The Non Alternating Knot 10₁₂₆

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Acknowledgement

KnotPlot

PD Presentation: X₄₂₅₁ X₈₄₉₃ X_5,14,6,15 X_15,20,16,1 X_9,16,10,17 X_11,18,12,19 X_17,10,18,11 X_19,12,20,13 X_13,6,14,7 X₂₈₃₇

Gauss Code: {1, -10, 2, -1, -3, 9, 10, -2, -5, 7, -6, 8, -9, 3, -4, 5, -7, 6, -8, 4}

DT (Dowker-Thistlethwaite) Code: 4 8 -14 2 -16 -18 -6 -20 -10 -12

Minimum Braid Representative:

Length is 10, 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 2 3 3 / NotAvailable 1

Alexander Polynomial: t^-3 - 2t^-2 + 4t^-1 - 5 + 4t - 2t² + t³

Conway Polynomial: 1 + 5z² + 4z⁴ + z⁶

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

Determinant and Signature: {19, -2}

Jones Polynomial: - q^-8 + q^-7 - 2q^-6 + 3q^-5 - 3q^-4 + 4q^-3 - 2q^-2 + 2q^-1 - 1

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

A2 (sl(3)) Invariant: - q^-24 - q^-22 - 2q^-20 - q^-18 + q^-16 + q^-14 + 3q^-12 + 2q^-10 + 2q^-8 + q^-6 - q^-4 - 1

HOMFLY-PT Polynomial: - 2a² - 3a²z² - a²z⁴ + 7a⁴ + 12a⁴z² + 6a⁴z⁴ + a⁴z⁶ - 4a⁶ - 4a⁶z² - a⁶z⁴

Kauffman Polynomial: - 2az + az³ + 2a² - 4a²z² + 2a²z⁴ - 6a³z + 11a³z³ - 5a³z⁵ + a³z⁷ + 7a⁴ - 16a⁴z² + 16a⁴z⁴ - 6a⁴z⁶ + a⁴z⁸ - 8a⁵z + 16a⁵z³ - 9a⁵z⁵ + 2a⁵z⁷ + 4a⁶ - 11a⁶z² + 11a⁶z⁴ - 5a⁶z⁶ + a⁶z⁸ - a⁷z + 2a⁷z³ - 3a⁷z⁵ + a⁷z⁷ + a⁸z² - 3a⁸z⁴ + a⁸z⁶ + 3a⁹z - 4a⁹z³ + a⁹z⁵

V₂ and V₃, the type 2 and 3 Vassiliev invariants: {5, -9}

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

t^rq^j r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1

j = 1 1

j = -1 1

j = -3 2 2

j = -5 2

j = -7 1 2

j = -9 2 2

j = -11 1

j = -13 1 2

j = -15

j = -17 1

n Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2 q^-23 - q^-22 + 2q^-20 - 3q^-19 - q^-18 + 5q^-17 - 4q^-16 - 4q^-15 + 8q^-14 - 3q^-13 - 7q^-12 + 10q^-11 - q^-10 - 9q^-9 + 10q^-8 - 7q^-6 + 6q^-5 + q^-4 - 4q^-3 + 2q^-2 + q^-1 - 1

3 - q^-45 + q^-44 - q^-41 + 2q^-40 - q^-38 - 2q^-37 + 4q^-36 + 2q^-35 - 3q^-34 - 6q^-33 + 4q^-32 + 7q^-31 - q^-30 - 10q^-29 - 2q^-28 + 10q^-27 + 5q^-26 - 9q^-25 - 10q^-24 + 9q^-23 + 11q^-22 - 6q^-21 - 16q^-20 + 8q^-19 + 14q^-18 - 3q^-17 - 19q^-16 + 8q^-15 + 13q^-14 - q^-13 - 16q^-12 + 3q^-11 + 10q^-10 + 3q^-9 - 9q^-8 - 2q^-7 + 4q^-6 + 4q^-5 - q^-4 - 3q^-3 - q^-2 + 2q^-1 + 2 - q - q² - q³ + q⁴

4 q^-74 - q^-73 - q^-70 + 2q^-69 - 2q^-68 + q^-67 + q^-66 - 3q^-65 + 3q^-64 - 4q^-63 + q^-62 + 5q^-61 - 3q^-60 + 5q^-59 - 9q^-58 - 3q^-57 + 7q^-56 + 14q^-54 - 11q^-53 - 10q^-52 - 4q^-50 + 26q^-49 - 3q^-48 - 7q^-47 - 9q^-46 - 21q^-45 + 27q^-44 + 9q^-43 + 8q^-42 - 8q^-41 - 41q^-40 + 16q^-39 + 14q^-38 + 26q^-37 - 54q^-35 + 5q^-34 + 14q^-33 + 36q^-32 + 6q^-31 - 59q^-30 - q^-29 + 14q^-28 + 40q^-27 + 8q^-26 - 60q^-25 - 3q^-24 + 13q^-23 + 39q^-22 + 11q^-21 - 51q^-20 - 10q^-19 + 6q^-18 + 34q^-17 + 18q^-16 - 33q^-15 - 13q^-14 - 7q^-13 + 19q^-12 + 22q^-11 - 10q^-10 - 7q^-9 - 15q^-8 + q^-7 + 15q^-6 + 4q^-5 + 3q^-4 - 10q^-3 - 8q^-2 + 4q^-1 + 4 + 5q - q² - 5q³ - q⁴ + 2q⁶ + q⁷ - q⁸

5 - q^-110 + q^-109 + q^-106 - 2q^-104 + q^-103 - q^-101 + 2q^-100 + 2q^-99 - 3q^-98 - 3q^-95 + q^-94 + 4q^-93 + q^-91 - 6q^-89 - 4q^-88 + q^-87 + 3q^-86 + 7q^-85 + 8q^-84 - 4q^-83 - 9q^-82 - 10q^-81 - 7q^-80 + 6q^-79 + 20q^-78 + 12q^-77 + 2q^-76 - 13q^-75 - 24q^-74 - 17q^-73 + 8q^-72 + 23q^-71 + 30q^-70 + 13q^-69 - 17q^-68 - 42q^-67 - 32q^-66 + q^-65 + 43q^-64 + 53q^-63 + 22q^-62 - 37q^-61 - 71q^-60 - 45q^-59 + 25q^-58 + 79q^-57 + 71q^-56 - 10q^-55 - 88q^-54 - 85q^-53 - 5q^-52 + 86q^-51 + 104q^-50 + 14q^-49 - 92q^-48 - 103q^-47 - 25q^-46 + 86q^-45 + 118q^-44 + 24q^-43 - 94q^-42 - 108q^-41 - 30q^-40 + 84q^-39 + 123q^-38 + 28q^-37 - 97q^-36 - 107q^-35 - 34q^-34 + 79q^-33 + 123q^-32 + 37q^-31 - 84q^-30 - 107q^-29 - 45q^-28 + 60q^-27 + 111q^-26 + 53q^-25 - 50q^-24 - 89q^-23 - 59q^-22 + 20q^-21 + 77q^-20 + 60q^-19 - 2q^-18 - 47q^-17 - 56q^-16 - 20q^-15 + 26q^-14 + 44q^-13 + 27q^-12 - q^-11 - 27q^-10 - 29q^-9 - 12q^-8 + 11q^-7 + 22q^-6 + 16q^-5 + 3q^-4 - 11q^-3 - 17q^-2 - 7q^-1 + 4 + 8q + 8q² + 3q³ - 5q⁴ - 6q⁵ - q⁶ + 2q⁸ + 2q⁹ - q¹¹

6 q^-153 - q^-152 - q^-149 + 3q^-146 - 2q^-145 + q^-143 - 2q^-142 - q^-141 - q^-140 + 6q^-139 - 2q^-138 - q^-137 + 3q^-136 - 3q^-135 - 2q^-134 - 4q^-133 + 8q^-132 - q^-131 - q^-130 + 7q^-129 - 2q^-128 - 3q^-127 - 11q^-126 + 7q^-125 - 3q^-124 - 3q^-123 + 13q^-122 + 6q^-121 + 5q^-120 - 15q^-119 + 4q^-118 - 14q^-117 - 17q^-116 + 7q^-115 + 12q^-114 + 23q^-113 - q^-112 + 24q^-111 - 14q^-110 - 36q^-109 - 22q^-108 - 13q^-107 + 14q^-106 + 5q^-105 + 67q^-104 + 28q^-103 - 9q^-102 - 28q^-101 - 52q^-100 - 47q^-99 - 52q^-98 + 67q^-97 + 72q^-96 + 70q^-95 + 42q^-94 - 25q^-93 - 95q^-92 - 156q^-91 - 21q^-90 + 39q^-89 + 120q^-88 + 149q^-87 + 91q^-86 - 56q^-85 - 222q^-84 - 142q^-83 - 73q^-82 + 88q^-81 + 216q^-80 + 228q^-79 + 47q^-78 - 216q^-77 - 225q^-76 - 191q^-75 + 11q^-74 + 226q^-73 + 322q^-72 + 142q^-71 - 180q^-70 - 258q^-69 - 263q^-68 - 49q^-67 + 213q^-66 + 366q^-65 + 191q^-64 - 157q^-63 - 267q^-62 - 289q^-61 - 72q^-60 + 203q^-59 + 380q^-58 + 205q^-57 - 152q^-56 - 270q^-55 - 295q^-54 - 76q^-53 + 202q^-52 + 384q^-51 + 210q^-50 - 148q^-49 - 273q^-48 - 300q^-47 - 84q^-46 + 195q^-45 + 380q^-44 + 223q^-43 - 117q^-42 - 263q^-41 - 303q^-40 - 113q^-39 + 154q^-38 + 349q^-37 + 241q^-36 - 46q^-35 - 203q^-34 - 278q^-33 - 160q^-32 + 59q^-31 + 259q^-30 + 234q^-29 + 47q^-28 - 84q^-27 - 194q^-26 - 179q^-25 - 51q^-24 + 117q^-23 + 165q^-22 + 95q^-21 + 35q^-20 - 65q^-19 - 125q^-18 - 97q^-17 - 6q^-16 + 55q^-15 + 60q^-14 + 73q^-13 + 30q^-12 - 32q^-11 - 57q^-10 - 41q^-9 - 15q^-8 - 5q^-7 + 33q^-6 + 40q^-5 + 18q^-4 - 3q^-3 - 14q^-2 - 14q^-1 - 25 - 5q + 10q² + 12q³ + 9q⁴ + 4q⁵ + 4q⁶ - 10q⁷ - 6q⁸ - 2q⁹ + q¹² + 5q¹³ - q¹⁴ - q¹⁸ - q¹⁹ + q²⁰

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[10, 126]]

Out[2]=
PD[X[4, 2, 5, 1], X[8, 4, 9, 3], X[5, 14, 6, 15], X[15, 20, 16, 1], > X[9, 16, 10, 17], X[11, 18, 12, 19], X[17, 10, 18, 11], X[19, 12, 20, 13], > X[13, 6, 14, 7], X[2, 8, 3, 7]]

In[3]:=
GaussCode[Knot[10, 126]]

Out[3]=
GaussCode[1, -10, 2, -1, -3, 9, 10, -2, -5, 7, -6, 8, -9, 3, -4, 5, -7, 6, -8, > 4]

In[4]:=
DTCode[Knot[10, 126]]

Out[4]=
DTCode[4, 8, -14, 2, -16, -18, -6, -20, -10, -12]

In[5]:=
br = BR[Knot[10, 126]]

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

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

Out[6]=
{3, 10}

In[7]:=
BraidIndex[Knot[10, 126]]

Out[7]=
3

In[8]:=
Show[DrawMorseLink[Knot[10, 126]]]

Out[8]=
-Graphics-

In[9]:=
#[Knot[10, 126]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}

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

In[10]:=
alex = Alexander[Knot[10, 126]][t]

Out[10]=
-3 2 4 2 3 -5 + t - -- + - + 4 t - 2 t + t 2 t t

In[11]:=
Conway[Knot[10, 126]][z]

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

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

Out[12]=
{Knot[10, 126]}

In[13]:=
{KnotDet[Knot[10, 126]], KnotSignature[Knot[10, 126]]}

Out[13]=
{19, -2}

In[14]:=
Jones[Knot[10, 126]][q]

Out[14]=
-8 -7 2 3 3 4 2 2 -1 - q + q - -- + -- - -- + -- - -- + - 6 5 4 3 2 q q q q q q

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

Out[15]=
{Knot[10, 126]}

In[16]:=
A2Invariant[Knot[10, 126]][q]

Out[16]=
-24 -22 2 -18 -16 -14 3 2 2 -6 -4 -1 - q - q - --- - q + q + q + --- + --- + -- + q - q 20 12 10 8 q q q q

In[17]:=
HOMFLYPT[Knot[10, 126]][a, z]

Out[17]=
2 4 6 2 2 4 2 6 2 2 4 4 4 6 4 -2 a + 7 a - 4 a - 3 a z + 12 a z - 4 a z - a z + 6 a z - a z + 4 6 > a z

In[18]:=
Kauffman[Knot[10, 126]][a, z]

Out[18]=
2 4 6 3 5 7 9 2 2 2 a + 7 a + 4 a - 2 a z - 6 a z - 8 a z - a z + 3 a z - 4 a z - 4 2 6 2 8 2 3 3 3 5 3 7 3 > 16 a z - 11 a z + a z + a z + 11 a z + 16 a z + 2 a z - 9 3 2 4 4 4 6 4 8 4 3 5 5 5 > 4 a z + 2 a z + 16 a z + 11 a z - 3 a z - 5 a z - 9 a z - 7 5 9 5 4 6 6 6 8 6 3 7 5 7 7 7 > 3 a z + a z - 6 a z - 5 a z + a z + a z + 2 a z + a z + 4 8 6 8 > a z + a z

In[19]:=
{Vassiliev[2][Knot[10, 126]], Vassiliev[3][Knot[10, 126]]}

Out[19]=
{5, -9}

In[20]:=
Kh[Knot[10, 126]][q, t]

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

In[21]:=
ColouredJones[Knot[10, 126], 2][q]

Out[21]=
-23 -22 2 3 -18 5 4 4 8 3 7 10 -1 + q - q + --- - --- - q + --- - --- - --- + --- - --- - --- + --- - 20 19 17 16 15 14 13 12 11 q q q q q q q q q -10 9 10 7 6 -4 4 2 1 > q - -- + -- - -- + -- + q - -- + -- + - 9 8 6 5 3 2 q q q q q q q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 10₁₂₆

10₁₂₅
10₁₂₇

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q^-23 - q^-22 + 2q^-20 - 3q^-19 - q^-18 + 5q^-17 - 4q^-16 - 4q^-15 + 8q^-14 - 3q^-13 - 7q^-12 + 10q^-11 - q^-10 - 9q^-9 + 10q^-8 - 7q^-6 + 6q^-5 + q^-4 - 4q^-3 + 2q^-2 + q^-1 - 1
3	- q^-45 + q^-44 - q^-41 + 2q^-40 - q^-38 - 2q^-37 + 4q^-36 + 2q^-35 - 3q^-34 - 6q^-33 + 4q^-32 + 7q^-31 - q^-30 - 10q^-29 - 2q^-28 + 10q^-27 + 5q^-26 - 9q^-25 - 10q^-24 + 9q^-23 + 11q^-22 - 6q^-21 - 16q^-20 + 8q^-19 + 14q^-18 - 3q^-17 - 19q^-16 + 8q^-15 + 13q^-14 - q^-13 - 16q^-12 + 3q^-11 + 10q^-10 + 3q^-9 - 9q^-8 - 2q^-7 + 4q^-6 + 4q^-5 - q^-4 - 3q^-3 - q^-2 + 2q^-1 + 2 - q - q² - q³ + q⁴
4	q^-74 - q^-73 - q^-70 + 2q^-69 - 2q^-68 + q^-67 + q^-66 - 3q^-65 + 3q^-64 - 4q^-63 + q^-62 + 5q^-61 - 3q^-60 + 5q^-59 - 9q^-58 - 3q^-57 + 7q^-56 + 14q^-54 - 11q^-53 - 10q^-52 - 4q^-50 + 26q^-49 - 3q^-48 - 7q^-47 - 9q^-46 - 21q^-45 + 27q^-44 + 9q^-43 + 8q^-42 - 8q^-41 - 41q^-40 + 16q^-39 + 14q^-38 + 26q^-37 - 54q^-35 + 5q^-34 + 14q^-33 + 36q^-32 + 6q^-31 - 59q^-30 - q^-29 + 14q^-28 + 40q^-27 + 8q^-26 - 60q^-25 - 3q^-24 + 13q^-23 + 39q^-22 + 11q^-21 - 51q^-20 - 10q^-19 + 6q^-18 + 34q^-17 + 18q^-16 - 33q^-15 - 13q^-14 - 7q^-13 + 19q^-12 + 22q^-11 - 10q^-10 - 7q^-9 - 15q^-8 + q^-7 + 15q^-6 + 4q^-5 + 3q^-4 - 10q^-3 - 8q^-2 + 4q^-1 + 4 + 5q - q² - 5q³ - q⁴ + 2q⁶ + q⁷ - q⁸
5	- q^-110 + q^-109 + q^-106 - 2q^-104 + q^-103 - q^-101 + 2q^-100 + 2q^-99 - 3q^-98 - 3q^-95 + q^-94 + 4q^-93 + q^-91 - 6q^-89 - 4q^-88 + q^-87 + 3q^-86 + 7q^-85 + 8q^-84 - 4q^-83 - 9q^-82 - 10q^-81 - 7q^-80 + 6q^-79 + 20q^-78 + 12q^-77 + 2q^-76 - 13q^-75 - 24q^-74 - 17q^-73 + 8q^-72 + 23q^-71 + 30q^-70 + 13q^-69 - 17q^-68 - 42q^-67 - 32q^-66 + q^-65 + 43q^-64 + 53q^-63 + 22q^-62 - 37q^-61 - 71q^-60 - 45q^-59 + 25q^-58 + 79q^-57 + 71q^-56 - 10q^-55 - 88q^-54 - 85q^-53 - 5q^-52 + 86q^-51 + 104q^-50 + 14q^-49 - 92q^-48 - 103q^-47 - 25q^-46 + 86q^-45 + 118q^-44 + 24q^-43 - 94q^-42 - 108q^-41 - 30q^-40 + 84q^-39 + 123q^-38 + 28q^-37 - 97q^-36 - 107q^-35 - 34q^-34 + 79q^-33 + 123q^-32 + 37q^-31 - 84q^-30 - 107q^-29 - 45q^-28 + 60q^-27 + 111q^-26 + 53q^-25 - 50q^-24 - 89q^-23 - 59q^-22 + 20q^-21 + 77q^-20 + 60q^-19 - 2q^-18 - 47q^-17 - 56q^-16 - 20q^-15 + 26q^-14 + 44q^-13 + 27q^-12 - q^-11 - 27q^-10 - 29q^-9 - 12q^-8 + 11q^-7 + 22q^-6 + 16q^-5 + 3q^-4 - 11q^-3 - 17q^-2 - 7q^-1 + 4 + 8q + 8q² + 3q³ - 5q⁴ - 6q⁵ - q⁶ + 2q⁸ + 2q⁹ - q¹¹
6	q^-153 - q^-152 - q^-149 + 3q^-146 - 2q^-145 + q^-143 - 2q^-142 - q^-141 - q^-140 + 6q^-139 - 2q^-138 - q^-137 + 3q^-136 - 3q^-135 - 2q^-134 - 4q^-133 + 8q^-132 - q^-131 - q^-130 + 7q^-129 - 2q^-128 - 3q^-127 - 11q^-126 + 7q^-125 - 3q^-124 - 3q^-123 + 13q^-122 + 6q^-121 + 5q^-120 - 15q^-119 + 4q^-118 - 14q^-117 - 17q^-116 + 7q^-115 + 12q^-114 + 23q^-113 - q^-112 + 24q^-111 - 14q^-110 - 36q^-109 - 22q^-108 - 13q^-107 + 14q^-106 + 5q^-105 + 67q^-104 + 28q^-103 - 9q^-102 - 28q^-101 - 52q^-100 - 47q^-99 - 52q^-98 + 67q^-97 + 72q^-96 + 70q^-95 + 42q^-94 - 25q^-93 - 95q^-92 - 156q^-91 - 21q^-90 + 39q^-89 + 120q^-88 + 149q^-87 + 91q^-86 - 56q^-85 - 222q^-84 - 142q^-83 - 73q^-82 + 88q^-81 + 216q^-80 + 228q^-79 + 47q^-78 - 216q^-77 - 225q^-76 - 191q^-75 + 11q^-74 + 226q^-73 + 322q^-72 + 142q^-71 - 180q^-70 - 258q^-69 - 263q^-68 - 49q^-67 + 213q^-66 + 366q^-65 + 191q^-64 - 157q^-63 - 267q^-62 - 289q^-61 - 72q^-60 + 203q^-59 + 380q^-58 + 205q^-57 - 152q^-56 - 270q^-55 - 295q^-54 - 76q^-53 + 202q^-52 + 384q^-51 + 210q^-50 - 148q^-49 - 273q^-48 - 300q^-47 - 84q^-46 + 195q^-45 + 380q^-44 + 223q^-43 - 117q^-42 - 263q^-41 - 303q^-40 - 113q^-39 + 154q^-38 + 349q^-37 + 241q^-36 - 46q^-35 - 203q^-34 - 278q^-33 - 160q^-32 + 59q^-31 + 259q^-30 + 234q^-29 + 47q^-28 - 84q^-27 - 194q^-26 - 179q^-25 - 51q^-24 + 117q^-23 + 165q^-22 + 95q^-21 + 35q^-20 - 65q^-19 - 125q^-18 - 97q^-17 - 6q^-16 + 55q^-15 + 60q^-14 + 73q^-13 + 30q^-12 - 32q^-11 - 57q^-10 - 41q^-9 - 15q^-8 - 5q^-7 + 33q^-6 + 40q^-5 + 18q^-4 - 3q^-3 - 14q^-2 - 14q^-1 - 25 - 5q + 10q² + 12q³ + 9q⁴ + 4q⁵ + 4q⁶ - 10q⁷ - 6q⁸ - 2q⁹ + q¹² + 5q¹³ - q¹⁴ - q¹⁸ - q¹⁹ + q²⁰

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	PD[Knot[10, 126]]
Out[2]=	PD[X[4, 2, 5, 1], X[8, 4, 9, 3], X[5, 14, 6, 15], X[15, 20, 16, 1], > X[9, 16, 10, 17], X[11, 18, 12, 19], X[17, 10, 18, 11], X[19, 12, 20, 13], > X[13, 6, 14, 7], X[2, 8, 3, 7]]
In[3]:=	GaussCode[Knot[10, 126]]
Out[3]=	GaussCode[1, -10, 2, -1, -3, 9, 10, -2, -5, 7, -6, 8, -9, 3, -4, 5, -7, 6, -8, > 4]
In[4]:=	DTCode[Knot[10, 126]]
Out[4]=	DTCode[4, 8, -14, 2, -16, -18, -6, -20, -10, -12]
In[5]:=	br = BR[Knot[10, 126]]
Out[5]=	BR[3, {-1, -1, -1, -1, -1, -2, 1, 1, 1, -2}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{3, 10}
In[7]:=	BraidIndex[Knot[10, 126]]
Out[7]=	3
In[8]:=	Show[DrawMorseLink[Knot[10, 126]]]

Out[8]=	-Graphics-
In[9]:=	#[Knot[10, 126]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 2, 3, 3, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 126]][t]
Out[10]=	-3 2 4 2 3 -5 + t - -- + - + 4 t - 2 t + t 2 t t
In[11]:=	Conway[Knot[10, 126]][z]
Out[11]=	2 4 6 1 + 5 z + 4 z + z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 126]}
In[13]:=	{KnotDet[Knot[10, 126]], KnotSignature[Knot[10, 126]]}
Out[13]=	{19, -2}
In[14]:=	Jones[Knot[10, 126]][q]
Out[14]=	-8 -7 2 3 3 4 2 2 -1 - q + q - -- + -- - -- + -- - -- + - 6 5 4 3 2 q q q q q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] \|\| (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[10, 126]}
In[16]:=	A2Invariant[Knot[10, 126]][q]
Out[16]=	-24 -22 2 -18 -16 -14 3 2 2 -6 -4 -1 - q - q - --- - q + q + q + --- + --- + -- + q - q 20 12 10 8 q q q q
In[17]:=	HOMFLYPT[Knot[10, 126]][a, z]
Out[17]=	2 4 6 2 2 4 2 6 2 2 4 4 4 6 4 -2 a + 7 a - 4 a - 3 a z + 12 a z - 4 a z - a z + 6 a z - a z + 4 6 > a z
In[18]:=	Kauffman[Knot[10, 126]][a, z]
Out[18]=	2 4 6 3 5 7 9 2 2 2 a + 7 a + 4 a - 2 a z - 6 a z - 8 a z - a z + 3 a z - 4 a z - 4 2 6 2 8 2 3 3 3 5 3 7 3 > 16 a z - 11 a z + a z + a z + 11 a z + 16 a z + 2 a z - 9 3 2 4 4 4 6 4 8 4 3 5 5 5 > 4 a z + 2 a z + 16 a z + 11 a z - 3 a z - 5 a z - 9 a z - 7 5 9 5 4 6 6 6 8 6 3 7 5 7 7 7 > 3 a z + a z - 6 a z - 5 a z + a z + a z + 2 a z + a z + 4 8 6 8 > a z + a z
In[19]:=	{Vassiliev[2][Knot[10, 126]], Vassiliev[3][Knot[10, 126]]}
Out[19]=	{5, -9}
In[20]:=	Kh[Knot[10, 126]][q, t]
Out[20]=	2 1 1 1 2 1 2 2 1 2 -- + - + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + 3 q 17 7 13 6 13 5 11 4 9 4 9 3 7 3 7 2 q q t q t q t q t q t q t q t q t 2 2 > ----- + ---- + q t 5 2 3 q t q t
In[21]:=	ColouredJones[Knot[10, 126], 2][q]
Out[21]=	-23 -22 2 3 -18 5 4 4 8 3 7 10 -1 + q - q + --- - --- - q + --- - --- - --- + --- - --- - --- + --- - 20 19 17 16 15 14 13 12 11 q q q q q q q q q -10 9 10 7 6 -4 4 2 1 > q - -- + -- - -- + -- + q - -- + -- + - 9 8 6 5 3 2 q q q q q q q

The Non Alternating Knot 10126

The Non Alternating Knot 10₁₂₆