The Alternating Knot 9₃₈

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Acknowledgement

KnotPlot

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

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

DT (Dowker-Thistlethwaite) Code: 6 10 14 18 4 16 2 8 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 2--3 2 3 / 4--7 2

Alexander Polynomial: 5t^-2 - 14t^-1 + 19 - 14t + 5t²

Conway Polynomial: 1 + 6z² + 5z⁴

Other knots with the same Alexander/Conway Polynomial: {10₆₃, ...}

Determinant and Signature: {57, -4}

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

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

A2 (sl(3)) Invariant: - q^-34 + q^-32 + q^-30 - 3q^-28 - 2q^-24 - q^-22 + 2q^-20 + 4q^-16 + q^-12 + 2q^-10 - 2q^-8 + q^-6

HOMFLY-PT Polynomial: a⁴z² + a⁴z⁴ + 4a⁶ + 7a⁶z² + 3a⁶z⁴ - 3a⁸ - a⁸z² + a⁸z⁴ - a¹⁰z²

Kauffman Polynomial: - a⁴z² + a⁴z⁴ - 2a⁵z³ + 3a⁵z⁵ - 4a⁶ + 9a⁶z² - 10a⁶z⁴ + 6a⁶z⁶ + 3a⁷z - 2a⁷z³ - 4a⁷z⁵ + 5a⁷z⁷ - 3a⁸ + 10a⁸z² - 15a⁸z⁴ + 6a⁸z⁶ + 2a⁸z⁸ + a⁹z + 5a⁹z³ - 15a⁹z⁵ + 9a⁹z⁷ + 3a¹⁰z² - 10a¹⁰z⁴ + 3a¹⁰z⁶ + 2a¹⁰z⁸ - a¹¹z + 3a¹¹z³ - 7a¹¹z⁵ + 4a¹¹z⁷ + 3a¹²z² - 6a¹²z⁴ + 3a¹²z⁶ + a¹³z - 2a¹³z³ + a¹³z⁵

V₂ and V₃, the type 2 and 3 Vassiliev invariants: {6, -14}

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=-4 is the signature of 9₃₈. 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 = -3 1

j = -5 3 1

j = -7 4

j = -9 4 3

j = -11 6 4

j = -13 4 4

j = -15 4 6

j = -17 2 4

j = -19 1 4

j = -21 2

j = -23 1

n Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2 q^-31 - 3q^-30 + q^-29 + 10q^-28 - 16q^-27 - 5q^-26 + 38q^-25 - 32q^-24 - 27q^-23 + 74q^-22 - 37q^-21 - 57q^-20 + 98q^-19 - 29q^-18 - 77q^-17 + 97q^-16 - 14q^-15 - 74q^-14 + 71q^-13 + 2q^-12 - 50q^-11 + 34q^-10 + 7q^-9 - 20q^-8 + 8q^-7 + 3q^-6 - 3q^-5 + q^-4

3 - q^-60 + 3q^-59 - q^-58 - 5q^-57 - 2q^-56 + 16q^-55 + 8q^-54 - 33q^-53 - 27q^-52 + 53q^-51 + 64q^-50 - 65q^-49 - 125q^-48 + 69q^-47 + 192q^-46 - 41q^-45 - 272q^-44 - 3q^-43 + 338q^-42 + 71q^-41 - 395q^-40 - 142q^-39 + 425q^-38 + 224q^-37 - 446q^-36 - 287q^-35 + 440q^-34 + 343q^-33 - 417q^-32 - 385q^-31 + 380q^-30 + 399q^-29 - 315q^-28 - 407q^-27 + 253q^-26 + 369q^-25 - 162q^-24 - 334q^-23 + 100q^-22 + 257q^-21 - 29q^-20 - 197q^-19 + 6q^-18 + 117q^-17 + 20q^-16 - 71q^-15 - 16q^-14 + 34q^-13 + 11q^-12 - 14q^-11 - 4q^-10 + 4q^-9 + 3q^-8 - 3q^-7 + q^-6

4 q^-98 - 3q^-97 + q^-96 + 5q^-95 - 3q^-94 + 2q^-93 - 19q^-92 + 4q^-91 + 35q^-90 + 5q^-89 + 7q^-88 - 102q^-87 - 40q^-86 + 112q^-85 + 111q^-84 + 118q^-83 - 281q^-82 - 284q^-81 + 77q^-80 + 330q^-79 + 572q^-78 - 328q^-77 - 743q^-76 - 364q^-75 + 365q^-74 + 1364q^-73 + 96q^-72 - 1052q^-71 - 1198q^-70 - 131q^-69 + 2086q^-68 + 956q^-67 - 862q^-66 - 2026q^-65 - 1071q^-64 + 2382q^-63 + 1859q^-62 - 252q^-61 - 2528q^-60 - 2054q^-59 + 2274q^-58 + 2498q^-57 + 459q^-56 - 2672q^-55 - 2790q^-54 + 1921q^-53 + 2809q^-52 + 1093q^-51 - 2515q^-50 - 3202q^-49 + 1373q^-48 + 2768q^-47 + 1606q^-46 - 2008q^-45 - 3221q^-44 + 627q^-43 + 2279q^-42 + 1900q^-41 - 1158q^-40 - 2731q^-39 - 107q^-38 + 1385q^-37 + 1751q^-36 - 269q^-35 - 1785q^-34 - 473q^-33 + 466q^-32 + 1158q^-31 + 219q^-30 - 809q^-29 - 377q^-28 - 38q^-27 + 503q^-26 + 236q^-25 - 231q^-24 - 139q^-23 - 108q^-22 + 138q^-21 + 95q^-20 - 45q^-19 - 17q^-18 - 44q^-17 + 25q^-16 + 21q^-15 - 10q^-14 + 2q^-13 - 8q^-12 + 4q^-11 + 3q^-10 - 3q^-9 + q^-8

5 - q^-145 + 3q^-144 - q^-143 - 5q^-142 + 3q^-141 + 3q^-140 + q^-139 + 7q^-138 - 6q^-137 - 30q^-136 - 8q^-135 + 29q^-134 + 46q^-133 + 54q^-132 - 18q^-131 - 136q^-130 - 163q^-129 - 14q^-128 + 219q^-127 + 370q^-126 + 221q^-125 - 263q^-124 - 707q^-123 - 642q^-122 + 100q^-121 + 1054q^-120 + 1356q^-119 + 451q^-118 - 1208q^-117 - 2289q^-116 - 1549q^-115 + 928q^-114 + 3212q^-113 + 3113q^-112 + 121q^-111 - 3758q^-110 - 5079q^-109 - 1944q^-108 + 3639q^-107 + 6930q^-106 + 4510q^-105 - 2518q^-104 - 8445q^-103 - 7483q^-102 + 522q^-101 + 9150q^-100 + 10483q^-99 + 2285q^-98 - 9040q^-97 - 13152q^-96 - 5441q^-95 + 8050q^-94 + 15262q^-93 + 8704q^-92 - 6553q^-91 - 16664q^-90 - 11685q^-89 + 4643q^-88 + 17498q^-87 + 14306q^-86 - 2746q^-85 - 17823q^-84 - 16377q^-83 + 828q^-82 + 17809q^-81 + 18076q^-80 + 891q^-79 - 17535q^-78 - 19340q^-77 - 2520q^-76 + 16953q^-75 + 20295q^-74 + 4156q^-73 - 16091q^-72 - 20919q^-71 - 5739q^-70 + 14737q^-69 + 21044q^-68 + 7498q^-67 - 12892q^-66 - 20712q^-65 - 9024q^-64 + 10432q^-63 + 19516q^-62 + 10499q^-61 - 7573q^-60 - 17656q^-59 - 11246q^-58 + 4438q^-57 + 14870q^-56 + 11480q^-55 - 1536q^-54 - 11714q^-53 - 10595q^-52 - 908q^-51 + 8197q^-50 + 9222q^-49 + 2424q^-48 - 5137q^-47 - 7067q^-46 - 3103q^-45 + 2512q^-44 + 5037q^-43 + 2975q^-42 - 904q^-41 - 3040q^-40 - 2379q^-39 - 80q^-38 + 1658q^-37 + 1631q^-36 + 373q^-35 - 729q^-34 - 953q^-33 - 394q^-32 + 270q^-31 + 477q^-30 + 267q^-29 - 54q^-28 - 224q^-27 - 145q^-26 + 11q^-25 + 82q^-24 + 55q^-23 + 22q^-22 - 37q^-21 - 33q^-20 + 7q^-19 + 12q^-18 + 6q^-16 - 2q^-15 - 8q^-14 + 4q^-13 + 3q^-12 - 3q^-11 + q^-10

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, 38]]

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

In[3]:=
GaussCode[Knot[9, 38]]

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

In[4]:=
DTCode[Knot[9, 38]]

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

In[5]:=
br = BR[Knot[9, 38]]

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

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

Out[6]=
{4, 11}

In[7]:=
BraidIndex[Knot[9, 38]]

Out[7]=
4

In[8]:=
Show[DrawMorseLink[Knot[9, 38]]]

Out[8]=
-Graphics-

In[9]:=
#[Knot[9, 38]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}

Out[9]=
{Reversible, {2, 3}, 2, 3, {4, 7}, 2}

In[10]:=
alex = Alexander[Knot[9, 38]][t]

Out[10]=
5 14 2 19 + -- - -- - 14 t + 5 t 2 t t

In[11]:=
Conway[Knot[9, 38]][z]

Out[11]=
2 4 1 + 6 z + 5 z

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

Out[12]=
{Knot[9, 38], Knot[10, 63]}

In[13]:=
{KnotDet[Knot[9, 38]], KnotSignature[Knot[9, 38]]}

Out[13]=
{57, -4}

In[14]:=
Jones[Knot[9, 38]][q]

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

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

Out[15]=
{Knot[9, 38]}

In[16]:=
A2Invariant[Knot[9, 38]][q]

Out[16]=
-34 -32 -30 3 2 -22 2 4 -12 2 2 -6 -q + q + q - --- - --- - q + --- + --- + q + --- - -- + q 28 24 20 16 10 8 q q q q q q

In[17]:=
HOMFLYPT[Knot[9, 38]][a, z]

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

In[18]:=
Kauffman[Knot[9, 38]][a, z]

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

In[19]:=
{Vassiliev[2][Knot[9, 38]], Vassiliev[3][Knot[9, 38]]}

Out[19]=
{6, -14}

In[20]:=
Kh[Knot[9, 38]][q, t]

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

In[21]:=
ColouredJones[Knot[9, 38], 2][q]

Out[21]=
-31 3 -29 10 16 5 38 32 27 74 37 57 98 q - --- + q + --- - --- - --- + --- - --- - --- + --- - --- - --- + --- - 30 28 27 26 25 24 23 22 21 20 19 q q q q q q q q q q q 29 77 97 14 74 71 2 50 34 7 20 8 3 > --- - --- + --- - --- - --- + --- + --- - --- + --- + -- - -- + -- + -- - 18 17 16 15 14 13 12 11 10 9 8 7 6 q q q q q q q q q q q q q 3 -4 > -- + q 5 q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 9₃₈

9₃₇
9₃₉

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q^-31 - 3q^-30 + q^-29 + 10q^-28 - 16q^-27 - 5q^-26 + 38q^-25 - 32q^-24 - 27q^-23 + 74q^-22 - 37q^-21 - 57q^-20 + 98q^-19 - 29q^-18 - 77q^-17 + 97q^-16 - 14q^-15 - 74q^-14 + 71q^-13 + 2q^-12 - 50q^-11 + 34q^-10 + 7q^-9 - 20q^-8 + 8q^-7 + 3q^-6 - 3q^-5 + q^-4
3	- q^-60 + 3q^-59 - q^-58 - 5q^-57 - 2q^-56 + 16q^-55 + 8q^-54 - 33q^-53 - 27q^-52 + 53q^-51 + 64q^-50 - 65q^-49 - 125q^-48 + 69q^-47 + 192q^-46 - 41q^-45 - 272q^-44 - 3q^-43 + 338q^-42 + 71q^-41 - 395q^-40 - 142q^-39 + 425q^-38 + 224q^-37 - 446q^-36 - 287q^-35 + 440q^-34 + 343q^-33 - 417q^-32 - 385q^-31 + 380q^-30 + 399q^-29 - 315q^-28 - 407q^-27 + 253q^-26 + 369q^-25 - 162q^-24 - 334q^-23 + 100q^-22 + 257q^-21 - 29q^-20 - 197q^-19 + 6q^-18 + 117q^-17 + 20q^-16 - 71q^-15 - 16q^-14 + 34q^-13 + 11q^-12 - 14q^-11 - 4q^-10 + 4q^-9 + 3q^-8 - 3q^-7 + q^-6
4	q^-98 - 3q^-97 + q^-96 + 5q^-95 - 3q^-94 + 2q^-93 - 19q^-92 + 4q^-91 + 35q^-90 + 5q^-89 + 7q^-88 - 102q^-87 - 40q^-86 + 112q^-85 + 111q^-84 + 118q^-83 - 281q^-82 - 284q^-81 + 77q^-80 + 330q^-79 + 572q^-78 - 328q^-77 - 743q^-76 - 364q^-75 + 365q^-74 + 1364q^-73 + 96q^-72 - 1052q^-71 - 1198q^-70 - 131q^-69 + 2086q^-68 + 956q^-67 - 862q^-66 - 2026q^-65 - 1071q^-64 + 2382q^-63 + 1859q^-62 - 252q^-61 - 2528q^-60 - 2054q^-59 + 2274q^-58 + 2498q^-57 + 459q^-56 - 2672q^-55 - 2790q^-54 + 1921q^-53 + 2809q^-52 + 1093q^-51 - 2515q^-50 - 3202q^-49 + 1373q^-48 + 2768q^-47 + 1606q^-46 - 2008q^-45 - 3221q^-44 + 627q^-43 + 2279q^-42 + 1900q^-41 - 1158q^-40 - 2731q^-39 - 107q^-38 + 1385q^-37 + 1751q^-36 - 269q^-35 - 1785q^-34 - 473q^-33 + 466q^-32 + 1158q^-31 + 219q^-30 - 809q^-29 - 377q^-28 - 38q^-27 + 503q^-26 + 236q^-25 - 231q^-24 - 139q^-23 - 108q^-22 + 138q^-21 + 95q^-20 - 45q^-19 - 17q^-18 - 44q^-17 + 25q^-16 + 21q^-15 - 10q^-14 + 2q^-13 - 8q^-12 + 4q^-11 + 3q^-10 - 3q^-9 + q^-8
5	- q^-145 + 3q^-144 - q^-143 - 5q^-142 + 3q^-141 + 3q^-140 + q^-139 + 7q^-138 - 6q^-137 - 30q^-136 - 8q^-135 + 29q^-134 + 46q^-133 + 54q^-132 - 18q^-131 - 136q^-130 - 163q^-129 - 14q^-128 + 219q^-127 + 370q^-126 + 221q^-125 - 263q^-124 - 707q^-123 - 642q^-122 + 100q^-121 + 1054q^-120 + 1356q^-119 + 451q^-118 - 1208q^-117 - 2289q^-116 - 1549q^-115 + 928q^-114 + 3212q^-113 + 3113q^-112 + 121q^-111 - 3758q^-110 - 5079q^-109 - 1944q^-108 + 3639q^-107 + 6930q^-106 + 4510q^-105 - 2518q^-104 - 8445q^-103 - 7483q^-102 + 522q^-101 + 9150q^-100 + 10483q^-99 + 2285q^-98 - 9040q^-97 - 13152q^-96 - 5441q^-95 + 8050q^-94 + 15262q^-93 + 8704q^-92 - 6553q^-91 - 16664q^-90 - 11685q^-89 + 4643q^-88 + 17498q^-87 + 14306q^-86 - 2746q^-85 - 17823q^-84 - 16377q^-83 + 828q^-82 + 17809q^-81 + 18076q^-80 + 891q^-79 - 17535q^-78 - 19340q^-77 - 2520q^-76 + 16953q^-75 + 20295q^-74 + 4156q^-73 - 16091q^-72 - 20919q^-71 - 5739q^-70 + 14737q^-69 + 21044q^-68 + 7498q^-67 - 12892q^-66 - 20712q^-65 - 9024q^-64 + 10432q^-63 + 19516q^-62 + 10499q^-61 - 7573q^-60 - 17656q^-59 - 11246q^-58 + 4438q^-57 + 14870q^-56 + 11480q^-55 - 1536q^-54 - 11714q^-53 - 10595q^-52 - 908q^-51 + 8197q^-50 + 9222q^-49 + 2424q^-48 - 5137q^-47 - 7067q^-46 - 3103q^-45 + 2512q^-44 + 5037q^-43 + 2975q^-42 - 904q^-41 - 3040q^-40 - 2379q^-39 - 80q^-38 + 1658q^-37 + 1631q^-36 + 373q^-35 - 729q^-34 - 953q^-33 - 394q^-32 + 270q^-31 + 477q^-30 + 267q^-29 - 54q^-28 - 224q^-27 - 145q^-26 + 11q^-25 + 82q^-24 + 55q^-23 + 22q^-22 - 37q^-21 - 33q^-20 + 7q^-19 + 12q^-18 + 6q^-16 - 2q^-15 - 8q^-14 + 4q^-13 + 3q^-12 - 3q^-11 + q^-10

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	PD[Knot[9, 38]]
Out[2]=	PD[X[1, 6, 2, 7], X[5, 14, 6, 15], X[7, 18, 8, 1], X[15, 8, 16, 9], > X[3, 10, 4, 11], X[9, 4, 10, 5], X[17, 12, 18, 13], X[11, 16, 12, 17], > X[13, 2, 14, 3]]
In[3]:=	GaussCode[Knot[9, 38]]
Out[3]=	GaussCode[-1, 9, -5, 6, -2, 1, -3, 4, -6, 5, -8, 7, -9, 2, -4, 8, -7, 3]
In[4]:=	DTCode[Knot[9, 38]]
Out[4]=	DTCode[6, 10, 14, 18, 4, 16, 2, 8, 12]
In[5]:=	br = BR[Knot[9, 38]]
Out[5]=	BR[4, {-1, -1, -2, -2, 3, -2, 1, -2, -3, -3, -2}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{4, 11}
In[7]:=	BraidIndex[Knot[9, 38]]
Out[7]=	4
In[8]:=	Show[DrawMorseLink[Knot[9, 38]]]

Out[8]=	-Graphics-
In[9]:=	#[Knot[9, 38]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, {2, 3}, 2, 3, {4, 7}, 2}
In[10]:=	alex = Alexander[Knot[9, 38]][t]
Out[10]=	5 14 2 19 + -- - -- - 14 t + 5 t 2 t t
In[11]:=	Conway[Knot[9, 38]][z]
Out[11]=	2 4 1 + 6 z + 5 z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[9, 38], Knot[10, 63]}
In[13]:=	{KnotDet[Knot[9, 38]], KnotSignature[Knot[9, 38]]}
Out[13]=	{57, -4}
In[14]:=	Jones[Knot[9, 38]][q]
Out[14]=	-11 3 6 8 10 10 8 7 3 -2 -q + --- - -- + -- - -- + -- - -- + -- - -- + q 10 9 8 7 6 5 4 3 q q q q q q q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] \|\| (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[9, 38]}
In[16]:=	A2Invariant[Knot[9, 38]][q]
Out[16]=	-34 -32 -30 3 2 -22 2 4 -12 2 2 -6 -q + q + q - --- - --- - q + --- + --- + q + --- - -- + q 28 24 20 16 10 8 q q q q q q
In[17]:=	HOMFLYPT[Knot[9, 38]][a, z]
Out[17]=	6 8 4 2 6 2 8 2 10 2 4 4 6 4 8 4 4 a - 3 a + a z + 7 a z - a z - a z + a z + 3 a z + a z
In[18]:=	Kauffman[Knot[9, 38]][a, z]
Out[18]=	6 8 7 9 11 13 4 2 6 2 8 2 -4 a - 3 a + 3 a z + a z - a z + a z - a z + 9 a z + 10 a z + 10 2 12 2 5 3 7 3 9 3 11 3 13 3 > 3 a z + 3 a z - 2 a z - 2 a z + 5 a z + 3 a z - 2 a z + 4 4 6 4 8 4 10 4 12 4 5 5 7 5 > a z - 10 a z - 15 a z - 10 a z - 6 a z + 3 a z - 4 a z - 9 5 11 5 13 5 6 6 8 6 10 6 12 6 > 15 a z - 7 a z + a z + 6 a z + 6 a z + 3 a z + 3 a z + 7 7 9 7 11 7 8 8 10 8 > 5 a z + 9 a z + 4 a z + 2 a z + 2 a z
In[19]:=	{Vassiliev[2][Knot[9, 38]], Vassiliev[3][Knot[9, 38]]}
Out[19]=	{6, -14}
In[20]:=	Kh[Knot[9, 38]][q, t]
Out[20]=	-5 -3 1 2 1 4 2 4 4 q + q + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 23 9 21 8 19 8 19 7 17 7 17 6 15 6 q t q t q t q t q t q t q t 6 4 4 6 4 4 3 4 3 > ------ + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ---- 15 5 13 5 13 4 11 4 11 3 9 3 9 2 7 2 5 q t q t q t q t q t q t q t q t q t
In[21]:=	ColouredJones[Knot[9, 38], 2][q]
Out[21]=	-31 3 -29 10 16 5 38 32 27 74 37 57 98 q - --- + q + --- - --- - --- + --- - --- - --- + --- - --- - --- + --- - 30 28 27 26 25 24 23 22 21 20 19 q q q q q q q q q q q 29 77 97 14 74 71 2 50 34 7 20 8 3 > --- - --- + --- - --- - --- + --- + --- - --- + --- + -- - -- + -- + -- - 18 17 16 15 14 13 12 11 10 9 8 7 6 q q q q q q q q q q q q q 3 -4 > -- + q 5 q

The Alternating Knot 938

The Alternating Knot 9₃₈