The Alternating Knot 8₁

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Acknowledgement

KnotPlot

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

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

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

Minimum Braid Representative:

Length is 10, width is 5
Braid index is 5

A Morse Link Presentation:

3D Invariants:

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

Reversible 1 1 2 / 4--5 1

Alexander Polynomial: - 3t^-1 + 7 - 3t

Conway Polynomial: 1 - 3z²

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

Determinant and Signature: {13, 0}

Jones Polynomial: q^-6 - q^-5 + q^-4 - 2q^-3 + 2q^-2 - 2q^-1 + 2 - q + q²

Other knots (up to mirrors) with the same Jones Polynomial: {K11n70, ...}

A2 (sl(3)) Invariant: q^-20 + q^-18 - q^-12 - q^-10 + q² + q⁶ + q⁸

HOMFLY-PT Polynomial: a^-2 - z² - a²z² - a⁴ - a⁴z² + a⁶

Kauffman Polynomial: - a^-2 + a^-2z² + a^-1z³ + z⁴ - az³ + az⁵ - 2a²z⁴ + a²z⁶ - 3a³z + 5a³z³ - 4a³z⁵ + a³z⁷ - a⁴ + 7a⁴z² - 8a⁴z⁴ + 2a⁴z⁶ - 3a⁵z + 7a⁵z³ - 5a⁵z⁵ + a⁵z⁷ - a⁶ + 6a⁶z² - 5a⁶z⁴ + a⁶z⁶

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

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=0 is the signature of 8₁. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

t^rq^j r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2

j = 5 1

j = 3

j = 1 2 1

j = -1 1 1

j = -3 1 1

j = -5 1 1

j = -7 1

j = -9 1 1

j = -11

j = -13 1

n Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2 q^-18 - q^-17 - q^-16 + 2q^-15 - q^-14 - 2q^-13 + 3q^-12 - 3q^-10 + 3q^-9 - 3q^-7 + 3q^-6 + q^-5 - 3q^-4 + 2q^-3 + q^-2 - 3q^-1 + 2 - 2q² + 2q³ - q⁵ + q⁶

3 q^-36 - q^-35 - q^-34 + 2q^-32 - 2q^-30 - q^-29 + 3q^-28 + q^-27 - 2q^-26 - 2q^-25 + 2q^-24 + 2q^-23 - 2q^-22 - 2q^-21 + 2q^-20 + 2q^-19 - 2q^-18 - q^-17 + 2q^-16 + q^-15 - 3q^-14 + 2q^-12 - 3q^-10 + q^-9 + 2q^-8 - q^-7 - 2q^-6 + 2q^-5 + 3q^-4 - 2q^-3 - 2q^-2 + 4 - q - 2q² - q³ + 3q⁴ - q⁶ - 2q⁷ + 2q⁸ - q¹¹ + q¹²

4 q^-60 - q^-59 - q^-58 + 3q^-55 - q^-54 - q^-53 - q^-52 - 2q^-51 + 5q^-50 - q^-48 - q^-47 - 4q^-46 + 5q^-45 - 5q^-41 + 5q^-40 - 5q^-36 + 6q^-35 - q^-33 - q^-32 - 6q^-31 + 8q^-30 + q^-29 - 2q^-28 - 2q^-27 - 7q^-26 + 10q^-25 + 3q^-24 - 3q^-23 - 3q^-22 - 8q^-21 + 11q^-20 + 5q^-19 - 4q^-18 - 4q^-17 - 8q^-16 + 11q^-15 + 6q^-14 - 5q^-13 - 5q^-12 - 8q^-11 + 12q^-10 + 7q^-9 - 6q^-8 - 6q^-7 - 8q^-6 + 12q^-5 + 7q^-4 - 4q^-3 - 5q^-2 - 9q^-1 + 10 + 6q - 2q² - 3q³ - 7q⁴ + 7q⁵ + 3q⁶ - q⁷ - q⁸ - 5q⁹ + 4q¹⁰ + q¹¹ - 3q¹⁴ + 2q¹⁵ - q¹⁹ + q²⁰

5 q^-90 - q^-89 - q^-88 + q^-85 + 2q^-84 - 2q^-82 - q^-81 - q^-80 + 3q^-78 + 2q^-77 - q^-76 - 2q^-75 - 2q^-74 - q^-73 + 2q^-72 + 3q^-71 - q^-69 - 2q^-68 - q^-67 + q^-66 + 2q^-65 + q^-64 - q^-63 - 2q^-62 - q^-61 + q^-60 + 3q^-59 + q^-58 - 2q^-57 - 3q^-56 - 2q^-55 + q^-54 + 5q^-53 + 3q^-52 - 2q^-51 - 5q^-50 - 4q^-49 + q^-48 + 6q^-47 + 6q^-46 - 7q^-44 - 7q^-43 + 6q^-41 + 8q^-40 + 2q^-39 - 7q^-38 - 9q^-37 - 2q^-36 + 6q^-35 + 10q^-34 + 3q^-33 - 6q^-32 - 10q^-31 - 4q^-30 + 5q^-29 + 11q^-28 + 4q^-27 - 5q^-26 - 10q^-25 - 6q^-24 + 4q^-23 + 12q^-22 + 5q^-21 - 4q^-20 - 11q^-19 - 7q^-18 + 5q^-17 + 13q^-16 + 7q^-15 - 5q^-14 - 13q^-13 - 7q^-12 + 5q^-11 + 12q^-10 + 9q^-9 - 5q^-8 - 13q^-7 - 7q^-6 + 4q^-5 + 9q^-4 + 9q^-3 - 3q^-2 - 11q^-1 - 5 + 3q + 7q² + 6q³ - 3q⁴ - 8q⁵ - 2q⁶ + 3q⁷ + 4q⁸ + 4q⁹ - 3q¹⁰ - 6q¹¹ + q¹² + 2q¹³ + 2q¹⁴ + 2q¹⁵ - q¹⁶ - 4q¹⁷ + q¹⁸ + q¹⁹ + q²¹ - 2q²³ + q²⁴ - q²⁹ + q³⁰

6 q^-126 - q^-125 - q^-124 + q^-121 + 3q^-119 - q^-118 - 2q^-117 - q^-116 - q^-115 - q^-113 + 6q^-112 - q^-110 - q^-109 - 2q^-108 - q^-107 - 4q^-106 + 7q^-105 + q^-104 - q^-101 - q^-100 - 6q^-99 + 7q^-98 - 7q^-92 + 7q^-91 + q^-89 - q^-87 - q^-86 - 8q^-85 + 7q^-84 + 4q^-82 + 2q^-81 - q^-80 - 2q^-79 - 11q^-78 + 5q^-77 - q^-76 + 7q^-75 + 5q^-74 + q^-73 - q^-72 - 14q^-71 + q^-70 - 4q^-69 + 8q^-68 + 7q^-67 + 4q^-66 + 2q^-65 - 15q^-64 - q^-63 - 7q^-62 + 7q^-61 + 7q^-60 + 6q^-59 + 6q^-58 - 15q^-57 - q^-56 - 9q^-55 + 5q^-54 + 5q^-53 + 5q^-52 + 9q^-51 - 13q^-50 - 9q^-48 + 4q^-47 + 2q^-46 + 3q^-45 + 11q^-44 - 10q^-43 + 2q^-42 - 9q^-41 + 3q^-40 - q^-39 + 14q^-37 - 7q^-36 + 3q^-35 - 9q^-34 + 2q^-33 - 4q^-32 - 2q^-31 + 16q^-30 - 5q^-29 + 3q^-28 - 10q^-27 + q^-26 - 5q^-25 - 2q^-24 + 18q^-23 - 4q^-22 + 2q^-21 - 11q^-20 - 5q^-18 - q^-17 + 20q^-16 - 3q^-15 + 3q^-14 - 12q^-13 - q^-12 - 6q^-11 - 2q^-10 + 18q^-9 - 3q^-8 + 6q^-7 - 9q^-6 - 6q^-4 - 3q^-3 + 13q^-2 - 5q^-1 + 6 - 6q + 3q² - 3q³ - 2q⁴ + 8q⁵ - 8q⁶ + 3q⁷ - 5q⁸ + 5q⁹ + q¹⁰ + 5q¹² - 9q¹³ + q¹⁴ - 4q¹⁵ + 5q¹⁶ + 2q¹⁷ + q¹⁸ + 3q¹⁹ - 7q²⁰ + q²¹ - 3q²² + 3q²³ + q²⁴ + q²⁵ + 2q²⁶ - 4q²⁷ + 2q²⁸ - 2q²⁹ + q³⁰ + q³³ - 2q³⁴ + 2q³⁵ - q³⁶ - q⁴¹ + q⁴²

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[8, 1]]

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

In[3]:=
GaussCode[Knot[8, 1]]

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

In[4]:=
DTCode[Knot[8, 1]]

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

In[5]:=
br = BR[Knot[8, 1]]

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

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

Out[6]=
{5, 10}

In[7]:=
BraidIndex[Knot[8, 1]]

Out[7]=
5

In[8]:=
Show[DrawMorseLink[Knot[8, 1]]]

Out[8]=
-Graphics-

In[9]:=
#[Knot[8, 1]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}

Out[9]=
{Reversible, 1, 1, 2, {4, 5}, 1}

In[10]:=
alex = Alexander[Knot[8, 1]][t]

Out[10]=
3 7 - - - 3 t t

In[11]:=
Conway[Knot[8, 1]][z]

Out[11]=
2 1 - 3 z

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

Out[12]=
{Knot[8, 1]}

In[13]:=
{KnotDet[Knot[8, 1]], KnotSignature[Knot[8, 1]]}

Out[13]=
{13, 0}

In[14]:=
Jones[Knot[8, 1]][q]

Out[14]=
-6 -5 -4 2 2 2 2 2 + q - q + q - -- + -- - - - q + q 3 2 q q q

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

Out[15]=
{Knot[8, 1], Knot[11, NonAlternating, 70]}

In[16]:=
A2Invariant[Knot[8, 1]][q]

Out[16]=
-20 -18 -12 -10 2 6 8 q + q - q - q + q + q + q

In[17]:=
HOMFLYPT[Knot[8, 1]][a, z]

Out[17]=
-2 4 6 2 2 2 4 2 a - a + a - z - a z - a z

In[18]:=
Kauffman[Knot[8, 1]][a, z]

Out[18]=
2 3 -2 4 6 3 5 z 4 2 6 2 z 3 -a - a - a - 3 a z - 3 a z + -- + 7 a z + 6 a z + -- - a z + 2 a a 3 3 5 3 4 2 4 4 4 6 4 5 3 5 > 5 a z + 7 a z + z - 2 a z - 8 a z - 5 a z + a z - 4 a z - 5 5 2 6 4 6 6 6 3 7 5 7 > 5 a z + a z + 2 a z + a z + a z + a z

In[19]:=
{Vassiliev[2][Knot[8, 1]], Vassiliev[3][Knot[8, 1]]}

Out[19]=
{-3, 3}

In[20]:=
Kh[Knot[8, 1]][q, t]

Out[20]=
1 1 1 1 1 1 1 1 1 1 - + 2 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + ---- + --- + q 13 6 9 5 9 4 7 3 5 3 5 2 3 2 3 q t q t q t q t q t q t q t q t q t 5 2 > q t + q t

In[21]:=
ColouredJones[Knot[8, 1], 2][q]

Out[21]=
-18 -17 -16 2 -14 2 3 3 3 3 3 -5 2 + q - q - q + --- - q - --- + --- - --- + -- - -- + -- + q - 15 13 12 10 9 7 6 q q q q q q q 3 2 -2 3 2 3 5 6 > -- + -- + q - - - 2 q + 2 q - q + q 4 3 q q q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 8₁

7₇
8₂

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q^-18 - q^-17 - q^-16 + 2q^-15 - q^-14 - 2q^-13 + 3q^-12 - 3q^-10 + 3q^-9 - 3q^-7 + 3q^-6 + q^-5 - 3q^-4 + 2q^-3 + q^-2 - 3q^-1 + 2 - 2q² + 2q³ - q⁵ + q⁶
3	q^-36 - q^-35 - q^-34 + 2q^-32 - 2q^-30 - q^-29 + 3q^-28 + q^-27 - 2q^-26 - 2q^-25 + 2q^-24 + 2q^-23 - 2q^-22 - 2q^-21 + 2q^-20 + 2q^-19 - 2q^-18 - q^-17 + 2q^-16 + q^-15 - 3q^-14 + 2q^-12 - 3q^-10 + q^-9 + 2q^-8 - q^-7 - 2q^-6 + 2q^-5 + 3q^-4 - 2q^-3 - 2q^-2 + 4 - q - 2q² - q³ + 3q⁴ - q⁶ - 2q⁷ + 2q⁸ - q¹¹ + q¹²
4	q^-60 - q^-59 - q^-58 + 3q^-55 - q^-54 - q^-53 - q^-52 - 2q^-51 + 5q^-50 - q^-48 - q^-47 - 4q^-46 + 5q^-45 - 5q^-41 + 5q^-40 - 5q^-36 + 6q^-35 - q^-33 - q^-32 - 6q^-31 + 8q^-30 + q^-29 - 2q^-28 - 2q^-27 - 7q^-26 + 10q^-25 + 3q^-24 - 3q^-23 - 3q^-22 - 8q^-21 + 11q^-20 + 5q^-19 - 4q^-18 - 4q^-17 - 8q^-16 + 11q^-15 + 6q^-14 - 5q^-13 - 5q^-12 - 8q^-11 + 12q^-10 + 7q^-9 - 6q^-8 - 6q^-7 - 8q^-6 + 12q^-5 + 7q^-4 - 4q^-3 - 5q^-2 - 9q^-1 + 10 + 6q - 2q² - 3q³ - 7q⁴ + 7q⁵ + 3q⁶ - q⁷ - q⁸ - 5q⁹ + 4q¹⁰ + q¹¹ - 3q¹⁴ + 2q¹⁵ - q¹⁹ + q²⁰
5	q^-90 - q^-89 - q^-88 + q^-85 + 2q^-84 - 2q^-82 - q^-81 - q^-80 + 3q^-78 + 2q^-77 - q^-76 - 2q^-75 - 2q^-74 - q^-73 + 2q^-72 + 3q^-71 - q^-69 - 2q^-68 - q^-67 + q^-66 + 2q^-65 + q^-64 - q^-63 - 2q^-62 - q^-61 + q^-60 + 3q^-59 + q^-58 - 2q^-57 - 3q^-56 - 2q^-55 + q^-54 + 5q^-53 + 3q^-52 - 2q^-51 - 5q^-50 - 4q^-49 + q^-48 + 6q^-47 + 6q^-46 - 7q^-44 - 7q^-43 + 6q^-41 + 8q^-40 + 2q^-39 - 7q^-38 - 9q^-37 - 2q^-36 + 6q^-35 + 10q^-34 + 3q^-33 - 6q^-32 - 10q^-31 - 4q^-30 + 5q^-29 + 11q^-28 + 4q^-27 - 5q^-26 - 10q^-25 - 6q^-24 + 4q^-23 + 12q^-22 + 5q^-21 - 4q^-20 - 11q^-19 - 7q^-18 + 5q^-17 + 13q^-16 + 7q^-15 - 5q^-14 - 13q^-13 - 7q^-12 + 5q^-11 + 12q^-10 + 9q^-9 - 5q^-8 - 13q^-7 - 7q^-6 + 4q^-5 + 9q^-4 + 9q^-3 - 3q^-2 - 11q^-1 - 5 + 3q + 7q² + 6q³ - 3q⁴ - 8q⁵ - 2q⁶ + 3q⁷ + 4q⁸ + 4q⁹ - 3q¹⁰ - 6q¹¹ + q¹² + 2q¹³ + 2q¹⁴ + 2q¹⁵ - q¹⁶ - 4q¹⁷ + q¹⁸ + q¹⁹ + q²¹ - 2q²³ + q²⁴ - q²⁹ + q³⁰
6	q^-126 - q^-125 - q^-124 + q^-121 + 3q^-119 - q^-118 - 2q^-117 - q^-116 - q^-115 - q^-113 + 6q^-112 - q^-110 - q^-109 - 2q^-108 - q^-107 - 4q^-106 + 7q^-105 + q^-104 - q^-101 - q^-100 - 6q^-99 + 7q^-98 - 7q^-92 + 7q^-91 + q^-89 - q^-87 - q^-86 - 8q^-85 + 7q^-84 + 4q^-82 + 2q^-81 - q^-80 - 2q^-79 - 11q^-78 + 5q^-77 - q^-76 + 7q^-75 + 5q^-74 + q^-73 - q^-72 - 14q^-71 + q^-70 - 4q^-69 + 8q^-68 + 7q^-67 + 4q^-66 + 2q^-65 - 15q^-64 - q^-63 - 7q^-62 + 7q^-61 + 7q^-60 + 6q^-59 + 6q^-58 - 15q^-57 - q^-56 - 9q^-55 + 5q^-54 + 5q^-53 + 5q^-52 + 9q^-51 - 13q^-50 - 9q^-48 + 4q^-47 + 2q^-46 + 3q^-45 + 11q^-44 - 10q^-43 + 2q^-42 - 9q^-41 + 3q^-40 - q^-39 + 14q^-37 - 7q^-36 + 3q^-35 - 9q^-34 + 2q^-33 - 4q^-32 - 2q^-31 + 16q^-30 - 5q^-29 + 3q^-28 - 10q^-27 + q^-26 - 5q^-25 - 2q^-24 + 18q^-23 - 4q^-22 + 2q^-21 - 11q^-20 - 5q^-18 - q^-17 + 20q^-16 - 3q^-15 + 3q^-14 - 12q^-13 - q^-12 - 6q^-11 - 2q^-10 + 18q^-9 - 3q^-8 + 6q^-7 - 9q^-6 - 6q^-4 - 3q^-3 + 13q^-2 - 5q^-1 + 6 - 6q + 3q² - 3q³ - 2q⁴ + 8q⁵ - 8q⁶ + 3q⁷ - 5q⁸ + 5q⁹ + q¹⁰ + 5q¹² - 9q¹³ + q¹⁴ - 4q¹⁵ + 5q¹⁶ + 2q¹⁷ + q¹⁸ + 3q¹⁹ - 7q²⁰ + q²¹ - 3q²² + 3q²³ + q²⁴ + q²⁵ + 2q²⁶ - 4q²⁷ + 2q²⁸ - 2q²⁹ + q³⁰ + q³³ - 2q³⁴ + 2q³⁵ - q³⁶ - q⁴¹ + q⁴²

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

Out[8]=	-Graphics-
In[9]:=	#[Knot[8, 1]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 1, 1, 2, {4, 5}, 1}
In[10]:=	alex = Alexander[Knot[8, 1]][t]
Out[10]=	3 7 - - - 3 t t
In[11]:=	Conway[Knot[8, 1]][z]
Out[11]=	2 1 - 3 z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[8, 1]}
In[13]:=	{KnotDet[Knot[8, 1]], KnotSignature[Knot[8, 1]]}
Out[13]=	{13, 0}
In[14]:=	Jones[Knot[8, 1]][q]
Out[14]=	-6 -5 -4 2 2 2 2 2 + q - q + q - -- + -- - - - q + q 3 2 q q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] \|\| (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[8, 1], Knot[11, NonAlternating, 70]}
In[16]:=	A2Invariant[Knot[8, 1]][q]
Out[16]=	-20 -18 -12 -10 2 6 8 q + q - q - q + q + q + q
In[17]:=	HOMFLYPT[Knot[8, 1]][a, z]
Out[17]=	-2 4 6 2 2 2 4 2 a - a + a - z - a z - a z
In[18]:=	Kauffman[Knot[8, 1]][a, z]
Out[18]=	2 3 -2 4 6 3 5 z 4 2 6 2 z 3 -a - a - a - 3 a z - 3 a z + -- + 7 a z + 6 a z + -- - a z + 2 a a 3 3 5 3 4 2 4 4 4 6 4 5 3 5 > 5 a z + 7 a z + z - 2 a z - 8 a z - 5 a z + a z - 4 a z - 5 5 2 6 4 6 6 6 3 7 5 7 > 5 a z + a z + 2 a z + a z + a z + a z
In[19]:=	{Vassiliev[2][Knot[8, 1]], Vassiliev[3][Knot[8, 1]]}
Out[19]=	{-3, 3}
In[20]:=	Kh[Knot[8, 1]][q, t]
Out[20]=	1 1 1 1 1 1 1 1 1 1 - + 2 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + ---- + --- + q 13 6 9 5 9 4 7 3 5 3 5 2 3 2 3 q t q t q t q t q t q t q t q t q t 5 2 > q t + q t
In[21]:=	ColouredJones[Knot[8, 1], 2][q]
Out[21]=	-18 -17 -16 2 -14 2 3 3 3 3 3 -5 2 + q - q - q + --- - q - --- + --- - --- + -- - -- + -- + q - 15 13 12 10 9 7 6 q q q q q q q 3 2 -2 3 2 3 5 6 > -- + -- + q - - - 2 q + 2 q - q + q 4 3 q q q

The Alternating Knot 81

The Alternating Knot 8₁