[11, Alternating, 245]}
In[13]:=	{KnotDet[Knot[10, 66]], KnotSignature[Knot[10, 66]]}
Out[13]=	{75, -6}
In[14]:=	Jones[Knot[10, 66]][q]
Out[14]=	-13 4 7 10 12 13 11 8 6 2 -3 q - --- + --- - --- + -- - -- + -- - -- + -- - -- + q 12 11 10 9 8 7 6 5 4 q q q q q q q q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] \|\| (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[10, 66]}
In[16]:=	A2Invariant[Knot[10, 66]][q]
Out[16]=	-40 2 -34 2 3 2 2 3 2 3 -12 -10 q - --- + q - --- - --- + --- - --- + --- + --- + --- - q + q 36 32 26 24 22 20 18 14 q q q q q q q q
In[17]:=	HOMFLYPT[Knot[10, 66]][a, z]
Out[17]=	6 8 10 12 6 2 8 2 10 2 12 2 6 4 2 a + 2 a - 4 a + a + 5 a z + 9 a z - 8 a z + a z + 4 a z + 8 4 10 4 6 6 8 6 > 8 a z - 3 a z + a z + 2 a z
In[18]:=	Kauffman[Knot[10, 66]][a, z]
Out[18]=	6 8 10 12 7 9 11 6 2 8 2 -2 a + 2 a + 4 a + a + a z - 5 a z - 6 a z + 5 a z - 6 a z - 10 2 12 2 14 2 7 3 9 3 11 3 13 3 > 8 a z + 5 a z + 2 a z + 2 a z + 20 a z + 22 a z + a z - 15 3 6 4 8 4 10 4 12 4 14 4 16 4 > 3 a z - 4 a z + 8 a z + 8 a z - 13 a z - 8 a z + a z - 7 5 9 5 11 5 13 5 15 5 6 6 8 6 > 5 a z - 22 a z - 28 a z - 7 a z + 4 a z + a z - 8 a z - 10 6 12 6 14 6 7 7 9 7 11 7 > 13 a z + 3 a z + 7 a z + 2 a z + 6 a z + 11 a z + 13 7 8 8 10 8 12 8 9 9 11 9 > 7 a z + 3 a z + 7 a z + 4 a z + a z + a z
In[19]:=	{Vassiliev[2][Knot[10, 66]], Vassiliev[3][Knot[10, 66]]}
Out[19]=	{7, -17}
In[20]:=	Kh[Knot[10, 66]][q, t]
Out[20]=	-7 -5 1 3 1 4 3 6 4 q + q + ------- + ------ + ------ + ------ + ------ + ------ + ------ + 27 10 25 9 23 9 23 8 21 8 21 7 19 7 q t q t q t q t q t q t q t 6 6 7 6 4 7 4 4 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 19 6 17 6 17 5 15 5 15 4 13 4 13 3 11 3 q t q t q t q t q t q t q t q t 2 4 2 > ------ + ----- + ---- 11 2 9 2 7 q t q t q t
In[21]:=	ColouredJones[Knot[10, 66], 2][q]
Out[21]=	-36 4 3 10 24 10 36 62 15 77 104 8 116 q - --- + --- + --- - --- + --- + --- - --- + --- + --- - --- + --- + --- - 35 34 33 32 31 30 29 28 27 26 25 24 q q q q q q q q q q q q 122 10 131 105 28 113 67 36 73 27 28 33 > --- - --- + --- - --- - --- + --- - --- - --- + --- - --- - --- + --- - 23 22 21 20 19 18 17 16 15 14 13 12 q q q q q q q q q q q q 4 12 8 -8 2 -6 > --- - --- + -- + q - -- + q 11 10 9 7 q q q q

The Alternating Knot 1066

The Alternating Knot 10₆₆