The Alternating Knot 9₆

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Acknowledgement

KnotPlot

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

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

DT (Dowker-Thistlethwaite) Code: 4 12 14 16 18 2 10 6 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 + 5t^-1 - 5 + 5t - 4t² + 2t³

Conway Polynomial: 1 + 7z² + 8z⁴ + 2z⁶

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

Determinant and Signature: {27, -6}

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

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

A2 (sl(3)) Invariant: - q^-36 - 2q^-26 - q^-22 + q^-20 + 2q^-18 + q^-16 + 2q^-14 + q^-10

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

Kauffman Polynomial: - 3a⁶ + 7a⁶z² - 5a⁶z⁴ + a⁶z⁶ + 2a⁷z - 3a⁷z⁵ + a⁷z⁷ - a⁸ + a⁸z² + a⁸z⁴ - 3a⁸z⁶ + a⁸z⁸ - a⁹z + 8a⁹z³ - 10a⁹z⁵ + 3a⁹z⁷ + a¹⁰ - 3a¹⁰z² + 2a¹⁰z⁴ - 2a¹⁰z⁶ + a¹⁰z⁸ - 2a¹¹z + 6a¹¹z³ - 5a¹¹z⁵ + 2a¹¹z⁷ + a¹²z² - 2a¹²z⁴ + 2a¹²z⁶ - a¹³z³ + 2a¹³z⁵ - 2a¹⁴z² + 2a¹⁴z⁴ - a¹⁵z + a¹⁵z³

V₂ and V₃, the type 2 and 3 Vassiliev invariants: {7, -18}

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 9₆. 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 1 1

j = -13 3 2

j = -15 2 1

j = -17 2 3

j = -19 1 2

j = -21 1 2

j = -23 1

j = -25 1

n Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2 q^-33 - 2q^-32 + 4q^-30 - 5q^-29 + q^-28 + 7q^-27 - 11q^-26 + 3q^-25 + 11q^-24 - 16q^-23 + 3q^-22 + 14q^-21 - 16q^-20 + q^-19 + 14q^-18 - 13q^-17 - 2q^-16 + 12q^-15 - 8q^-14 - 4q^-13 + 9q^-12 - 3q^-11 - 3q^-10 + 4q^-9 - q^-7 + q^-6

3 - q^-63 + 2q^-62 - q^-60 - 3q^-59 + 3q^-58 + 3q^-57 - 2q^-56 - 4q^-55 + 2q^-54 + 3q^-53 - 2q^-52 - q^-51 + 3q^-50 - 3q^-49 - 4q^-48 + 6q^-47 + 8q^-46 - 9q^-45 - 10q^-44 + 9q^-43 + 14q^-42 - 10q^-41 - 13q^-40 + 5q^-39 + 16q^-38 - 4q^-37 - 16q^-36 + 15q^-34 + 5q^-33 - 17q^-32 - 6q^-31 + 13q^-30 + 14q^-29 - 16q^-28 - 12q^-27 + 8q^-26 + 18q^-25 - 9q^-24 - 14q^-23 + 17q^-21 - 2q^-20 - 9q^-19 - 4q^-18 + 9q^-17 + 2q^-16 - 3q^-15 - 3q^-14 + 3q^-13 + q^-12 - q^-10 + q^-9

4 q^-102 - 2q^-101 + q^-99 + 5q^-97 - 7q^-96 - q^-95 + 16q^-92 - 12q^-91 - 3q^-90 - 7q^-89 - 4q^-88 + 32q^-87 - 10q^-86 - 2q^-85 - 22q^-84 - 18q^-83 + 49q^-82 + 3q^-81 + 10q^-80 - 39q^-79 - 47q^-78 + 56q^-77 + 21q^-76 + 34q^-75 - 48q^-74 - 76q^-73 + 52q^-72 + 26q^-71 + 57q^-70 - 42q^-69 - 91q^-68 + 45q^-67 + 19q^-66 + 66q^-65 - 34q^-64 - 88q^-63 + 44q^-62 + 7q^-61 + 60q^-60 - 24q^-59 - 75q^-58 + 42q^-57 - 6q^-56 + 48q^-55 - 13q^-54 - 56q^-53 + 37q^-52 - 22q^-51 + 34q^-50 - q^-49 - 32q^-48 + 35q^-47 - 36q^-46 + 15q^-45 + 4q^-44 - 9q^-43 + 40q^-42 - 40q^-41 - 4q^-40 - q^-39 + 2q^-38 + 45q^-37 - 27q^-36 - 13q^-35 - 13q^-34 - q^-33 + 41q^-32 - 9q^-31 - 8q^-30 - 16q^-29 - 8q^-28 + 25q^-27 + q^-26 + q^-25 - 9q^-24 - 9q^-23 + 10q^-22 + q^-21 + 3q^-20 - 2q^-19 - 4q^-18 + 3q^-17 + q^-15 - q^-13 + q^-12

5 - q^-150 + 2q^-149 - q^-147 - 2q^-145 - q^-144 + 5q^-143 + 3q^-142 - 3q^-141 - q^-140 - 7q^-139 - 5q^-138 + 10q^-137 + 11q^-136 + 3q^-135 - 5q^-134 - 17q^-133 - 16q^-132 + 10q^-131 + 28q^-130 + 21q^-129 - 7q^-128 - 38q^-127 - 39q^-126 + 4q^-125 + 54q^-124 + 61q^-123 + 2q^-122 - 71q^-121 - 90q^-120 - 15q^-119 + 89q^-118 + 125q^-117 + 40q^-116 - 107q^-115 - 167q^-114 - 66q^-113 + 117q^-112 + 203q^-111 + 102q^-110 - 116q^-109 - 241q^-108 - 134q^-107 + 111q^-106 + 258q^-105 + 166q^-104 - 97q^-103 - 271q^-102 - 184q^-101 + 81q^-100 + 267q^-99 + 202q^-98 - 70q^-97 - 265q^-96 - 196q^-95 + 55q^-94 + 251q^-93 + 201q^-92 - 52q^-91 - 242q^-90 - 187q^-89 + 42q^-88 + 224q^-87 + 184q^-86 - 36q^-85 - 208q^-84 - 174q^-83 + 25q^-82 + 188q^-81 + 163q^-80 - 10q^-79 - 159q^-78 - 159q^-77 - q^-76 + 139q^-75 + 132q^-74 + 18q^-73 - 99q^-72 - 127q^-71 - 24q^-70 + 82q^-69 + 88q^-68 + 29q^-67 - 42q^-66 - 78q^-65 - 23q^-64 + 35q^-63 + 34q^-62 + 16q^-61 - 10q^-60 - 27q^-59 + q^-58 + 21q^-57 - 8q^-56 - 12q^-55 - 9q^-54 - 5q^-53 + 26q^-52 + 34q^-51 - 11q^-50 - 29q^-49 - 26q^-48 - 12q^-47 + 23q^-46 + 47q^-45 + 7q^-44 - 17q^-43 - 30q^-42 - 25q^-41 + 4q^-40 + 33q^-39 + 18q^-38 + 3q^-37 - 14q^-36 - 22q^-35 - 9q^-34 + 13q^-33 + 9q^-32 + 9q^-31 - 9q^-29 - 8q^-28 + 4q^-27 + q^-26 + 3q^-25 + 3q^-24 - 2q^-23 - 3q^-22 + 2q^-21 + q^-18 - q^-16 + q^-15

6 q^-207 - 2q^-206 + q^-204 + 2q^-202 - 2q^-201 + 3q^-200 - 7q^-199 + 4q^-197 - q^-196 + 7q^-195 - 2q^-194 + 7q^-193 - 20q^-192 - 3q^-191 + 5q^-190 - 2q^-189 + 18q^-188 + 6q^-187 + 17q^-186 - 41q^-185 - 11q^-184 - 2q^-183 - 6q^-182 + 35q^-181 + 26q^-180 + 32q^-179 - 72q^-178 - 26q^-177 - 14q^-176 - 7q^-175 + 66q^-174 + 58q^-173 + 41q^-172 - 123q^-171 - 63q^-170 - 30q^-169 + 13q^-168 + 136q^-167 + 122q^-166 + 41q^-165 - 222q^-164 - 157q^-163 - 74q^-162 + 58q^-161 + 275q^-160 + 251q^-159 + 63q^-158 - 357q^-157 - 329q^-156 - 184q^-155 + 80q^-154 + 450q^-153 + 453q^-152 + 150q^-151 - 457q^-150 - 519q^-149 - 352q^-148 + 27q^-147 + 566q^-146 + 646q^-145 + 293q^-144 - 464q^-143 - 628q^-142 - 499q^-141 - 81q^-140 + 578q^-139 + 743q^-138 + 411q^-137 - 410q^-136 - 634q^-135 - 559q^-134 - 165q^-133 + 529q^-132 + 744q^-131 + 459q^-130 - 360q^-129 - 588q^-128 - 550q^-127 - 198q^-126 + 474q^-125 + 698q^-124 + 468q^-123 - 319q^-122 - 535q^-121 - 523q^-120 - 218q^-119 + 410q^-118 + 640q^-117 + 480q^-116 - 253q^-115 - 465q^-114 - 502q^-113 - 262q^-112 + 313q^-111 + 565q^-110 + 506q^-109 - 149q^-108 - 362q^-107 - 471q^-106 - 323q^-105 + 179q^-104 + 457q^-103 + 517q^-102 - 24q^-101 - 222q^-100 - 405q^-99 - 365q^-98 + 32q^-97 + 310q^-96 + 478q^-95 + 74q^-94 - 68q^-93 - 286q^-92 - 344q^-91 - 85q^-90 + 144q^-89 + 371q^-88 + 99q^-87 + 49q^-86 - 139q^-85 - 243q^-84 - 123q^-83 + 16q^-82 + 229q^-81 + 43q^-80 + 82q^-79 - 32q^-78 - 112q^-77 - 77q^-76 - 25q^-75 + 127q^-74 - 34q^-73 + 38q^-72 - 9q^-71 - 33q^-70 - 15q^-69 + 5q^-68 + 101q^-67 - 55q^-66 - 9q^-65 - 40q^-64 - 29q^-63 - 6q^-62 + 29q^-61 + 110q^-60 - 19q^-59 - 5q^-58 - 49q^-57 - 46q^-56 - 35q^-55 + 9q^-54 + 88q^-53 + 14q^-52 + 23q^-51 - 19q^-50 - 32q^-49 - 47q^-48 - 18q^-47 + 42q^-46 + 10q^-45 + 28q^-44 + 6q^-43 - 4q^-42 - 28q^-41 - 20q^-40 + 13q^-39 - 3q^-38 + 12q^-37 + 8q^-36 + 6q^-35 - 9q^-34 - 9q^-33 + 5q^-32 - 4q^-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, 6]]

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

In[3]:=
GaussCode[Knot[9, 6]]

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

In[4]:=
DTCode[Knot[9, 6]]

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

In[5]:=
br = BR[Knot[9, 6]]

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

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

Out[6]=
{3, 10}

In[7]:=
BraidIndex[Knot[9, 6]]

Out[7]=
3

In[8]:=
Show[DrawMorseLink[Knot[9, 6]]]

Out[8]=
-Graphics-

In[9]:=
#[Knot[9, 6]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}

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

In[10]:=
alex = Alexander[Knot[9, 6]][t]

Out[10]=
2 4 5 2 3 -5 + -- - -- + - + 5 t - 4 t + 2 t 3 2 t t t

In[11]:=
Conway[Knot[9, 6]][z]

Out[11]=
2 4 6 1 + 7 z + 8 z + 2 z

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

Out[12]=
{Knot[9, 6]}

In[13]:=
{KnotDet[Knot[9, 6]], KnotSignature[Knot[9, 6]]}

Out[13]=
{27, -6}

In[14]:=
Jones[Knot[9, 6]][q]

Out[14]=
-12 2 3 4 5 4 3 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, 6]}

In[16]:=
A2Invariant[Knot[9, 6]][q]

Out[16]=
-36 2 -22 -20 2 -16 2 -10 -q - --- - q + q + --- + q + --- + q 26 18 14 q q q

In[17]:=
HOMFLYPT[Knot[9, 6]][a, z]

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

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

Out[19]=
{7, -18}

In[20]:=
Kh[Knot[9, 6]][q, t]

Out[20]=
-7 -5 1 1 1 2 1 2 2 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 1 3 2 1 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, 6], 2][q]

Out[21]=
-33 2 4 5 -28 7 11 3 11 16 3 14 16 q - --- + --- - --- + q + --- - --- + --- + --- - --- + --- + --- - --- + 32 30 29 27 26 25 24 23 22 21 20 q q q q q q q q q q q -19 14 13 2 12 8 4 9 3 3 4 -7 -6 > q + --- - --- - --- + --- - --- - --- + --- - --- - --- + -- - q + q 18 17 16 15 14 13 12 11 10 9 q q q q q q q q q q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 9₆

9₅
9₇

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q^-33 - 2q^-32 + 4q^-30 - 5q^-29 + q^-28 + 7q^-27 - 11q^-26 + 3q^-25 + 11q^-24 - 16q^-23 + 3q^-22 + 14q^-21 - 16q^-20 + q^-19 + 14q^-18 - 13q^-17 - 2q^-16 + 12q^-15 - 8q^-14 - 4q^-13 + 9q^-12 - 3q^-11 - 3q^-10 + 4q^-9 - q^-7 + q^-6
3	- q^-63 + 2q^-62 - q^-60 - 3q^-59 + 3q^-58 + 3q^-57 - 2q^-56 - 4q^-55 + 2q^-54 + 3q^-53 - 2q^-52 - q^-51 + 3q^-50 - 3q^-49 - 4q^-48 + 6q^-47 + 8q^-46 - 9q^-45 - 10q^-44 + 9q^-43 + 14q^-42 - 10q^-41 - 13q^-40 + 5q^-39 + 16q^-38 - 4q^-37 - 16q^-36 + 15q^-34 + 5q^-33 - 17q^-32 - 6q^-31 + 13q^-30 + 14q^-29 - 16q^-28 - 12q^-27 + 8q^-26 + 18q^-25 - 9q^-24 - 14q^-23 + 17q^-21 - 2q^-20 - 9q^-19 - 4q^-18 + 9q^-17 + 2q^-16 - 3q^-15 - 3q^-14 + 3q^-13 + q^-12 - q^-10 + q^-9
4	q^-102 - 2q^-101 + q^-99 + 5q^-97 - 7q^-96 - q^-95 + 16q^-92 - 12q^-91 - 3q^-90 - 7q^-89 - 4q^-88 + 32q^-87 - 10q^-86 - 2q^-85 - 22q^-84 - 18q^-83 + 49q^-82 + 3q^-81 + 10q^-80 - 39q^-79 - 47q^-78 + 56q^-77 + 21q^-76 + 34q^-75 - 48q^-74 - 76q^-73 + 52q^-72 + 26q^-71 + 57q^-70 - 42q^-69 - 91q^-68 + 45q^-67 + 19q^-66 + 66q^-65 - 34q^-64 - 88q^-63 + 44q^-62 + 7q^-61 + 60q^-60 - 24q^-59 - 75q^-58 + 42q^-57 - 6q^-56 + 48q^-55 - 13q^-54 - 56q^-53 + 37q^-52 - 22q^-51 + 34q^-50 - q^-49 - 32q^-48 + 35q^-47 - 36q^-46 + 15q^-45 + 4q^-44 - 9q^-43 + 40q^-42 - 40q^-41 - 4q^-40 - q^-39 + 2q^-38 + 45q^-37 - 27q^-36 - 13q^-35 - 13q^-34 - q^-33 + 41q^-32 - 9q^-31 - 8q^-30 - 16q^-29 - 8q^-28 + 25q^-27 + q^-26 + q^-25 - 9q^-24 - 9q^-23 + 10q^-22 + q^-21 + 3q^-20 - 2q^-19 - 4q^-18 + 3q^-17 + q^-15 - q^-13 + q^-12
5	- q^-150 + 2q^-149 - q^-147 - 2q^-145 - q^-144 + 5q^-143 + 3q^-142 - 3q^-141 - q^-140 - 7q^-139 - 5q^-138 + 10q^-137 + 11q^-136 + 3q^-135 - 5q^-134 - 17q^-133 - 16q^-132 + 10q^-131 + 28q^-130 + 21q^-129 - 7q^-128 - 38q^-127 - 39q^-126 + 4q^-125 + 54q^-124 + 61q^-123 + 2q^-122 - 71q^-121 - 90q^-120 - 15q^-119 + 89q^-118 + 125q^-117 + 40q^-116 - 107q^-115 - 167q^-114 - 66q^-113 + 117q^-112 + 203q^-111 + 102q^-110 - 116q^-109 - 241q^-108 - 134q^-107 + 111q^-106 + 258q^-105 + 166q^-104 - 97q^-103 - 271q^-102 - 184q^-101 + 81q^-100 + 267q^-99 + 202q^-98 - 70q^-97 - 265q^-96 - 196q^-95 + 55q^-94 + 251q^-93 + 201q^-92 - 52q^-91 - 242q^-90 - 187q^-89 + 42q^-88 + 224q^-87 + 184q^-86 - 36q^-85 - 208q^-84 - 174q^-83 + 25q^-82 + 188q^-81 + 163q^-80 - 10q^-79 - 159q^-78 - 159q^-77 - q^-76 + 139q^-75 + 132q^-74 + 18q^-73 - 99q^-72 - 127q^-71 - 24q^-70 + 82q^-69 + 88q^-68 + 29q^-67 - 42q^-66 - 78q^-65 - 23q^-64 + 35q^-63 + 34q^-62 + 16q^-61 - 10q^-60 - 27q^-59 + q^-58 + 21q^-57 - 8q^-56 - 12q^-55 - 9q^-54 - 5q^-53 + 26q^-52 + 34q^-51 - 11q^-50 - 29q^-49 - 26q^-48 - 12q^-47 + 23q^-46 + 47q^-45 + 7q^-44 - 17q^-43 - 30q^-42 - 25q^-41 + 4q^-40 + 33q^-39 + 18q^-38 + 3q^-37 - 14q^-36 - 22q^-35 - 9q^-34 + 13q^-33 + 9q^-32 + 9q^-31 - 9q^-29 - 8q^-28 + 4q^-27 + q^-26 + 3q^-25 + 3q^-24 - 2q^-23 - 3q^-22 + 2q^-21 + q^-18 - q^-16 + q^-15
6	q^-207 - 2q^-206 + q^-204 + 2q^-202 - 2q^-201 + 3q^-200 - 7q^-199 + 4q^-197 - q^-196 + 7q^-195 - 2q^-194 + 7q^-193 - 20q^-192 - 3q^-191 + 5q^-190 - 2q^-189 + 18q^-188 + 6q^-187 + 17q^-186 - 41q^-185 - 11q^-184 - 2q^-183 - 6q^-182 + 35q^-181 + 26q^-180 + 32q^-179 - 72q^-178 - 26q^-177 - 14q^-176 - 7q^-175 + 66q^-174 + 58q^-173 + 41q^-172 - 123q^-171 - 63q^-170 - 30q^-169 + 13q^-168 + 136q^-167 + 122q^-166 + 41q^-165 - 222q^-164 - 157q^-163 - 74q^-162 + 58q^-161 + 275q^-160 + 251q^-159 + 63q^-158 - 357q^-157 - 329q^-156 - 184q^-155 + 80q^-154 + 450q^-153 + 453q^-152 + 150q^-151 - 457q^-150 - 519q^-149 - 352q^-148 + 27q^-147 + 566q^-146 + 646q^-145 + 293q^-144 - 464q^-143 - 628q^-142 - 499q^-141 - 81q^-140 + 578q^-139 + 743q^-138 + 411q^-137 - 410q^-136 - 634q^-135 - 559q^-134 - 165q^-133 + 529q^-132 + 744q^-131 + 459q^-130 - 360q^-129 - 588q^-128 - 550q^-127 - 198q^-126 + 474q^-125 + 698q^-124 + 468q^-123 - 319q^-122 - 535q^-121 - 523q^-120 - 218q^-119 + 410q^-118 + 640q^-117 + 480q^-116 - 253q^-115 - 465q^-114 - 502q^-113 - 262q^-112 + 313q^-111 + 565q^-110 + 506q^-109 - 149q^-108 - 362q^-107 - 471q^-106 - 323q^-105 + 179q^-104 + 457q^-103 + 517q^-102 - 24q^-101 - 222q^-100 - 405q^-99 - 365q^-98 + 32q^-97 + 310q^-96 + 478q^-95 + 74q^-94 - 68q^-93 - 286q^-92 - 344q^-91 - 85q^-90 + 144q^-89 + 371q^-88 + 99q^-87 + 49q^-86 - 139q^-85 - 243q^-84 - 123q^-83 + 16q^-82 + 229q^-81 + 43q^-80 + 82q^-79 - 32q^-78 - 112q^-77 - 77q^-76 - 25q^-75 + 127q^-74 - 34q^-73 + 38q^-72 - 9q^-71 - 33q^-70 - 15q^-69 + 5q^-68 + 101q^-67 - 55q^-66 - 9q^-65 - 40q^-64 - 29q^-63 - 6q^-62 + 29q^-61 + 110q^-60 - 19q^-59 - 5q^-58 - 49q^-57 - 46q^-56 - 35q^-55 + 9q^-54 + 88q^-53 + 14q^-52 + 23q^-51 - 19q^-50 - 32q^-49 - 47q^-48 - 18q^-47 + 42q^-46 + 10q^-45 + 28q^-44 + 6q^-43 - 4q^-42 - 28q^-41 - 20q^-40 + 13q^-39 - 3q^-38 + 12q^-37 + 8q^-36 + 6q^-35 - 9q^-34 - 9q^-33 + 5q^-32 - 4q^-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, 6]]
Out[2]=	PD[X[1, 4, 2, 5], X[3, 12, 4, 13], X[5, 14, 6, 15], X[7, 16, 8, 17], > X[9, 18, 10, 1], X[15, 6, 16, 7], X[17, 8, 18, 9], X[13, 10, 14, 11], > X[11, 2, 12, 3]]
In[3]:=	GaussCode[Knot[9, 6]]
Out[3]=	GaussCode[-1, 9, -2, 1, -3, 6, -4, 7, -5, 8, -9, 2, -8, 3, -6, 4, -7, 5]
In[4]:=	DTCode[Knot[9, 6]]
Out[4]=	DTCode[4, 12, 14, 16, 18, 2, 10, 6, 8]
In[5]:=	br = BR[Knot[9, 6]]
Out[5]=	BR[3, {-1, -1, -1, -1, -1, -1, -2, 1, -2, -2}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{3, 10}
In[7]:=	BraidIndex[Knot[9, 6]]
Out[7]=	3
In[8]:=	Show[DrawMorseLink[Knot[9, 6]]]

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

The Alternating Knot 96

The Alternating Knot 9₆