The Alternating Knot 8₃

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Acknowledgement

KnotPlot

PD Presentation: X₆₂₇₁ X_14,10,15,9 X_10,5,11,6 X_12,3,13,4 X_4,11,5,12 X_2,13,3,14 X_16,8,1,7 X_8,16,9,15

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

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

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

FullyAmphicheiral 2 1 2 / 4--6 1

Alexander Polynomial: - 4t^-1 + 9 - 4t

Conway Polynomial: 1 - 4z²

Other knots with the same Alexander/Conway Polynomial: {10₁, ...}

Determinant and Signature: {17, 0}

Jones Polynomial: q^-4 - q^-3 + 2q^-2 - 3q^-1 + 3 - 3q + 2q² - q³ + q⁴

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

A2 (sl(3)) Invariant: q^-14 + q^-12 + q^-8 - q^-4 - 1 - q⁴ + q⁸ + q¹² + q¹⁴

HOMFLY-PT Polynomial: a^-4 - a^-2z² - 1 - 2z² - a²z² + a⁴

Kauffman Polynomial: a^-4 - 3a^-4z² + a^-4z⁴ - 2a^-3z³ + a^-3z⁵ + a^-2z² - 2a^-2z⁴ + a^-2z⁶ - 4a^-1z + 8a^-1z³ - 4a^-1z⁵ + a^-1z⁷ - 1 + 8z² - 6z⁴ + 2z⁶ - 4az + 8az³ - 4az⁵ + az⁷ + a²z² - 2a²z⁴ + a²z⁶ - 2a³z³ + a³z⁵ + a⁴ - 3a⁴z² + a⁴z⁴

V₂ and V₃, the type 2 and 3 Vassiliev invariants: {-4, 0}

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

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

j = 9 1

j = 7

j = 5 2 1

j = 3 1

j = 1 2 2

j = -1 2 2

j = -3 1

j = -5 1 2

j = -7

j = -9 1

n Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2 q^-12 - q^-11 + 2q^-9 - 3q^-8 - q^-7 + 5q^-6 - 4q^-5 - 3q^-4 + 9q^-3 - 5q^-2 - 5q^-1 + 11 - 5q - 5q² + 9q³ - 3q⁴ - 4q⁵ + 5q⁶ - q⁷ - 3q⁸ + 2q⁹ - q¹¹ + q¹²

3 q^-24 - q^-23 + q^-20 - 2q^-19 + q^-17 + 2q^-16 - 4q^-15 - q^-14 + 3q^-13 + 5q^-12 - 4q^-11 - 6q^-10 + 3q^-9 + 9q^-8 - 2q^-7 - 12q^-6 + 2q^-5 + 13q^-4 - 15q^-2 + 15 - 15q² + 13q⁴ + 2q⁵ - 12q⁶ - 2q⁷ + 9q⁸ + 3q⁹ - 6q¹⁰ - 4q¹¹ + 5q¹² + 3q¹³ - q¹⁴ - 4q¹⁵ + 2q¹⁶ + q¹⁷ - 2q¹⁹ + q²⁰ - q²³ + q²⁴

4 q^-40 - q^-39 - q^-36 + 2q^-35 - 2q^-34 + q^-33 + q^-32 - 3q^-31 + 3q^-30 - 4q^-29 + 2q^-28 + 5q^-27 - 4q^-26 + 4q^-25 - 9q^-24 + q^-23 + 7q^-22 - 2q^-21 + 10q^-20 - 14q^-19 - 4q^-18 + 4q^-17 - q^-16 + 21q^-15 - 14q^-14 - 9q^-13 - 3q^-12 - 4q^-11 + 34q^-10 - 12q^-9 - 12q^-8 - 9q^-7 - 8q^-6 + 42q^-5 - 10q^-4 - 12q^-3 - 12q^-2 - 10q^-1 + 45 - 10q - 12q² - 12q³ - 10q⁴ + 42q⁵ - 8q⁶ - 9q⁷ - 12q⁸ - 12q⁹ + 34q¹⁰ - 4q¹¹ - 3q¹² - 9q¹³ - 14q¹⁴ + 21q¹⁵ - q¹⁶ + 4q¹⁷ - 4q¹⁸ - 14q¹⁹ + 10q²⁰ - 2q²¹ + 7q²² + q²³ - 9q²⁴ + 4q²⁵ - 4q²⁶ + 5q²⁷ + 2q²⁸ - 4q²⁹ + 3q³⁰ - 3q³¹ + q³² + q³³ - 2q³⁴ + 2q³⁵ - q³⁶ - q³⁹ + q⁴⁰

5 q^-60 - q^-59 - q^-56 + 2q^-54 - q^-53 + q^-51 - 2q^-50 - 2q^-49 + 3q^-48 + q^-46 + 3q^-45 - 2q^-44 - 5q^-43 + 2q^-40 + 8q^-39 - 5q^-37 - 4q^-36 - 6q^-35 - q^-34 + 10q^-33 + 6q^-32 + 3q^-31 - 2q^-30 - 11q^-29 - 12q^-28 + 2q^-27 + 9q^-26 + 14q^-25 + 10q^-24 - 7q^-23 - 21q^-22 - 16q^-21 + 2q^-20 + 22q^-19 + 26q^-18 + 5q^-17 - 23q^-16 - 35q^-15 - 10q^-14 + 25q^-13 + 38q^-12 + 16q^-11 - 22q^-10 - 45q^-9 - 19q^-8 + 24q^-7 + 44q^-6 + 22q^-5 - 22q^-4 - 47q^-3 - 22q^-2 + 22q^-1 + 47 + 22q - 22q² - 47q³ - 22q⁴ + 22q⁵ + 44q⁶ + 24q⁷ - 19q⁸ - 45q⁹ - 22q¹⁰ + 16q¹¹ + 38q¹² + 25q¹³ - 10q¹⁴ - 35q¹⁵ - 23q¹⁶ + 5q¹⁷ + 26q¹⁸ + 22q¹⁹ + 2q²⁰ - 16q²¹ - 21q²² - 7q²³ + 10q²⁴ + 14q²⁵ + 9q²⁶ + 2q²⁷ - 12q²⁸ - 11q²⁹ - 2q³⁰ + 3q³¹ + 6q³² + 10q³³ - q³⁴ - 6q³⁵ - 4q³⁶ - 5q³⁷ + 8q³⁹ + 2q⁴⁰ - 5q⁴³ - 2q⁴⁴ + 3q⁴⁵ + q⁴⁶ + 3q⁴⁸ - 2q⁴⁹ - 2q⁵⁰ + q⁵¹ - q⁵³ + 2q⁵⁴ - q⁵⁶ - q⁵⁹ + q⁶⁰

6 q^-84 - q^-83 - q^-80 + 3q^-77 - 2q^-76 + q^-74 - 2q^-73 - q^-72 - q^-71 + 6q^-70 - 2q^-69 + 3q^-67 - 4q^-66 - 3q^-65 - 4q^-64 + 9q^-63 - q^-62 + q^-61 + 7q^-60 - 4q^-59 - 7q^-58 - 11q^-57 + 10q^-56 - 2q^-55 + 2q^-54 + 15q^-53 + 2q^-52 - 6q^-51 - 17q^-50 + 7q^-49 - 13q^-48 - 5q^-47 + 20q^-46 + 12q^-45 + 7q^-44 - 11q^-43 + 13q^-42 - 29q^-41 - 25q^-40 + 8q^-39 + 12q^-38 + 21q^-37 + 10q^-36 + 39q^-35 - 32q^-34 - 44q^-33 - 21q^-32 - 8q^-31 + 19q^-30 + 29q^-29 + 82q^-28 - 15q^-27 - 50q^-26 - 50q^-25 - 40q^-24 + q^-23 + 36q^-22 + 124q^-21 + 9q^-20 - 45q^-19 - 68q^-18 - 65q^-17 - 19q^-16 + 34q^-15 + 151q^-14 + 25q^-13 - 38q^-12 - 76q^-11 - 76q^-10 - 31q^-9 + 31q^-8 + 163q^-7 + 29q^-6 - 34q^-5 - 78q^-4 - 78q^-3 - 34q^-2 + 29q^-1 + 167 + 29q - 34q² - 78q³ - 78q⁴ - 34q⁵ + 29q⁶ + 163q⁷ + 31q⁸ - 31q⁹ - 76q¹⁰ - 76q¹¹ - 38q¹² + 25q¹³ + 151q¹⁴ + 34q¹⁵ - 19q¹⁶ - 65q¹⁷ - 68q¹⁸ - 45q¹⁹ + 9q²⁰ + 124q²¹ + 36q²² + q²³ - 40q²⁴ - 50q²⁵ - 50q²⁶ - 15q²⁷ + 82q²⁸ + 29q²⁹ + 19q³⁰ - 8q³¹ - 21q³² - 44q³³ - 32q³⁴ + 39q³⁵ + 10q³⁶ + 21q³⁷ + 12q³⁸ + 8q³⁹ - 25q⁴⁰ - 29q⁴¹ + 13q⁴² - 11q⁴³ + 7q⁴⁴ + 12q⁴⁵ + 20q⁴⁶ - 5q⁴⁷ - 13q⁴⁸ + 7q⁴⁹ - 17q⁵⁰ - 6q⁵¹ + 2q⁵² + 15q⁵³ + 2q⁵⁴ - 2q⁵⁵ + 10q⁵⁶ - 11q⁵⁷ - 7q⁵⁸ - 4q⁵⁹ + 7q⁶⁰ + q⁶¹ - q⁶² + 9q⁶³ - 4q⁶⁴ - 3q⁶⁵ - 4q⁶⁶ + 3q⁶⁷ - 2q⁶⁹ + 6q⁷⁰ - q⁷¹ - q⁷² - 2q⁷³ + q⁷⁴ - 2q⁷⁶ + 3q⁷⁷ - q⁸⁰ - q⁸³ + q⁸⁴

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, 3]]

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

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

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

In[4]:=
DTCode[Knot[8, 3]]

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

In[5]:=
br = BR[Knot[8, 3]]

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

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

Out[6]=
{5, 10}

In[7]:=
BraidIndex[Knot[8, 3]]

Out[7]=
5

In[8]:=
Show[DrawMorseLink[Knot[8, 3]]]

Out[8]=
-Graphics-

In[9]:=
#[Knot[8, 3]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}

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

In[10]:=
alex = Alexander[Knot[8, 3]][t]

Out[10]=
4 9 - - - 4 t t

In[11]:=
Conway[Knot[8, 3]][z]

Out[11]=
2 1 - 4 z

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

Out[12]=
{Knot[8, 3], Knot[10, 1]}

In[13]:=
{KnotDet[Knot[8, 3]], KnotSignature[Knot[8, 3]]}

Out[13]=
{17, 0}

In[14]:=
Jones[Knot[8, 3]][q]

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

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

Out[15]=
{Knot[8, 3]}

In[16]:=
A2Invariant[Knot[8, 3]][q]

Out[16]=
-14 -12 -8 -4 4 8 12 14 -1 + q + q + q - q - q + q + q + q

In[17]:=
HOMFLYPT[Knot[8, 3]][a, z]

Out[17]=
2 -4 4 2 z 2 2 -1 + a + a - 2 z - -- - a z 2 a

In[18]:=
Kauffman[Knot[8, 3]][a, z]

Out[18]=
2 2 3 -4 4 4 z 2 3 z z 2 2 4 2 2 z -1 + a + a - --- - 4 a z + 8 z - ---- + -- + a z - 3 a z - ---- + a 4 2 3 a a a 3 4 4 5 5 8 z 3 3 3 4 z 2 z 2 4 4 4 z 4 z > ---- + 8 a z - 2 a z - 6 z + -- - ---- - 2 a z + a z + -- - ---- - a 4 2 3 a a a a 6 7 5 3 5 6 z 2 6 z 7 > 4 a z + a z + 2 z + -- + a z + -- + a z 2 a a

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

Out[19]=
{-4, 0}

In[20]:=
Kh[Knot[8, 3]][q, t]

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

In[21]:=
ColouredJones[Knot[8, 3], 2][q]

Out[21]=
-12 -11 2 3 -7 5 4 3 9 5 5 2 11 + q - q + -- - -- - q + -- - -- - -- + -- - -- - - - 5 q - 5 q + 9 8 6 5 4 3 2 q q q q q q q q 3 4 5 6 7 8 9 11 12 > 9 q - 3 q - 4 q + 5 q - q - 3 q + 2 q - q + q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 8₃

8₂
8₄

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q^-12 - q^-11 + 2q^-9 - 3q^-8 - q^-7 + 5q^-6 - 4q^-5 - 3q^-4 + 9q^-3 - 5q^-2 - 5q^-1 + 11 - 5q - 5q² + 9q³ - 3q⁴ - 4q⁵ + 5q⁶ - q⁷ - 3q⁸ + 2q⁹ - q¹¹ + q¹²
3	q^-24 - q^-23 + q^-20 - 2q^-19 + q^-17 + 2q^-16 - 4q^-15 - q^-14 + 3q^-13 + 5q^-12 - 4q^-11 - 6q^-10 + 3q^-9 + 9q^-8 - 2q^-7 - 12q^-6 + 2q^-5 + 13q^-4 - 15q^-2 + 15 - 15q² + 13q⁴ + 2q⁵ - 12q⁶ - 2q⁷ + 9q⁸ + 3q⁹ - 6q¹⁰ - 4q¹¹ + 5q¹² + 3q¹³ - q¹⁴ - 4q¹⁵ + 2q¹⁶ + q¹⁷ - 2q¹⁹ + q²⁰ - q²³ + q²⁴
4	q^-40 - q^-39 - q^-36 + 2q^-35 - 2q^-34 + q^-33 + q^-32 - 3q^-31 + 3q^-30 - 4q^-29 + 2q^-28 + 5q^-27 - 4q^-26 + 4q^-25 - 9q^-24 + q^-23 + 7q^-22 - 2q^-21 + 10q^-20 - 14q^-19 - 4q^-18 + 4q^-17 - q^-16 + 21q^-15 - 14q^-14 - 9q^-13 - 3q^-12 - 4q^-11 + 34q^-10 - 12q^-9 - 12q^-8 - 9q^-7 - 8q^-6 + 42q^-5 - 10q^-4 - 12q^-3 - 12q^-2 - 10q^-1 + 45 - 10q - 12q² - 12q³ - 10q⁴ + 42q⁵ - 8q⁶ - 9q⁷ - 12q⁸ - 12q⁹ + 34q¹⁰ - 4q¹¹ - 3q¹² - 9q¹³ - 14q¹⁴ + 21q¹⁵ - q¹⁶ + 4q¹⁷ - 4q¹⁸ - 14q¹⁹ + 10q²⁰ - 2q²¹ + 7q²² + q²³ - 9q²⁴ + 4q²⁵ - 4q²⁶ + 5q²⁷ + 2q²⁸ - 4q²⁹ + 3q³⁰ - 3q³¹ + q³² + q³³ - 2q³⁴ + 2q³⁵ - q³⁶ - q³⁹ + q⁴⁰
5	q^-60 - q^-59 - q^-56 + 2q^-54 - q^-53 + q^-51 - 2q^-50 - 2q^-49 + 3q^-48 + q^-46 + 3q^-45 - 2q^-44 - 5q^-43 + 2q^-40 + 8q^-39 - 5q^-37 - 4q^-36 - 6q^-35 - q^-34 + 10q^-33 + 6q^-32 + 3q^-31 - 2q^-30 - 11q^-29 - 12q^-28 + 2q^-27 + 9q^-26 + 14q^-25 + 10q^-24 - 7q^-23 - 21q^-22 - 16q^-21 + 2q^-20 + 22q^-19 + 26q^-18 + 5q^-17 - 23q^-16 - 35q^-15 - 10q^-14 + 25q^-13 + 38q^-12 + 16q^-11 - 22q^-10 - 45q^-9 - 19q^-8 + 24q^-7 + 44q^-6 + 22q^-5 - 22q^-4 - 47q^-3 - 22q^-2 + 22q^-1 + 47 + 22q - 22q² - 47q³ - 22q⁴ + 22q⁵ + 44q⁶ + 24q⁷ - 19q⁸ - 45q⁹ - 22q¹⁰ + 16q¹¹ + 38q¹² + 25q¹³ - 10q¹⁴ - 35q¹⁵ - 23q¹⁶ + 5q¹⁷ + 26q¹⁸ + 22q¹⁹ + 2q²⁰ - 16q²¹ - 21q²² - 7q²³ + 10q²⁴ + 14q²⁵ + 9q²⁶ + 2q²⁷ - 12q²⁸ - 11q²⁹ - 2q³⁰ + 3q³¹ + 6q³² + 10q³³ - q³⁴ - 6q³⁵ - 4q³⁶ - 5q³⁷ + 8q³⁹ + 2q⁴⁰ - 5q⁴³ - 2q⁴⁴ + 3q⁴⁵ + q⁴⁶ + 3q⁴⁸ - 2q⁴⁹ - 2q⁵⁰ + q⁵¹ - q⁵³ + 2q⁵⁴ - q⁵⁶ - q⁵⁹ + q⁶⁰
6	q^-84 - q^-83 - q^-80 + 3q^-77 - 2q^-76 + q^-74 - 2q^-73 - q^-72 - q^-71 + 6q^-70 - 2q^-69 + 3q^-67 - 4q^-66 - 3q^-65 - 4q^-64 + 9q^-63 - q^-62 + q^-61 + 7q^-60 - 4q^-59 - 7q^-58 - 11q^-57 + 10q^-56 - 2q^-55 + 2q^-54 + 15q^-53 + 2q^-52 - 6q^-51 - 17q^-50 + 7q^-49 - 13q^-48 - 5q^-47 + 20q^-46 + 12q^-45 + 7q^-44 - 11q^-43 + 13q^-42 - 29q^-41 - 25q^-40 + 8q^-39 + 12q^-38 + 21q^-37 + 10q^-36 + 39q^-35 - 32q^-34 - 44q^-33 - 21q^-32 - 8q^-31 + 19q^-30 + 29q^-29 + 82q^-28 - 15q^-27 - 50q^-26 - 50q^-25 - 40q^-24 + q^-23 + 36q^-22 + 124q^-21 + 9q^-20 - 45q^-19 - 68q^-18 - 65q^-17 - 19q^-16 + 34q^-15 + 151q^-14 + 25q^-13 - 38q^-12 - 76q^-11 - 76q^-10 - 31q^-9 + 31q^-8 + 163q^-7 + 29q^-6 - 34q^-5 - 78q^-4 - 78q^-3 - 34q^-2 + 29q^-1 + 167 + 29q - 34q² - 78q³ - 78q⁴ - 34q⁵ + 29q⁶ + 163q⁷ + 31q⁸ - 31q⁹ - 76q¹⁰ - 76q¹¹ - 38q¹² + 25q¹³ + 151q¹⁴ + 34q¹⁵ - 19q¹⁶ - 65q¹⁷ - 68q¹⁸ - 45q¹⁹ + 9q²⁰ + 124q²¹ + 36q²² + q²³ - 40q²⁴ - 50q²⁵ - 50q²⁶ - 15q²⁷ + 82q²⁸ + 29q²⁹ + 19q³⁰ - 8q³¹ - 21q³² - 44q³³ - 32q³⁴ + 39q³⁵ + 10q³⁶ + 21q³⁷ + 12q³⁸ + 8q³⁹ - 25q⁴⁰ - 29q⁴¹ + 13q⁴² - 11q⁴³ + 7q⁴⁴ + 12q⁴⁵ + 20q⁴⁶ - 5q⁴⁷ - 13q⁴⁸ + 7q⁴⁹ - 17q⁵⁰ - 6q⁵¹ + 2q⁵² + 15q⁵³ + 2q⁵⁴ - 2q⁵⁵ + 10q⁵⁶ - 11q⁵⁷ - 7q⁵⁸ - 4q⁵⁹ + 7q⁶⁰ + q⁶¹ - q⁶² + 9q⁶³ - 4q⁶⁴ - 3q⁶⁵ - 4q⁶⁶ + 3q⁶⁷ - 2q⁶⁹ + 6q⁷⁰ - q⁷¹ - q⁷² - 2q⁷³ + q⁷⁴ - 2q⁷⁶ + 3q⁷⁷ - q⁸⁰ - q⁸³ + q⁸⁴

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

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

The Alternating Knot 83

The Alternating Knot 8₃