The Alternating Knot 6₂

I like to call this knot "The Miller Institute Knot", as it is the logo of the Miller Institute for Basic Research.

Visit 6₂'s page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 6₂'s page at Knotilus!

Acknowledgement

KnotPlot

Further views:
The Miller Institute Mug

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

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

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

Minimum Braid Representative:

Length is 6, 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 1 2 2 / 3--4 1

Alexander Polynomial: - t^-2 + 3t^-1 - 3 + 3t - t²

Conway Polynomial: 1 - z² - z⁴

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

Determinant and Signature: {11, -2}

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

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

A2 (sl(3)) Invariant: q^-16 - q^-8 - q^-4 + q^-2 + 1 + q² + q⁴

HOMFLY-PT Polynomial: 2 + z² - 2a² - 3a²z² - a²z⁴ + a⁴ + a⁴z²

Kauffman Polynomial: 2 - 3z² + z⁴ - 2az³ + az⁵ + 2a² - 6a²z² + 3a²z⁴ - a³z + a³z⁵ + a⁴ - 2a⁴z² + 2a⁴z⁴ - a⁵z + 2a⁵z³ + a⁶z²

V₂ and V₃, the type 2 and 3 Vassiliev invariants: {-1, 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=-2 is the signature of 6₂. 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 = 3 1

j = 1

j = -1 2 1

j = -3 1 1

j = -5 1 1

j = -7 1 1

j = -9 1

j = -11 1

n Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2 q^-14 - 2q^-13 + 4q^-11 - 5q^-10 + 6q^-8 - 6q^-7 + 6q^-5 - 5q^-4 - q^-3 + 5q^-2 - 3q^-1 - 1 + 3q - q² - q³ + q⁴

3 q^-27 - 2q^-26 + 2q^-24 + q^-23 - 4q^-22 - 2q^-21 + 7q^-20 + 2q^-19 - 8q^-18 - 3q^-17 + 10q^-16 + 3q^-15 - 10q^-14 - 4q^-13 + 10q^-12 + 4q^-11 - 10q^-10 - 4q^-9 + 8q^-8 + 5q^-7 - 7q^-6 - 4q^-5 + 5q^-4 + 6q^-3 - 5q^-2 - 4q^-1 + 2 + 5q - 2q² - 3q³ + 3q⁵ - q⁷ - q⁸ + q⁹

4 q^-44 - 2q^-43 + 2q^-41 - q^-40 + 2q^-39 - 6q^-38 + 2q^-37 + 6q^-36 - 2q^-35 + 3q^-34 - 14q^-33 + 3q^-32 + 13q^-31 - q^-30 + 2q^-29 - 22q^-28 + 3q^-27 + 18q^-26 + q^-25 + 3q^-24 - 26q^-23 + 2q^-22 + 20q^-21 + 2q^-20 + 4q^-19 - 26q^-18 + q^-17 + 17q^-16 + 2q^-15 + 6q^-14 - 24q^-13 + 13q^-11 + 2q^-10 + 8q^-9 - 19q^-8 - 2q^-7 + 8q^-6 + 3q^-5 + 10q^-4 - 14q^-3 - 4q^-2 + 3q^-1 + 3 + 10q - 8q² - 4q³ - q⁴ + q⁵ + 8q⁶ - 3q⁷ - 2q⁸ - 2q⁹ - q¹⁰ + 4q¹¹ - q¹⁴ - q¹⁵ + q¹⁶

5 q^-65 - 2q^-64 + 2q^-62 - q^-61 - 2q^-58 + q^-57 + 5q^-56 - 4q^-54 - 5q^-53 - 2q^-52 + 6q^-51 + 9q^-50 + 4q^-49 - 9q^-48 - 16q^-47 - 4q^-46 + 11q^-45 + 19q^-44 + 10q^-43 - 13q^-42 - 25q^-41 - 11q^-40 + 14q^-39 + 26q^-38 + 15q^-37 - 13q^-36 - 30q^-35 - 16q^-34 + 13q^-33 + 29q^-32 + 18q^-31 - 12q^-30 - 29q^-29 - 18q^-28 + 11q^-27 + 28q^-26 + 17q^-25 - 10q^-24 - 25q^-23 - 17q^-22 + 8q^-21 + 24q^-20 + 15q^-19 - 6q^-18 - 19q^-17 - 17q^-16 + 4q^-15 + 19q^-14 + 13q^-13 - q^-12 - 13q^-11 - 16q^-10 - 2q^-9 + 13q^-8 + 12q^-7 + 4q^-6 - 4q^-5 - 15q^-4 - 7q^-3 + 6q^-2 + 9q^-1 + 8 + 2q - 10q² - 9q³ - q⁴ + 4q⁵ + 8q⁶ + 6q⁷ - 4q⁸ - 6q⁹ - 4q¹⁰ - q¹¹ + 4q¹² + 6q¹³ - 2q¹⁵ - 2q¹⁶ - 3q¹⁷ + 3q¹⁹ + q²⁰ - q²³ - q²⁴ + q²⁵

6 q^-90 - 2q^-89 + 2q^-87 - q^-86 - 2q^-84 + 4q^-83 - 3q^-82 + 7q^-80 - 4q^-79 - 4q^-78 - 6q^-77 + 7q^-76 - 3q^-75 + 6q^-74 + 16q^-73 - 9q^-72 - 14q^-71 - 17q^-70 + 11q^-69 - q^-68 + 18q^-67 + 33q^-66 - 13q^-65 - 26q^-64 - 36q^-63 + 7q^-62 + 32q^-60 + 54q^-59 - 11q^-58 - 34q^-57 - 53q^-56 - q^-55 - 4q^-54 + 40q^-53 + 70q^-52 - 5q^-51 - 35q^-50 - 63q^-49 - 7q^-48 - 10q^-47 + 41q^-46 + 77q^-45 - 32q^-43 - 65q^-42 - 8q^-41 - 14q^-40 + 39q^-39 + 76q^-38 + q^-37 - 29q^-36 - 61q^-35 - 6q^-34 - 16q^-33 + 35q^-32 + 70q^-31 + q^-30 - 26q^-29 - 53q^-28 - 3q^-27 - 19q^-26 + 29q^-25 + 60q^-24 + 4q^-23 - 19q^-22 - 44q^-21 - 2q^-20 - 24q^-19 + 21q^-18 + 50q^-17 + 9q^-16 - 9q^-15 - 33q^-14 - 2q^-13 - 29q^-12 + 10q^-11 + 38q^-10 + 13q^-9 + 2q^-8 - 20q^-7 - 32q^-5 - q^-4 + 23q^-3 + 13q^-2 + 10q^-1 - 6 + 5q - 29q² - 9q³ + 8q⁴ + 8q⁵ + 11q⁶ + 4q⁷ + 11q⁸ - 19q⁹ - 10q¹⁰ - 3q¹¹ + q¹² + 5q¹³ + 6q¹⁴ + 14q¹⁵ - 8q¹⁶ - 5q¹⁷ - 5q¹⁸ - 3q¹⁹ - q²⁰ + 2q²¹ + 10q²² - q²³ - 2q²⁵ - 2q²⁶ - 3q²⁷ - q²⁸ + 4q²⁹ + q³¹ - q³⁴ - q³⁵ + q³⁶

7 q^-119 - 2q^-118 + 2q^-116 - q^-115 - 2q^-113 + 2q^-112 + 3q^-111 - 4q^-110 + 2q^-109 + 3q^-108 - 4q^-107 - 2q^-106 - 6q^-105 + 2q^-104 + 10q^-103 - q^-102 + 7q^-101 + 3q^-100 - 13q^-99 - 8q^-98 - 16q^-97 + q^-96 + 23q^-95 + 14q^-94 + 22q^-93 + 2q^-92 - 31q^-91 - 24q^-90 - 35q^-89 - 8q^-88 + 41q^-87 + 39q^-86 + 51q^-85 + 9q^-84 - 47q^-83 - 47q^-82 - 67q^-81 - 23q^-80 + 50q^-79 + 63q^-78 + 83q^-77 + 28q^-76 - 52q^-75 - 66q^-74 - 94q^-73 - 43q^-72 + 49q^-71 + 72q^-70 + 107q^-69 + 49q^-68 - 47q^-67 - 71q^-66 - 111q^-65 - 57q^-64 + 41q^-63 + 71q^-62 + 116q^-61 + 61q^-60 - 40q^-59 - 68q^-58 - 115q^-57 - 63q^-56 + 36q^-55 + 66q^-54 + 114q^-53 + 63q^-52 - 36q^-51 - 63q^-50 - 112q^-49 - 61q^-48 + 35q^-47 + 61q^-46 + 106q^-45 + 60q^-44 - 34q^-43 - 58q^-42 - 102q^-41 - 56q^-40 + 34q^-39 + 52q^-38 + 95q^-37 + 57q^-36 - 30q^-35 - 49q^-34 - 90q^-33 - 52q^-32 + 27q^-31 + 38q^-30 + 82q^-29 + 55q^-28 - 19q^-27 - 36q^-26 - 75q^-25 - 51q^-24 + 14q^-23 + 20q^-22 + 67q^-21 + 55q^-20 - 4q^-19 - 17q^-18 - 59q^-17 - 48q^-16 - q^-15 + q^-14 + 46q^-13 + 51q^-12 + 12q^-11 + 2q^-10 - 38q^-9 - 41q^-8 - 11q^-7 - 18q^-6 + 22q^-5 + 40q^-4 + 19q^-3 + 19q^-2 - 15q^-1 - 25 - 14q - 29q² + 21q⁴ + 14q⁵ + 25q⁶ + 6q⁷ - 7q⁸ - 6q⁹ - 28q¹⁰ - 12q¹¹ + 3q¹² + q¹³ + 18q¹⁴ + 13q¹⁵ + 7q¹⁶ + 6q¹⁷ - 16q¹⁸ - 12q¹⁹ - 5q²⁰ - 9q²¹ + 5q²² + 7q²³ + 9q²⁴ + 12q²⁵ - 4q²⁶ - 5q²⁷ - 3q²⁸ - 8q²⁹ - 3q³⁰ - q³¹ + 3q³² + 9q³³ + q³⁴ + q³⁶ - 3q³⁷ - 2q³⁸ - 3q³⁹ - q⁴⁰ + 3q⁴¹ + q⁴² + q⁴⁴ - q⁴⁷ - q⁴⁸ + q⁴⁹

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, 2]]

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

In[3]:=
GaussCode[Knot[6, 2]]

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

In[4]:=
DTCode[Knot[6, 2]]

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

In[5]:=
br = BR[Knot[6, 2]]

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

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

Out[6]=
{3, 6}

In[7]:=
BraidIndex[Knot[6, 2]]

Out[7]=
3

In[8]:=
Show[DrawMorseLink[Knot[6, 2]]]

Out[8]=
-Graphics-

In[9]:=
#[Knot[6, 2]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}

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

In[10]:=
alex = Alexander[Knot[6, 2]][t]

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

In[11]:=
Conway[Knot[6, 2]][z]

Out[11]=
2 4 1 - z - z

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

Out[12]=
{Knot[6, 2]}

In[13]:=
{KnotDet[Knot[6, 2]], KnotSignature[Knot[6, 2]]}

Out[13]=
{11, -2}

In[14]:=
Jones[Knot[6, 2]][q]

Out[14]=
-5 2 2 2 2 -1 + q - -- + -- - -- + - + q 4 3 2 q q q q

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

Out[15]=
{Knot[6, 2]}

In[16]:=
A2Invariant[Knot[6, 2]][q]

Out[16]=
-16 -8 -4 -2 2 4 1 + q - q - q + q + q + q

In[17]:=
HOMFLYPT[Knot[6, 2]][a, z]

Out[17]=
2 4 2 2 2 4 2 2 4 2 - 2 a + a + z - 3 a z + a z - a z

In[18]:=
Kauffman[Knot[6, 2]][a, z]

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

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

Out[19]=
{-1, 1}

In[20]:=
Kh[Knot[6, 2]][q, t]

Out[20]=
-3 2 1 1 1 1 1 1 1 t 3 2 q + - + ------ + ----- + ----- + ----- + ----- + ---- + ---- + - + q t q 11 4 9 3 7 3 7 2 5 2 5 3 q q t q t q t q t q t q t q t

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

Out[21]=
-14 2 4 5 6 6 6 5 -3 5 3 2 -1 + q - --- + --- - --- + -- - -- + -- - -- - q + -- - - + 3 q - q - 13 11 10 8 7 5 4 2 q q q q q q q q q 3 4 > q + q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 6₂

6₁
6₃

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q^-14 - 2q^-13 + 4q^-11 - 5q^-10 + 6q^-8 - 6q^-7 + 6q^-5 - 5q^-4 - q^-3 + 5q^-2 - 3q^-1 - 1 + 3q - q² - q³ + q⁴
3	q^-27 - 2q^-26 + 2q^-24 + q^-23 - 4q^-22 - 2q^-21 + 7q^-20 + 2q^-19 - 8q^-18 - 3q^-17 + 10q^-16 + 3q^-15 - 10q^-14 - 4q^-13 + 10q^-12 + 4q^-11 - 10q^-10 - 4q^-9 + 8q^-8 + 5q^-7 - 7q^-6 - 4q^-5 + 5q^-4 + 6q^-3 - 5q^-2 - 4q^-1 + 2 + 5q - 2q² - 3q³ + 3q⁵ - q⁷ - q⁸ + q⁹
4	q^-44 - 2q^-43 + 2q^-41 - q^-40 + 2q^-39 - 6q^-38 + 2q^-37 + 6q^-36 - 2q^-35 + 3q^-34 - 14q^-33 + 3q^-32 + 13q^-31 - q^-30 + 2q^-29 - 22q^-28 + 3q^-27 + 18q^-26 + q^-25 + 3q^-24 - 26q^-23 + 2q^-22 + 20q^-21 + 2q^-20 + 4q^-19 - 26q^-18 + q^-17 + 17q^-16 + 2q^-15 + 6q^-14 - 24q^-13 + 13q^-11 + 2q^-10 + 8q^-9 - 19q^-8 - 2q^-7 + 8q^-6 + 3q^-5 + 10q^-4 - 14q^-3 - 4q^-2 + 3q^-1 + 3 + 10q - 8q² - 4q³ - q⁴ + q⁵ + 8q⁶ - 3q⁷ - 2q⁸ - 2q⁹ - q¹⁰ + 4q¹¹ - q¹⁴ - q¹⁵ + q¹⁶
5	q^-65 - 2q^-64 + 2q^-62 - q^-61 - 2q^-58 + q^-57 + 5q^-56 - 4q^-54 - 5q^-53 - 2q^-52 + 6q^-51 + 9q^-50 + 4q^-49 - 9q^-48 - 16q^-47 - 4q^-46 + 11q^-45 + 19q^-44 + 10q^-43 - 13q^-42 - 25q^-41 - 11q^-40 + 14q^-39 + 26q^-38 + 15q^-37 - 13q^-36 - 30q^-35 - 16q^-34 + 13q^-33 + 29q^-32 + 18q^-31 - 12q^-30 - 29q^-29 - 18q^-28 + 11q^-27 + 28q^-26 + 17q^-25 - 10q^-24 - 25q^-23 - 17q^-22 + 8q^-21 + 24q^-20 + 15q^-19 - 6q^-18 - 19q^-17 - 17q^-16 + 4q^-15 + 19q^-14 + 13q^-13 - q^-12 - 13q^-11 - 16q^-10 - 2q^-9 + 13q^-8 + 12q^-7 + 4q^-6 - 4q^-5 - 15q^-4 - 7q^-3 + 6q^-2 + 9q^-1 + 8 + 2q - 10q² - 9q³ - q⁴ + 4q⁵ + 8q⁶ + 6q⁷ - 4q⁸ - 6q⁹ - 4q¹⁰ - q¹¹ + 4q¹² + 6q¹³ - 2q¹⁵ - 2q¹⁶ - 3q¹⁷ + 3q¹⁹ + q²⁰ - q²³ - q²⁴ + q²⁵
6	q^-90 - 2q^-89 + 2q^-87 - q^-86 - 2q^-84 + 4q^-83 - 3q^-82 + 7q^-80 - 4q^-79 - 4q^-78 - 6q^-77 + 7q^-76 - 3q^-75 + 6q^-74 + 16q^-73 - 9q^-72 - 14q^-71 - 17q^-70 + 11q^-69 - q^-68 + 18q^-67 + 33q^-66 - 13q^-65 - 26q^-64 - 36q^-63 + 7q^-62 + 32q^-60 + 54q^-59 - 11q^-58 - 34q^-57 - 53q^-56 - q^-55 - 4q^-54 + 40q^-53 + 70q^-52 - 5q^-51 - 35q^-50 - 63q^-49 - 7q^-48 - 10q^-47 + 41q^-46 + 77q^-45 - 32q^-43 - 65q^-42 - 8q^-41 - 14q^-40 + 39q^-39 + 76q^-38 + q^-37 - 29q^-36 - 61q^-35 - 6q^-34 - 16q^-33 + 35q^-32 + 70q^-31 + q^-30 - 26q^-29 - 53q^-28 - 3q^-27 - 19q^-26 + 29q^-25 + 60q^-24 + 4q^-23 - 19q^-22 - 44q^-21 - 2q^-20 - 24q^-19 + 21q^-18 + 50q^-17 + 9q^-16 - 9q^-15 - 33q^-14 - 2q^-13 - 29q^-12 + 10q^-11 + 38q^-10 + 13q^-9 + 2q^-8 - 20q^-7 - 32q^-5 - q^-4 + 23q^-3 + 13q^-2 + 10q^-1 - 6 + 5q - 29q² - 9q³ + 8q⁴ + 8q⁵ + 11q⁶ + 4q⁷ + 11q⁸ - 19q⁹ - 10q¹⁰ - 3q¹¹ + q¹² + 5q¹³ + 6q¹⁴ + 14q¹⁵ - 8q¹⁶ - 5q¹⁷ - 5q¹⁸ - 3q¹⁹ - q²⁰ + 2q²¹ + 10q²² - q²³ - 2q²⁵ - 2q²⁶ - 3q²⁷ - q²⁸ + 4q²⁹ + q³¹ - q³⁴ - q³⁵ + q³⁶
7	q^-119 - 2q^-118 + 2q^-116 - q^-115 - 2q^-113 + 2q^-112 + 3q^-111 - 4q^-110 + 2q^-109 + 3q^-108 - 4q^-107 - 2q^-106 - 6q^-105 + 2q^-104 + 10q^-103 - q^-102 + 7q^-101 + 3q^-100 - 13q^-99 - 8q^-98 - 16q^-97 + q^-96 + 23q^-95 + 14q^-94 + 22q^-93 + 2q^-92 - 31q^-91 - 24q^-90 - 35q^-89 - 8q^-88 + 41q^-87 + 39q^-86 + 51q^-85 + 9q^-84 - 47q^-83 - 47q^-82 - 67q^-81 - 23q^-80 + 50q^-79 + 63q^-78 + 83q^-77 + 28q^-76 - 52q^-75 - 66q^-74 - 94q^-73 - 43q^-72 + 49q^-71 + 72q^-70 + 107q^-69 + 49q^-68 - 47q^-67 - 71q^-66 - 111q^-65 - 57q^-64 + 41q^-63 + 71q^-62 + 116q^-61 + 61q^-60 - 40q^-59 - 68q^-58 - 115q^-57 - 63q^-56 + 36q^-55 + 66q^-54 + 114q^-53 + 63q^-52 - 36q^-51 - 63q^-50 - 112q^-49 - 61q^-48 + 35q^-47 + 61q^-46 + 106q^-45 + 60q^-44 - 34q^-43 - 58q^-42 - 102q^-41 - 56q^-40 + 34q^-39 + 52q^-38 + 95q^-37 + 57q^-36 - 30q^-35 - 49q^-34 - 90q^-33 - 52q^-32 + 27q^-31 + 38q^-30 + 82q^-29 + 55q^-28 - 19q^-27 - 36q^-26 - 75q^-25 - 51q^-24 + 14q^-23 + 20q^-22 + 67q^-21 + 55q^-20 - 4q^-19 - 17q^-18 - 59q^-17 - 48q^-16 - q^-15 + q^-14 + 46q^-13 + 51q^-12 + 12q^-11 + 2q^-10 - 38q^-9 - 41q^-8 - 11q^-7 - 18q^-6 + 22q^-5 + 40q^-4 + 19q^-3 + 19q^-2 - 15q^-1 - 25 - 14q - 29q² + 21q⁴ + 14q⁵ + 25q⁶ + 6q⁷ - 7q⁸ - 6q⁹ - 28q¹⁰ - 12q¹¹ + 3q¹² + q¹³ + 18q¹⁴ + 13q¹⁵ + 7q¹⁶ + 6q¹⁷ - 16q¹⁸ - 12q¹⁹ - 5q²⁰ - 9q²¹ + 5q²² + 7q²³ + 9q²⁴ + 12q²⁵ - 4q²⁶ - 5q²⁷ - 3q²⁸ - 8q²⁹ - 3q³⁰ - q³¹ + 3q³² + 9q³³ + q³⁴ + q³⁶ - 3q³⁷ - 2q³⁸ - 3q³⁹ - q⁴⁰ + 3q⁴¹ + q⁴² + q⁴⁴ - q⁴⁷ - q⁴⁸ + q⁴⁹

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

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

The Alternating Knot 62

The Alternating Knot 6₂