The Alternating Knot 6₁

Also known as "Stevedore's Knot". See e.g. 1.

Visit 6₁'s page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 6₁'s page at Knotilus!

Acknowledgement

KnotPlot

PD Presentation: X₁₄₂₅ X_7,10,8,11 X₃₉₄₈ X_9,3,10,2 X_5,12,6,1 X_11,6,12,7

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

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

Minimum Braid Representative:

Length is 7, 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 1 1 2 / 3--4 1

Alexander Polynomial: - 2t^-1 + 5 - 2t

Conway Polynomial: 1 - 2z²

Other knots with the same Alexander/Conway Polynomial: {9₄₆, K11n67, K11n97, K11n139, ...}

Determinant and Signature: {9, 0}

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

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

A2 (sl(3)) Invariant: q^-14 + q^-12 - q^-6 - q^-4 + q² + q⁶ + q⁸

HOMFLY-PT Polynomial: a^-2 - z² - a² - a²z² + a⁴

Kauffman Polynomial: - a^-2 + a^-2z² + a^-1z³ + z⁴ + 2az - 2az³ + az⁵ + a² - 4a²z² + 2a²z⁴ + 2a³z - 3a³z³ + a³z⁵ + a⁴ - 3a⁴z² + a⁴z⁴

V₂ and V₃, the type 2 and 3 Vassiliev invariants: {-2, 1}

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 6₁. 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

j = 5 1

j = 3

j = 1 2 1

j = -1 1 1

j = -3 1

j = -5 1 1

j = -7

j = -9 1

n Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2 q^-12 - q^-11 - q^-10 + 2q^-9 - q^-8 - 2q^-7 + 3q^-6 - 3q^-4 + 4q^-3 - 4q^-1 + 4 - 3q² + 2q³ - q⁵ + q⁶

3 q^-24 - q^-23 - q^-22 + 2q^-20 - 2q^-18 - q^-17 + 3q^-16 + q^-15 - 2q^-14 - 2q^-13 + 2q^-12 + 3q^-11 - 2q^-10 - 3q^-9 + q^-8 + 4q^-7 - 2q^-6 - 4q^-5 + q^-4 + 5q^-3 - q^-2 - 5q^-1 + 2 + 4q - 4q³ + q⁴ + 2q⁵ - 2q⁷ + q⁸ - q¹¹ + q¹²

4 q^-40 - q^-39 - q^-38 + 3q^-35 - q^-34 - q^-33 - q^-32 - 2q^-31 + 5q^-30 - q^-28 - q^-27 - 4q^-26 + 5q^-25 + q^-23 - 6q^-21 + 4q^-20 - q^-19 + 3q^-18 + q^-17 - 7q^-16 + 3q^-15 - 2q^-14 + 5q^-13 + 3q^-12 - 8q^-11 + 3q^-10 - 3q^-9 + 6q^-8 + 4q^-7 - 10q^-6 + 2q^-5 - 3q^-4 + 7q^-3 + 4q^-2 - 10q^-1 + 2 - 3q + 6q² + 4q³ - 7q⁴ + q⁵ - 3q⁶ + 4q⁷ + 3q⁸ - 4q⁹ + 2q¹⁰ - 2q¹¹ + q¹² + q¹³ - 2q¹⁴ + 2q¹⁵ - q¹⁶ - q¹⁹ + q²⁰

5 q^-60 - q^-59 - q^-58 + q^-55 + 2q^-54 - 2q^-52 - q^-51 - q^-50 + 3q^-48 + 2q^-47 - q^-46 - 2q^-45 - 2q^-44 - q^-43 + 2q^-42 + 3q^-41 - 2q^-38 - 2q^-37 + q^-35 + q^-34 + 2q^-33 - 2q^-31 - 2q^-30 - q^-29 + q^-28 + 3q^-27 + 3q^-26 - 4q^-24 - 4q^-23 + q^-22 + 3q^-21 + 5q^-20 + q^-19 - 5q^-18 - 7q^-17 + 4q^-15 + 7q^-14 + 2q^-13 - 6q^-12 - 8q^-11 + 6q^-9 + 8q^-8 + q^-7 - 7q^-6 - 8q^-5 - q^-4 + 8q^-3 + 8q^-2 - 5 - 8q - 3q² + 7q³ + 8q⁴ + q⁵ - 4q⁶ - 7q⁷ - 4q⁸ + 5q⁹ + 5q¹⁰ + 2q¹¹ - q¹² - 4q¹³ - 3q¹⁴ + 2q¹⁵ + 2q¹⁶ + 2q¹⁸ - q¹⁹ - 2q²⁰ + q²¹ - q²³ + 2q²⁴ - q²⁶ - q²⁹ + q³⁰

6 q^-84 - q^-83 - q^-82 + q^-79 + 3q^-77 - q^-76 - 2q^-75 - q^-74 - q^-73 - q^-71 + 6q^-70 - q^-68 - q^-67 - 2q^-66 - q^-65 - 4q^-64 + 7q^-63 + q^-62 - q^-58 - 7q^-57 + 6q^-56 - q^-55 + 4q^-52 + 2q^-51 - 8q^-50 + 5q^-49 - 4q^-48 - 2q^-47 - q^-46 + 7q^-45 + 5q^-44 - 6q^-43 + 6q^-42 - 7q^-41 - 6q^-40 - 4q^-39 + 9q^-38 + 7q^-37 - 3q^-36 + 8q^-35 - 9q^-34 - 9q^-33 - 8q^-32 + 10q^-31 + 9q^-30 + 11q^-28 - 10q^-27 - 12q^-26 - 11q^-25 + 11q^-24 + 11q^-23 + 2q^-22 + 13q^-21 - 11q^-20 - 15q^-19 - 14q^-18 + 13q^-17 + 13q^-16 + 2q^-15 + 13q^-14 - 12q^-13 - 16q^-12 - 15q^-11 + 15q^-10 + 14q^-9 + 2q^-8 + 12q^-7 - 13q^-6 - 16q^-5 - 15q^-4 + 15q^-3 + 14q^-2 + 3q^-1 + 13 - 12q - 16q² - 15q³ + 12q⁴ + 13q⁵ + 5q⁶ + 12q⁷ - 9q⁸ - 14q⁹ - 14q¹⁰ + 8q¹¹ + 9q¹² + 4q¹³ + 11q¹⁴ - 5q¹⁵ - 9q¹⁶ - 10q¹⁷ + 4q¹⁸ + 4q¹⁹ + 8q²¹ - q²² - 4q²³ - 5q²⁴ + 2q²⁵ + q²⁶ - 2q²⁷ + 5q²⁸ - q³⁰ - 2q³¹ + q³² - 2q³⁴ + 3q³⁵ - q³⁸ - q⁴¹ + q⁴²

7 q^-112 - q^-111 - q^-110 + q^-107 + q^-105 + 2q^-104 - q^-103 - 2q^-102 - q^-101 - 2q^-100 + q^-99 + 5q^-96 + q^-95 - q^-94 - q^-93 - 4q^-92 - q^-90 - 2q^-89 + 5q^-88 + 2q^-87 + q^-86 + 2q^-85 - 5q^-84 + q^-83 - q^-82 - 5q^-81 + 3q^-80 + q^-78 + 3q^-77 - 3q^-76 + 4q^-75 + 2q^-74 - 5q^-73 + 2q^-72 - 4q^-71 - 3q^-70 + 2q^-69 - 3q^-68 + 7q^-67 + 6q^-66 - 2q^-65 + 4q^-64 - 6q^-63 - 8q^-62 - 2q^-61 - 6q^-60 + 7q^-59 + 10q^-58 + 2q^-57 + 9q^-56 - 5q^-55 - 11q^-54 - 6q^-53 - 11q^-52 + 4q^-51 + 12q^-50 + 7q^-49 + 14q^-48 - 2q^-47 - 12q^-46 - 8q^-45 - 17q^-44 - q^-43 + 13q^-42 + 11q^-41 + 19q^-40 + q^-39 - 12q^-38 - 10q^-37 - 22q^-36 - 5q^-35 + 12q^-34 + 14q^-33 + 24q^-32 + 4q^-31 - 14q^-30 - 12q^-29 - 26q^-28 - 7q^-27 + 13q^-26 + 16q^-25 + 28q^-24 + 5q^-23 - 15q^-22 - 14q^-21 - 28q^-20 - 7q^-19 + 15q^-18 + 17q^-17 + 29q^-16 + 5q^-15 - 16q^-14 - 16q^-13 - 28q^-12 - 6q^-11 + 17q^-10 + 16q^-9 + 29q^-8 + 6q^-7 - 17q^-6 - 18q^-5 - 28q^-4 - 5q^-3 + 16q^-2 + 15q^-1 + 30 + 8q - 15q² - 19q³ - 28q⁴ - 7q⁵ + 14q⁶ + 14q⁷ + 28q⁸ + 10q⁹ - 11q¹⁰ - 16q¹¹ - 25q¹² - 8q¹³ + 8q¹⁴ + 11q¹⁵ + 23q¹⁶ + 10q¹⁷ - 7q¹⁸ - 9q¹⁹ - 17q²⁰ - 7q²¹ + 3q²² + 4q²³ + 15q²⁴ + 6q²⁵ - 3q²⁶ - 3q²⁷ - 9q²⁸ - 2q²⁹ - q³¹ + 7q³² + 4q³³ - 2q³⁴ - q³⁵ - 4q³⁶ + q³⁷ - 3q³⁹ + 4q⁴⁰ + 2q⁴¹ - q⁴² - 2q⁴⁴ + q⁴⁵ - 2q⁴⁷ + 2q⁴⁸ + q⁴⁹ - q⁵² - q⁵⁵ + q⁵⁶

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[6, 1]]

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

In[3]:=
GaussCode[Knot[6, 1]]

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

In[4]:=
DTCode[Knot[6, 1]]

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

In[5]:=
br = BR[Knot[6, 1]]

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

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

Out[6]=
{4, 7}

In[7]:=
BraidIndex[Knot[6, 1]]

Out[7]=
4

In[8]:=
Show[DrawMorseLink[Knot[6, 1]]]

Out[8]=
-Graphics-

In[9]:=
#[Knot[6, 1]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}

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

In[10]:=
alex = Alexander[Knot[6, 1]][t]

Out[10]=
2 5 - - - 2 t t

In[11]:=
Conway[Knot[6, 1]][z]

Out[11]=
2 1 - 2 z

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

Out[12]=
{Knot[6, 1], Knot[9, 46], Knot[11, NonAlternating, 67], > Knot[11, NonAlternating, 97], Knot[11, NonAlternating, 139]}

In[13]:=
{KnotDet[Knot[6, 1]], KnotSignature[Knot[6, 1]]}

Out[13]=
{9, 0}

In[14]:=
Jones[Knot[6, 1]][q]

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

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

Out[15]=
{Knot[6, 1]}

In[16]:=
A2Invariant[Knot[6, 1]][q]

Out[16]=
-14 -12 -6 -4 2 6 8 q + q - q - q + q + q + q

In[17]:=
HOMFLYPT[Knot[6, 1]][a, z]

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

In[18]:=
Kauffman[Knot[6, 1]][a, z]

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

In[19]:=
{Vassiliev[2][Knot[6, 1]], Vassiliev[3][Knot[6, 1]]}

Out[19]=
{-2, 1}

In[20]:=
Kh[Knot[6, 1]][q, t]

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

In[21]:=
ColouredJones[Knot[6, 1], 2][q]

Out[21]=
-12 -11 -10 2 -8 2 3 3 4 4 2 3 5 4 + q - q - q + -- - q - -- + -- - -- + -- - - - 3 q + 2 q - q + 9 7 6 4 3 q q q q q q 6 > q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 6₁

5₂
6₂

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q^-12 - q^-11 - q^-10 + 2q^-9 - q^-8 - 2q^-7 + 3q^-6 - 3q^-4 + 4q^-3 - 4q^-1 + 4 - 3q² + 2q³ - q⁵ + q⁶
3	q^-24 - q^-23 - q^-22 + 2q^-20 - 2q^-18 - q^-17 + 3q^-16 + q^-15 - 2q^-14 - 2q^-13 + 2q^-12 + 3q^-11 - 2q^-10 - 3q^-9 + q^-8 + 4q^-7 - 2q^-6 - 4q^-5 + q^-4 + 5q^-3 - q^-2 - 5q^-1 + 2 + 4q - 4q³ + q⁴ + 2q⁵ - 2q⁷ + q⁸ - q¹¹ + q¹²
4	q^-40 - q^-39 - q^-38 + 3q^-35 - q^-34 - q^-33 - q^-32 - 2q^-31 + 5q^-30 - q^-28 - q^-27 - 4q^-26 + 5q^-25 + q^-23 - 6q^-21 + 4q^-20 - q^-19 + 3q^-18 + q^-17 - 7q^-16 + 3q^-15 - 2q^-14 + 5q^-13 + 3q^-12 - 8q^-11 + 3q^-10 - 3q^-9 + 6q^-8 + 4q^-7 - 10q^-6 + 2q^-5 - 3q^-4 + 7q^-3 + 4q^-2 - 10q^-1 + 2 - 3q + 6q² + 4q³ - 7q⁴ + q⁵ - 3q⁶ + 4q⁷ + 3q⁸ - 4q⁹ + 2q¹⁰ - 2q¹¹ + q¹² + q¹³ - 2q¹⁴ + 2q¹⁵ - q¹⁶ - q¹⁹ + q²⁰
5	q^-60 - q^-59 - q^-58 + q^-55 + 2q^-54 - 2q^-52 - q^-51 - q^-50 + 3q^-48 + 2q^-47 - q^-46 - 2q^-45 - 2q^-44 - q^-43 + 2q^-42 + 3q^-41 - 2q^-38 - 2q^-37 + q^-35 + q^-34 + 2q^-33 - 2q^-31 - 2q^-30 - q^-29 + q^-28 + 3q^-27 + 3q^-26 - 4q^-24 - 4q^-23 + q^-22 + 3q^-21 + 5q^-20 + q^-19 - 5q^-18 - 7q^-17 + 4q^-15 + 7q^-14 + 2q^-13 - 6q^-12 - 8q^-11 + 6q^-9 + 8q^-8 + q^-7 - 7q^-6 - 8q^-5 - q^-4 + 8q^-3 + 8q^-2 - 5 - 8q - 3q² + 7q³ + 8q⁴ + q⁵ - 4q⁶ - 7q⁷ - 4q⁸ + 5q⁹ + 5q¹⁰ + 2q¹¹ - q¹² - 4q¹³ - 3q¹⁴ + 2q¹⁵ + 2q¹⁶ + 2q¹⁸ - q¹⁹ - 2q²⁰ + q²¹ - q²³ + 2q²⁴ - q²⁶ - q²⁹ + q³⁰
6	q^-84 - q^-83 - q^-82 + q^-79 + 3q^-77 - q^-76 - 2q^-75 - q^-74 - q^-73 - q^-71 + 6q^-70 - q^-68 - q^-67 - 2q^-66 - q^-65 - 4q^-64 + 7q^-63 + q^-62 - q^-58 - 7q^-57 + 6q^-56 - q^-55 + 4q^-52 + 2q^-51 - 8q^-50 + 5q^-49 - 4q^-48 - 2q^-47 - q^-46 + 7q^-45 + 5q^-44 - 6q^-43 + 6q^-42 - 7q^-41 - 6q^-40 - 4q^-39 + 9q^-38 + 7q^-37 - 3q^-36 + 8q^-35 - 9q^-34 - 9q^-33 - 8q^-32 + 10q^-31 + 9q^-30 + 11q^-28 - 10q^-27 - 12q^-26 - 11q^-25 + 11q^-24 + 11q^-23 + 2q^-22 + 13q^-21 - 11q^-20 - 15q^-19 - 14q^-18 + 13q^-17 + 13q^-16 + 2q^-15 + 13q^-14 - 12q^-13 - 16q^-12 - 15q^-11 + 15q^-10 + 14q^-9 + 2q^-8 + 12q^-7 - 13q^-6 - 16q^-5 - 15q^-4 + 15q^-3 + 14q^-2 + 3q^-1 + 13 - 12q - 16q² - 15q³ + 12q⁴ + 13q⁵ + 5q⁶ + 12q⁷ - 9q⁸ - 14q⁹ - 14q¹⁰ + 8q¹¹ + 9q¹² + 4q¹³ + 11q¹⁴ - 5q¹⁵ - 9q¹⁶ - 10q¹⁷ + 4q¹⁸ + 4q¹⁹ + 8q²¹ - q²² - 4q²³ - 5q²⁴ + 2q²⁵ + q²⁶ - 2q²⁷ + 5q²⁸ - q³⁰ - 2q³¹ + q³² - 2q³⁴ + 3q³⁵ - q³⁸ - q⁴¹ + q⁴²
7	q^-112 - q^-111 - q^-110 + q^-107 + q^-105 + 2q^-104 - q^-103 - 2q^-102 - q^-101 - 2q^-100 + q^-99 + 5q^-96 + q^-95 - q^-94 - q^-93 - 4q^-92 - q^-90 - 2q^-89 + 5q^-88 + 2q^-87 + q^-86 + 2q^-85 - 5q^-84 + q^-83 - q^-82 - 5q^-81 + 3q^-80 + q^-78 + 3q^-77 - 3q^-76 + 4q^-75 + 2q^-74 - 5q^-73 + 2q^-72 - 4q^-71 - 3q^-70 + 2q^-69 - 3q^-68 + 7q^-67 + 6q^-66 - 2q^-65 + 4q^-64 - 6q^-63 - 8q^-62 - 2q^-61 - 6q^-60 + 7q^-59 + 10q^-58 + 2q^-57 + 9q^-56 - 5q^-55 - 11q^-54 - 6q^-53 - 11q^-52 + 4q^-51 + 12q^-50 + 7q^-49 + 14q^-48 - 2q^-47 - 12q^-46 - 8q^-45 - 17q^-44 - q^-43 + 13q^-42 + 11q^-41 + 19q^-40 + q^-39 - 12q^-38 - 10q^-37 - 22q^-36 - 5q^-35 + 12q^-34 + 14q^-33 + 24q^-32 + 4q^-31 - 14q^-30 - 12q^-29 - 26q^-28 - 7q^-27 + 13q^-26 + 16q^-25 + 28q^-24 + 5q^-23 - 15q^-22 - 14q^-21 - 28q^-20 - 7q^-19 + 15q^-18 + 17q^-17 + 29q^-16 + 5q^-15 - 16q^-14 - 16q^-13 - 28q^-12 - 6q^-11 + 17q^-10 + 16q^-9 + 29q^-8 + 6q^-7 - 17q^-6 - 18q^-5 - 28q^-4 - 5q^-3 + 16q^-2 + 15q^-1 + 30 + 8q - 15q² - 19q³ - 28q⁴ - 7q⁵ + 14q⁶ + 14q⁷ + 28q⁸ + 10q⁹ - 11q¹⁰ - 16q¹¹ - 25q¹² - 8q¹³ + 8q¹⁴ + 11q¹⁵ + 23q¹⁶ + 10q¹⁷ - 7q¹⁸ - 9q¹⁹ - 17q²⁰ - 7q²¹ + 3q²² + 4q²³ + 15q²⁴ + 6q²⁵ - 3q²⁶ - 3q²⁷ - 9q²⁸ - 2q²⁹ - q³¹ + 7q³² + 4q³³ - 2q³⁴ - q³⁵ - 4q³⁶ + q³⁷ - 3q³⁹ + 4q⁴⁰ + 2q⁴¹ - q⁴² - 2q⁴⁴ + q⁴⁵ - 2q⁴⁷ + 2q⁴⁸ + q⁴⁹ - q⁵² - q⁵⁵ + q⁵⁶

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

Out[8]=	-Graphics-
In[9]:=	#[Knot[6, 1]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 1, 1, 2, {3, 4}, 1}
In[10]:=	alex = Alexander[Knot[6, 1]][t]
Out[10]=	2 5 - - - 2 t t
In[11]:=	Conway[Knot[6, 1]][z]
Out[11]=	2 1 - 2 z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[6, 1], Knot[9, 46], Knot[11, NonAlternating, 67], > Knot[11, NonAlternating, 97], Knot[11, NonAlternating, 139]}
In[13]:=	{KnotDet[Knot[6, 1]], KnotSignature[Knot[6, 1]]}
Out[13]=	{9, 0}
In[14]:=	Jones[Knot[6, 1]][q]
Out[14]=	-4 -3 -2 2 2 2 + q - q + q - - - q + q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] \|\| (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[6, 1]}
In[16]:=	A2Invariant[Knot[6, 1]][q]
Out[16]=	-14 -12 -6 -4 2 6 8 q + q - q - q + q + q + q
In[17]:=	HOMFLYPT[Knot[6, 1]][a, z]
Out[17]=	-2 2 4 2 2 2 a - a + a - z - a z
In[18]:=	Kauffman[Knot[6, 1]][a, z]
Out[18]=	2 3 -2 2 4 3 z 2 2 4 2 z 3 -a + a + a + 2 a z + 2 a z + -- - 4 a z - 3 a z + -- - 2 a z - 2 a a 3 3 4 2 4 4 4 5 3 5 > 3 a z + z + 2 a z + a z + a z + a z
In[19]:=	{Vassiliev[2][Knot[6, 1]], Vassiliev[3][Knot[6, 1]]}
Out[19]=	{-2, 1}
In[20]:=	Kh[Knot[6, 1]][q, t]
Out[20]=	1 1 1 1 1 1 5 2 - + 2 q + ----- + ----- + ----- + ---- + --- + q t + q t q 9 4 5 3 5 2 3 q t q t q t q t q t
In[21]:=	ColouredJones[Knot[6, 1], 2][q]
Out[21]=	-12 -11 -10 2 -8 2 3 3 4 4 2 3 5 4 + q - q - q + -- - q - -- + -- - -- + -- - - - 3 q + 2 q - q + 9 7 6 4 3 q q q q q q 6 > q

The Alternating Knot 61

The Alternating Knot 6₁