The Alternating Knot 9₉

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Acknowledgement

KnotPlot

PD Presentation: X₁₆₂₇ X_3,12,4,13 X_7,16,8,17 X_9,18,10,1 X_17,8,18,9 X_15,10,16,11 X_5,14,6,15 X_11,2,12,3 X_13,4,14,5

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

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

Minimum Braid Representative:

Length is 10, width is 3
Braid index is 3

A Morse Link Presentation:

3D Invariants:

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

Reversible 3 3 2 / 4--6 1

Alexander Polynomial: 2t^-3 - 4t^-2 + 6t^-1 - 7 + 6t - 4t² + 2t³

Conway Polynomial: 1 + 8z² + 8z⁴ + 2z⁶

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

Determinant and Signature: {31, -6}

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

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

A2 (sl(3)) Invariant: - q^-36 - q^-32 - q^-30 - q^-26 + 2q^-24 + q^-20 + q^-18 + 2q^-14 + q^-10

HOMFLY-PT Polynomial: 2a⁶ + 7a⁶z² + 5a⁶z⁴ + a⁶z⁶ + a⁸ + 4a⁸z² + 4a⁸z⁴ + a⁸z⁶ - 2a¹⁰ - 3a¹⁰z² - a¹⁰z⁴

Kauffman Polynomial: - 2a⁶ + 7a⁶z² - 5a⁶z⁴ + a⁶z⁶ + a⁷z + a⁷z³ - 3a⁷z⁵ + a⁷z⁷ + a⁸ - 3a⁸z² + 3a⁸z⁴ - 3a⁸z⁶ + a⁸z⁸ - 2a⁹z + 5a⁹z³ - 8a⁹z⁵ + 3a⁹z⁷ + 2a¹⁰ - 6a¹⁰z² + 2a¹⁰z⁴ - a¹⁰z⁶ + a¹⁰z⁸ - 2a¹¹z⁵ + 2a¹¹z⁷ + 3a¹²z² - 4a¹²z⁴ + 3a¹²z⁶ + 2a¹³z - 3a¹³z³ + 3a¹³z⁵ - a¹⁴z² + 2a¹⁴z⁴ - a¹⁵z + a¹⁵z³

V₂ and V₃, the type 2 and 3 Vassiliev invariants: {8, -22}

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 9₉. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

t^rq^j 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 1 1

j = -9 2

j = -11 2 1

j = -13 3 2

j = -15 2 2

j = -17 3 3

j = -19 1 2

j = -21 1 3

j = -23 1

j = -25 1

n Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2 q^-33 - 2q^-32 + q^-31 + 4q^-30 - 8q^-29 + 3q^-28 + 10q^-27 - 18q^-26 + 6q^-25 + 17q^-24 - 25q^-23 + 6q^-22 + 20q^-21 - 25q^-20 + 2q^-19 + 21q^-18 - 19q^-17 - 2q^-16 + 17q^-15 - 11q^-14 - 4q^-13 + 10q^-12 - 4q^-11 - 3q^-10 + 4q^-9 - q^-7 + q^-6

3 - q^-63 + 2q^-62 - q^-61 - q^-60 - q^-59 + 5q^-58 - q^-57 - 6q^-56 + q^-55 + 11q^-54 - 6q^-53 - 13q^-52 + 7q^-51 + 23q^-50 - 15q^-49 - 28q^-48 + 18q^-47 + 37q^-46 - 21q^-45 - 44q^-44 + 22q^-43 + 48q^-42 - 18q^-41 - 53q^-40 + 15q^-39 + 51q^-38 - 7q^-37 - 51q^-36 + q^-35 + 46q^-34 + 8q^-33 - 45q^-32 - 11q^-31 + 36q^-30 + 19q^-29 - 33q^-28 - 19q^-27 + 22q^-26 + 24q^-25 - 18q^-24 - 18q^-23 + 6q^-22 + 20q^-21 - 5q^-20 - 11q^-19 - 3q^-18 + 10q^-17 + q^-16 - 3q^-15 - 3q^-14 + 3q^-13 + q^-12 - q^-10 + q^-9

4 q^-102 - 2q^-101 + q^-100 + q^-99 - 2q^-98 + 4q^-97 - 7q^-96 + 3q^-95 + 4q^-94 - 6q^-93 + 11q^-92 - 16q^-91 + 8q^-90 + 7q^-89 - 19q^-88 + 21q^-87 - 20q^-86 + 24q^-85 + 6q^-84 - 50q^-83 + 27q^-82 - 17q^-81 + 60q^-80 + 12q^-79 - 100q^-78 + 14q^-77 - 14q^-76 + 113q^-75 + 33q^-74 - 144q^-73 - 12q^-72 - 27q^-71 + 152q^-70 + 65q^-69 - 157q^-68 - 32q^-67 - 54q^-66 + 158q^-65 + 87q^-64 - 138q^-63 - 30q^-62 - 82q^-61 + 137q^-60 + 94q^-59 - 104q^-58 - 16q^-57 - 103q^-56 + 103q^-55 + 91q^-54 - 61q^-53 - q^-52 - 119q^-51 + 63q^-50 + 85q^-49 - 17q^-48 + 15q^-47 - 121q^-46 + 19q^-45 + 65q^-44 + 17q^-43 + 38q^-42 - 102q^-41 - 17q^-40 + 31q^-39 + 28q^-38 + 54q^-37 - 61q^-36 - 28q^-35 - 3q^-34 + 15q^-33 + 51q^-32 - 21q^-31 - 16q^-30 - 16q^-29 - 3q^-28 + 30q^-27 - 2q^-26 - q^-25 - 10q^-24 - 8q^-23 + 11q^-22 + 3q^-20 - 2q^-19 - 4q^-18 + 3q^-17 + q^-15 - q^-13 + q^-12

5 - q^-150 + 2q^-149 - q^-148 - q^-147 + 2q^-146 - q^-145 - 2q^-144 + 5q^-143 - q^-142 - 5q^-141 + 3q^-140 - q^-139 - 4q^-138 + 10q^-137 + q^-136 - 5q^-135 - 5q^-133 - 9q^-132 + 6q^-131 + 13q^-130 + 11q^-129 + 8q^-128 - 20q^-127 - 40q^-126 - 15q^-125 + 29q^-124 + 65q^-123 + 53q^-122 - 40q^-121 - 118q^-120 - 83q^-119 + 37q^-118 + 157q^-117 + 160q^-116 - 29q^-115 - 221q^-114 - 218q^-113 - 3q^-112 + 254q^-111 + 305q^-110 + 48q^-109 - 288q^-108 - 372q^-107 - 98q^-106 + 288q^-105 + 424q^-104 + 158q^-103 - 275q^-102 - 458q^-101 - 200q^-100 + 249q^-99 + 458q^-98 + 233q^-97 - 207q^-96 - 453q^-95 - 248q^-94 + 181q^-93 + 416q^-92 + 251q^-91 - 139q^-90 - 388q^-89 - 246q^-88 + 115q^-87 + 340q^-86 + 240q^-85 - 74q^-84 - 308q^-83 - 231q^-82 + 47q^-81 + 253q^-80 + 225q^-79 + 3q^-78 - 215q^-77 - 218q^-76 - 29q^-75 + 146q^-74 + 201q^-73 + 87q^-72 - 103q^-71 - 178q^-70 - 104q^-69 + 26q^-68 + 143q^-67 + 139q^-66 + 15q^-65 - 95q^-64 - 128q^-63 - 80q^-62 + 47q^-61 + 126q^-60 + 92q^-59 + 10q^-58 - 80q^-57 - 125q^-56 - 45q^-55 + 56q^-54 + 90q^-53 + 77q^-52 + 4q^-51 - 89q^-50 - 79q^-49 - 17q^-48 + 33q^-47 + 70q^-46 + 55q^-45 - 22q^-44 - 49q^-43 - 40q^-42 - 15q^-41 + 25q^-40 + 46q^-39 + 12q^-38 - 7q^-37 - 20q^-36 - 23q^-35 - 4q^-34 + 18q^-33 + 8q^-32 + 7q^-31 - q^-30 - 10q^-29 - 7q^-28 + 5q^-27 + 3q^-25 + 3q^-24 - 2q^-23 - 3q^-22 + 2q^-21 + q^-18 - q^-16 + q^-15

6 q^-207 - 2q^-206 + q^-205 + q^-204 - 2q^-203 + q^-202 - q^-201 + 4q^-200 - 7q^-199 + 2q^-198 + 8q^-197 - 7q^-196 + 2q^-195 - 2q^-194 + 7q^-193 - 18q^-192 + 2q^-191 + 22q^-190 - 12q^-189 + 7q^-188 + q^-187 + 12q^-186 - 46q^-185 - 8q^-184 + 38q^-183 - 17q^-182 + 33q^-181 + 25q^-180 + 25q^-179 - 102q^-178 - 58q^-177 + 36q^-176 - 18q^-175 + 110q^-174 + 113q^-173 + 62q^-172 - 207q^-171 - 199q^-170 - 37q^-169 - 25q^-168 + 280q^-167 + 333q^-166 + 181q^-165 - 357q^-164 - 483q^-163 - 256q^-162 - 103q^-161 + 528q^-160 + 722q^-159 + 465q^-158 - 451q^-157 - 860q^-156 - 653q^-155 - 353q^-154 + 713q^-153 + 1189q^-152 + 923q^-151 - 358q^-150 - 1139q^-149 - 1099q^-148 - 767q^-147 + 681q^-146 + 1506q^-145 + 1395q^-144 - 86q^-143 - 1168q^-142 - 1371q^-141 - 1164q^-140 + 467q^-139 + 1543q^-138 + 1662q^-137 + 183q^-136 - 1002q^-135 - 1377q^-134 - 1359q^-133 + 244q^-132 + 1385q^-131 + 1676q^-130 + 306q^-129 - 800q^-128 - 1223q^-127 - 1348q^-126 + 109q^-125 + 1178q^-124 + 1556q^-123 + 315q^-122 - 623q^-121 - 1040q^-120 - 1256q^-119 + 11q^-118 + 964q^-117 + 1413q^-116 + 327q^-115 - 425q^-114 - 853q^-113 - 1175q^-112 - 133q^-111 + 707q^-110 + 1255q^-109 + 388q^-108 - 162q^-107 - 618q^-106 - 1077q^-105 - 325q^-104 + 374q^-103 + 1024q^-102 + 440q^-101 + 139q^-100 - 304q^-99 - 885q^-98 - 478q^-97 + 5q^-96 + 683q^-95 + 381q^-94 + 373q^-93 + 50q^-92 - 558q^-91 - 481q^-90 - 282q^-89 + 284q^-88 + 167q^-87 + 420q^-86 + 313q^-85 - 164q^-84 - 291q^-83 - 357q^-82 - 26q^-81 - 124q^-80 + 253q^-79 + 355q^-78 + 126q^-77 - 13q^-76 - 206q^-75 - 113q^-74 - 308q^-73 + 5q^-72 + 185q^-71 + 182q^-70 + 156q^-69 + 14q^-68 - 5q^-67 - 276q^-66 - 129q^-65 - 18q^-64 + 64q^-63 + 129q^-62 + 113q^-61 + 117q^-60 - 120q^-59 - 93q^-58 - 89q^-57 - 43q^-56 + 16q^-55 + 70q^-54 + 124q^-53 - 9q^-52 - 7q^-51 - 45q^-50 - 48q^-49 - 42q^-48 + 4q^-47 + 61q^-46 + 8q^-45 + 24q^-44 - 11q^-42 - 31q^-41 - 15q^-40 + 18q^-39 - 4q^-38 + 12q^-37 + 7q^-36 + 5q^-35 - 10q^-34 - 8q^-33 + 6q^-32 - 5q^-31 + 2q^-30 + 2q^-29 + 4q^-28 - 2q^-27 - 3q^-26 + 3q^-25 - q^-24 + q^-21 - q^-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[9, 9]]

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

In[3]:=
GaussCode[Knot[9, 9]]

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

In[4]:=
DTCode[Knot[9, 9]]

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

In[5]:=
br = BR[Knot[9, 9]]

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

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

Out[6]=
{3, 10}

In[7]:=
BraidIndex[Knot[9, 9]]

Out[7]=
3

In[8]:=
Show[DrawMorseLink[Knot[9, 9]]]

Out[8]=
-Graphics-

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

Out[9]=
{Reversible, 3, 3, 2, {4, 6}, 1}

In[10]:=
alex = Alexander[Knot[9, 9]][t]

Out[10]=
2 4 6 2 3 -7 + -- - -- + - + 6 t - 4 t + 2 t 3 2 t t t

In[11]:=
Conway[Knot[9, 9]][z]

Out[11]=
2 4 6 1 + 8 z + 8 z + 2 z

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

Out[12]=
{Knot[9, 9]}

In[13]:=
{KnotDet[Knot[9, 9]], KnotSignature[Knot[9, 9]]}

Out[13]=
{31, -6}

In[14]:=
Jones[Knot[9, 9]][q]

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

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

Out[15]=
{Knot[9, 9]}

In[16]:=
A2Invariant[Knot[9, 9]][q]

Out[16]=
-36 -32 -30 -26 2 -20 -18 2 -10 -q - q - q - q + --- + q + q + --- + q 24 14 q q

In[17]:=
HOMFLYPT[Knot[9, 9]][a, z]

Out[17]=
6 8 10 6 2 8 2 10 2 6 4 8 4 10 4 2 a + a - 2 a + 7 a z + 4 a z - 3 a z + 5 a z + 4 a z - a z + 6 6 8 6 > a z + a z

In[18]:=
Kauffman[Knot[9, 9]][a, z]

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

In[19]:=
{Vassiliev[2][Knot[9, 9]], Vassiliev[3][Knot[9, 9]]}

Out[19]=
{8, -22}

In[20]:=
Kh[Knot[9, 9]][q, t]

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

In[21]:=
ColouredJones[Knot[9, 9], 2][q]

Out[21]=
-33 2 -31 4 8 3 10 18 6 17 25 6 20 q - --- + q + --- - --- + --- + --- - --- + --- + --- - --- + --- + --- - 32 30 29 28 27 26 25 24 23 22 21 q q q q q q q q q q q 25 2 21 19 2 17 11 4 10 4 3 4 > --- + --- + --- - --- - --- + --- - --- - --- + --- - --- - --- + -- - 20 19 18 17 16 15 14 13 12 11 10 9 q q q q q q q q q q q q -7 -6 > q + q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 9₉

9₈
9₁₀

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q^-33 - 2q^-32 + q^-31 + 4q^-30 - 8q^-29 + 3q^-28 + 10q^-27 - 18q^-26 + 6q^-25 + 17q^-24 - 25q^-23 + 6q^-22 + 20q^-21 - 25q^-20 + 2q^-19 + 21q^-18 - 19q^-17 - 2q^-16 + 17q^-15 - 11q^-14 - 4q^-13 + 10q^-12 - 4q^-11 - 3q^-10 + 4q^-9 - q^-7 + q^-6
3	- q^-63 + 2q^-62 - q^-61 - q^-60 - q^-59 + 5q^-58 - q^-57 - 6q^-56 + q^-55 + 11q^-54 - 6q^-53 - 13q^-52 + 7q^-51 + 23q^-50 - 15q^-49 - 28q^-48 + 18q^-47 + 37q^-46 - 21q^-45 - 44q^-44 + 22q^-43 + 48q^-42 - 18q^-41 - 53q^-40 + 15q^-39 + 51q^-38 - 7q^-37 - 51q^-36 + q^-35 + 46q^-34 + 8q^-33 - 45q^-32 - 11q^-31 + 36q^-30 + 19q^-29 - 33q^-28 - 19q^-27 + 22q^-26 + 24q^-25 - 18q^-24 - 18q^-23 + 6q^-22 + 20q^-21 - 5q^-20 - 11q^-19 - 3q^-18 + 10q^-17 + q^-16 - 3q^-15 - 3q^-14 + 3q^-13 + q^-12 - q^-10 + q^-9
4	q^-102 - 2q^-101 + q^-100 + q^-99 - 2q^-98 + 4q^-97 - 7q^-96 + 3q^-95 + 4q^-94 - 6q^-93 + 11q^-92 - 16q^-91 + 8q^-90 + 7q^-89 - 19q^-88 + 21q^-87 - 20q^-86 + 24q^-85 + 6q^-84 - 50q^-83 + 27q^-82 - 17q^-81 + 60q^-80 + 12q^-79 - 100q^-78 + 14q^-77 - 14q^-76 + 113q^-75 + 33q^-74 - 144q^-73 - 12q^-72 - 27q^-71 + 152q^-70 + 65q^-69 - 157q^-68 - 32q^-67 - 54q^-66 + 158q^-65 + 87q^-64 - 138q^-63 - 30q^-62 - 82q^-61 + 137q^-60 + 94q^-59 - 104q^-58 - 16q^-57 - 103q^-56 + 103q^-55 + 91q^-54 - 61q^-53 - q^-52 - 119q^-51 + 63q^-50 + 85q^-49 - 17q^-48 + 15q^-47 - 121q^-46 + 19q^-45 + 65q^-44 + 17q^-43 + 38q^-42 - 102q^-41 - 17q^-40 + 31q^-39 + 28q^-38 + 54q^-37 - 61q^-36 - 28q^-35 - 3q^-34 + 15q^-33 + 51q^-32 - 21q^-31 - 16q^-30 - 16q^-29 - 3q^-28 + 30q^-27 - 2q^-26 - q^-25 - 10q^-24 - 8q^-23 + 11q^-22 + 3q^-20 - 2q^-19 - 4q^-18 + 3q^-17 + q^-15 - q^-13 + q^-12
5	- q^-150 + 2q^-149 - q^-148 - q^-147 + 2q^-146 - q^-145 - 2q^-144 + 5q^-143 - q^-142 - 5q^-141 + 3q^-140 - q^-139 - 4q^-138 + 10q^-137 + q^-136 - 5q^-135 - 5q^-133 - 9q^-132 + 6q^-131 + 13q^-130 + 11q^-129 + 8q^-128 - 20q^-127 - 40q^-126 - 15q^-125 + 29q^-124 + 65q^-123 + 53q^-122 - 40q^-121 - 118q^-120 - 83q^-119 + 37q^-118 + 157q^-117 + 160q^-116 - 29q^-115 - 221q^-114 - 218q^-113 - 3q^-112 + 254q^-111 + 305q^-110 + 48q^-109 - 288q^-108 - 372q^-107 - 98q^-106 + 288q^-105 + 424q^-104 + 158q^-103 - 275q^-102 - 458q^-101 - 200q^-100 + 249q^-99 + 458q^-98 + 233q^-97 - 207q^-96 - 453q^-95 - 248q^-94 + 181q^-93 + 416q^-92 + 251q^-91 - 139q^-90 - 388q^-89 - 246q^-88 + 115q^-87 + 340q^-86 + 240q^-85 - 74q^-84 - 308q^-83 - 231q^-82 + 47q^-81 + 253q^-80 + 225q^-79 + 3q^-78 - 215q^-77 - 218q^-76 - 29q^-75 + 146q^-74 + 201q^-73 + 87q^-72 - 103q^-71 - 178q^-70 - 104q^-69 + 26q^-68 + 143q^-67 + 139q^-66 + 15q^-65 - 95q^-64 - 128q^-63 - 80q^-62 + 47q^-61 + 126q^-60 + 92q^-59 + 10q^-58 - 80q^-57 - 125q^-56 - 45q^-55 + 56q^-54 + 90q^-53 + 77q^-52 + 4q^-51 - 89q^-50 - 79q^-49 - 17q^-48 + 33q^-47 + 70q^-46 + 55q^-45 - 22q^-44 - 49q^-43 - 40q^-42 - 15q^-41 + 25q^-40 + 46q^-39 + 12q^-38 - 7q^-37 - 20q^-36 - 23q^-35 - 4q^-34 + 18q^-33 + 8q^-32 + 7q^-31 - q^-30 - 10q^-29 - 7q^-28 + 5q^-27 + 3q^-25 + 3q^-24 - 2q^-23 - 3q^-22 + 2q^-21 + q^-18 - q^-16 + q^-15
6	q^-207 - 2q^-206 + q^-205 + q^-204 - 2q^-203 + q^-202 - q^-201 + 4q^-200 - 7q^-199 + 2q^-198 + 8q^-197 - 7q^-196 + 2q^-195 - 2q^-194 + 7q^-193 - 18q^-192 + 2q^-191 + 22q^-190 - 12q^-189 + 7q^-188 + q^-187 + 12q^-186 - 46q^-185 - 8q^-184 + 38q^-183 - 17q^-182 + 33q^-181 + 25q^-180 + 25q^-179 - 102q^-178 - 58q^-177 + 36q^-176 - 18q^-175 + 110q^-174 + 113q^-173 + 62q^-172 - 207q^-171 - 199q^-170 - 37q^-169 - 25q^-168 + 280q^-167 + 333q^-166 + 181q^-165 - 357q^-164 - 483q^-163 - 256q^-162 - 103q^-161 + 528q^-160 + 722q^-159 + 465q^-158 - 451q^-157 - 860q^-156 - 653q^-155 - 353q^-154 + 713q^-153 + 1189q^-152 + 923q^-151 - 358q^-150 - 1139q^-149 - 1099q^-148 - 767q^-147 + 681q^-146 + 1506q^-145 + 1395q^-144 - 86q^-143 - 1168q^-142 - 1371q^-141 - 1164q^-140 + 467q^-139 + 1543q^-138 + 1662q^-137 + 183q^-136 - 1002q^-135 - 1377q^-134 - 1359q^-133 + 244q^-132 + 1385q^-131 + 1676q^-130 + 306q^-129 - 800q^-128 - 1223q^-127 - 1348q^-126 + 109q^-125 + 1178q^-124 + 1556q^-123 + 315q^-122 - 623q^-121 - 1040q^-120 - 1256q^-119 + 11q^-118 + 964q^-117 + 1413q^-116 + 327q^-115 - 425q^-114 - 853q^-113 - 1175q^-112 - 133q^-111 + 707q^-110 + 1255q^-109 + 388q^-108 - 162q^-107 - 618q^-106 - 1077q^-105 - 325q^-104 + 374q^-103 + 1024q^-102 + 440q^-101 + 139q^-100 - 304q^-99 - 885q^-98 - 478q^-97 + 5q^-96 + 683q^-95 + 381q^-94 + 373q^-93 + 50q^-92 - 558q^-91 - 481q^-90 - 282q^-89 + 284q^-88 + 167q^-87 + 420q^-86 + 313q^-85 - 164q^-84 - 291q^-83 - 357q^-82 - 26q^-81 - 124q^-80 + 253q^-79 + 355q^-78 + 126q^-77 - 13q^-76 - 206q^-75 - 113q^-74 - 308q^-73 + 5q^-72 + 185q^-71 + 182q^-70 + 156q^-69 + 14q^-68 - 5q^-67 - 276q^-66 - 129q^-65 - 18q^-64 + 64q^-63 + 129q^-62 + 113q^-61 + 117q^-60 - 120q^-59 - 93q^-58 - 89q^-57 - 43q^-56 + 16q^-55 + 70q^-54 + 124q^-53 - 9q^-52 - 7q^-51 - 45q^-50 - 48q^-49 - 42q^-48 + 4q^-47 + 61q^-46 + 8q^-45 + 24q^-44 - 11q^-42 - 31q^-41 - 15q^-40 + 18q^-39 - 4q^-38 + 12q^-37 + 7q^-36 + 5q^-35 - 10q^-34 - 8q^-33 + 6q^-32 - 5q^-31 + 2q^-30 + 2q^-29 + 4q^-28 - 2q^-27 - 3q^-26 + 3q^-25 - q^-24 + q^-21 - q^-19 + q^-18

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

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

The Alternating Knot 99

The Alternating Knot 9₉