The Non Alternating Knot 10₁₄₁

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Acknowledgement

KnotPlot

PD Presentation: X₁₄₂₅ X_3,10,4,11 X_14,6,15,5 X_16,8,17,7 X_6,16,7,15 X_17,20,18,1 X_11,18,12,19 X_19,12,20,13 X_8,14,9,13 X_9,2,10,3

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

DT (Dowker-Thistlethwaite) Code: 4 10 -14 -16 2 18 -8 -6 20 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 1 3 3 / NotAvailable 1

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

Conway Polynomial: 1 - z² - 3z⁴ - z⁶

Other knots with the same Alexander/Conway Polynomial: {8₅, ...}

Determinant and Signature: {21, 0}

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

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

A2 (sl(3)) Invariant: q^-18 - q^-12 - q^-10 + q^-8 + q^-4 + q² + q⁶

HOMFLY-PT Polynomial: 2 + 3z² + z⁴ - 2a² - 7a²z² - 5a²z⁴ - a²z⁶ + a⁴ + 3a⁴z² + a⁴z⁴

Kauffman Polynomial: a^-2z² - a^-1z + 2a^-1z³ + 2 - 4z² + 3z⁴ - 3az + 5az³ - 3az⁵ + az⁷ + 2a² - 9a²z² + 8a²z⁴ - 4a²z⁶ + a²z⁸ - 4a³z + 13a³z³ - 12a³z⁵ + 3a³z⁷ + a⁴ - a⁴z² + a⁴z⁴ - 3a⁴z⁶ + a⁴z⁸ - 2a⁵z + 10a⁵z³ - 9a⁵z⁵ + 2a⁵z⁷ + 3a⁶z² - 4a⁶z⁴ + a⁶z⁶

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=0 is the signature of 10₁₄₁. 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 1

j = 1 2 1

j = -1 2 2

j = -3 2 1

j = -5 1 2

j = -7 1 2

j = -9 1 1

j = -11 1

j = -13 1

n Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2 q^-18 - 2q^-17 - q^-16 + 5q^-15 - 3q^-14 - 4q^-13 + 8q^-12 - q^-11 - 8q^-10 + 7q^-9 + q^-8 - 9q^-7 + 6q^-6 + 4q^-5 - 8q^-4 + 3q^-3 + 6q^-2 - 6q^-1 + 1 + 3q - 3q² + q³ - q⁵ + q⁶

3 q^-36 - 2q^-35 - q^-34 + 2q^-33 + 4q^-32 - 2q^-31 - 7q^-30 + q^-29 + 9q^-28 + 2q^-27 - 10q^-26 - 6q^-25 + 8q^-24 + 9q^-23 - 6q^-22 - 8q^-21 + 2q^-20 + 8q^-19 + q^-18 - 4q^-17 - 3q^-16 - q^-15 + 4q^-14 + 4q^-13 - 4q^-12 - 10q^-11 + 7q^-10 + 11q^-9 - 6q^-8 - 16q^-7 + 9q^-6 + 17q^-5 - 6q^-4 - 20q^-3 + 6q^-2 + 18q^-1 - 17q - 3q² + 12q³ + 6q⁴ - 7q⁵ - 6q⁶ + 2q⁷ + 6q⁸ - q⁹ - 2q¹⁰ - 2q¹¹ + 2q¹²

4 q^-60 - 2q^-59 - q^-58 + 2q^-57 + q^-56 + 5q^-55 - 6q^-54 - 5q^-53 + 17q^-50 - 4q^-49 - 9q^-48 - 7q^-47 - 10q^-46 + 24q^-45 + 5q^-44 - 6q^-42 - 25q^-41 + 16q^-40 + 3q^-39 + 8q^-38 + 8q^-37 - 21q^-36 + 9q^-35 - 14q^-34 - 3q^-33 + 18q^-32 - q^-31 + 23q^-30 - 24q^-29 - 30q^-28 + 9q^-27 + 15q^-26 + 51q^-25 - 19q^-24 - 57q^-23 - 12q^-22 + 20q^-21 + 75q^-20 - 4q^-19 - 75q^-18 - 30q^-17 + 21q^-16 + 89q^-15 + 8q^-14 - 87q^-13 - 42q^-12 + 25q^-11 + 100q^-10 + 16q^-9 - 95q^-8 - 54q^-7 + 26q^-6 + 104q^-5 + 28q^-4 - 87q^-3 - 63q^-2 + 7q^-1 + 91 + 43q - 57q² - 56q³ - 16q⁴ + 57q⁵ + 40q⁶ - 20q⁷ - 29q⁸ - 23q⁹ + 21q¹⁰ + 21q¹¹ - q¹² - 7q¹³ - 13q¹⁴ + 5q¹⁵ + 6q¹⁶ + q¹⁷ - q¹⁸ - 4q¹⁹ + q²⁰ + q²¹

5 q^-90 - 2q^-89 - q^-88 + 2q^-87 + q^-86 + 2q^-85 + q^-84 - 4q^-83 - 7q^-82 + 4q^-80 + 7q^-79 + 8q^-78 - 12q^-76 - 15q^-75 - 3q^-74 + 8q^-73 + 17q^-72 + 17q^-71 - q^-70 - 19q^-69 - 21q^-68 - 9q^-67 + 6q^-66 + 21q^-65 + 21q^-64 + 2q^-63 - 11q^-62 - 17q^-61 - 13q^-60 - 4q^-59 + 6q^-58 + 9q^-57 + 16q^-56 + 17q^-55 + 6q^-54 - 14q^-53 - 36q^-52 - 36q^-51 - 2q^-50 + 46q^-49 + 68q^-48 + 31q^-47 - 39q^-46 - 92q^-45 - 78q^-44 + 19q^-43 + 111q^-42 + 112q^-41 + 16q^-40 - 104q^-39 - 154q^-38 - 58q^-37 + 101q^-36 + 176q^-35 + 96q^-34 - 74q^-33 - 195q^-32 - 138q^-31 + 56q^-30 + 205q^-29 + 165q^-28 - 29q^-27 - 208q^-26 - 196q^-25 + 11q^-24 + 214q^-23 + 212q^-22 + 2q^-21 - 214q^-20 - 232q^-19 - 12q^-18 + 226q^-17 + 241q^-16 + 15q^-15 - 227q^-14 - 258q^-13 - 21q^-12 + 234q^-11 + 266q^-10 + 37q^-9 - 228q^-8 - 279q^-7 - 55q^-6 + 214q^-5 + 275q^-4 + 82q^-3 - 178q^-2 - 267q^-1 - 104 + 137q + 234q² + 115q³ - 86q⁴ - 186q⁵ - 113q⁶ + 42q⁷ + 131q⁸ + 97q⁹ - 14q¹⁰ - 81q¹¹ - 64q¹² - 6q¹³ + 40q¹⁴ + 43q¹⁵ + 7q¹⁶ - 22q¹⁷ - 17q¹⁸ - 3q¹⁹ + 5q²⁰ + 11q²¹ + q²² - 8q²³ - 2q²⁴ + 2q²⁵ + 3q²⁶ + 2q²⁷ - 4q²⁹ + q³²

6 q^-126 - 2q^-125 - q^-124 + 2q^-123 + q^-122 + 2q^-121 - 2q^-120 + 3q^-119 - 6q^-118 - 7q^-117 + 3q^-116 + 3q^-115 + 8q^-114 + q^-113 + 13q^-112 - 10q^-111 - 17q^-110 - 8q^-109 - 6q^-108 + 9q^-107 + 4q^-106 + 40q^-105 + 5q^-104 - 13q^-103 - 17q^-102 - 28q^-101 - 15q^-100 - 20q^-99 + 49q^-98 + 27q^-97 + 21q^-96 + 9q^-95 - 15q^-94 - 26q^-93 - 61q^-92 + 13q^-91 - 2q^-90 + 21q^-89 + 29q^-88 + 27q^-87 + 24q^-86 - 32q^-85 + 12q^-84 - 38q^-83 - 36q^-82 - 39q^-81 - 12q^-80 + 43q^-79 + 38q^-78 + 106q^-77 + 42q^-76 - 13q^-75 - 110q^-74 - 145q^-73 - 76q^-72 - 15q^-71 + 159q^-70 + 194q^-69 + 162q^-68 - 20q^-67 - 195q^-66 - 236q^-65 - 221q^-64 + 27q^-63 + 226q^-62 + 345q^-61 + 208q^-60 - 49q^-59 - 247q^-58 - 415q^-57 - 226q^-56 + 65q^-55 + 371q^-54 + 404q^-53 + 202q^-52 - 74q^-51 - 456q^-50 - 444q^-49 - 186q^-48 + 242q^-47 + 467q^-46 + 417q^-45 + 171q^-44 - 366q^-43 - 556q^-42 - 410q^-41 + 60q^-40 + 435q^-39 + 547q^-38 + 387q^-37 - 243q^-36 - 597q^-35 - 561q^-34 - 90q^-33 + 383q^-32 + 617q^-31 + 528q^-30 - 154q^-29 - 617q^-28 - 648q^-27 - 173q^-26 + 360q^-25 + 662q^-24 + 598q^-23 - 123q^-22 - 643q^-21 - 700q^-20 - 207q^-19 + 372q^-18 + 709q^-17 + 645q^-16 - 109q^-15 - 673q^-14 - 760q^-13 - 260q^-12 + 357q^-11 + 748q^-10 + 719q^-9 - 25q^-8 - 632q^-7 - 805q^-6 - 379q^-5 + 224q^-4 + 683q^-3 + 768q^-2 + 151q^-1 - 430 - 711q - 467q² - 13q³ + 435q⁴ + 646q⁵ + 281q⁶ - 127q⁷ - 426q⁸ - 377q⁹ - 177q¹⁰ + 120q¹¹ + 352q¹² + 228q¹³ + 67q¹⁴ - 122q¹⁵ - 159q¹⁶ - 151q¹⁷ - 45q¹⁸ + 92q¹⁹ + 77q²⁰ + 71q²¹ + 11q²² - 5q²³ - 49q²⁴ - 41q²⁵ - q²⁶ - 9q²⁷ + 17q²⁸ + 10q²⁹ + 23q³⁰ + q³¹ - 6q³² - 3q³³ - 14q³⁴ - 2q³⁵ - 3q³⁶ + 9q³⁷ + 4q³⁸ + 2q³⁹ + q⁴⁰ - 4q⁴¹ - q⁴² - 2q⁴³ + q⁴⁴ + q⁴⁵

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, 141]]

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

In[3]:=
GaussCode[Knot[10, 141]]

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

In[4]:=
DTCode[Knot[10, 141]]

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

In[5]:=
br = BR[Knot[10, 141]]

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

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

Out[6]=
{3, 10}

In[7]:=
BraidIndex[Knot[10, 141]]

Out[7]=
3

In[8]:=
Show[DrawMorseLink[Knot[10, 141]]]

Out[8]=
-Graphics-

In[9]:=
#[Knot[10, 141]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}

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

In[10]:=
alex = Alexander[Knot[10, 141]][t]

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

In[11]:=
Conway[Knot[10, 141]][z]

Out[11]=
2 4 6 1 - z - 3 z - z

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

Out[12]=
{Knot[8, 5], Knot[10, 141]}

In[13]:=
{KnotDet[Knot[10, 141]], KnotSignature[Knot[10, 141]]}

Out[13]=
{21, 0}

In[14]:=
Jones[Knot[10, 141]][q]

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

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

Out[15]=
{Knot[10, 141]}

In[16]:=
A2Invariant[Knot[10, 141]][q]

Out[16]=
-18 -12 -10 -8 -4 2 6 q - q - q + q + q + q + q

In[17]:=
HOMFLYPT[Knot[10, 141]][a, z]

Out[17]=
2 4 2 2 2 4 2 4 2 4 4 4 2 6 2 - 2 a + a + 3 z - 7 a z + 3 a z + z - 5 a z + a z - a z

In[18]:=
Kauffman[Knot[10, 141]][a, z]

Out[18]=
2 2 4 z 3 5 2 z 2 2 4 2 2 + 2 a + a - - - 3 a z - 4 a z - 2 a z - 4 z + -- - 9 a z - a z + a 2 a 3 6 2 2 z 3 3 3 5 3 4 2 4 4 4 > 3 a z + ---- + 5 a z + 13 a z + 10 a z + 3 z + 8 a z + a z - a 6 4 5 3 5 5 5 2 6 4 6 6 6 7 > 4 a z - 3 a z - 12 a z - 9 a z - 4 a z - 3 a z + a z + a z + 3 7 5 7 2 8 4 8 > 3 a z + 2 a z + a z + a z

In[19]:=
{Vassiliev[2][Knot[10, 141]], Vassiliev[3][Knot[10, 141]]}

Out[19]=
{-1, 1}

In[20]:=
Kh[Knot[10, 141]][q, t]

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

In[21]:=
ColouredJones[Knot[10, 141], 2][q]

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

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 10₁₄₁

10₁₄₀
10₁₄₂

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q^-18 - 2q^-17 - q^-16 + 5q^-15 - 3q^-14 - 4q^-13 + 8q^-12 - q^-11 - 8q^-10 + 7q^-9 + q^-8 - 9q^-7 + 6q^-6 + 4q^-5 - 8q^-4 + 3q^-3 + 6q^-2 - 6q^-1 + 1 + 3q - 3q² + q³ - q⁵ + q⁶
3	q^-36 - 2q^-35 - q^-34 + 2q^-33 + 4q^-32 - 2q^-31 - 7q^-30 + q^-29 + 9q^-28 + 2q^-27 - 10q^-26 - 6q^-25 + 8q^-24 + 9q^-23 - 6q^-22 - 8q^-21 + 2q^-20 + 8q^-19 + q^-18 - 4q^-17 - 3q^-16 - q^-15 + 4q^-14 + 4q^-13 - 4q^-12 - 10q^-11 + 7q^-10 + 11q^-9 - 6q^-8 - 16q^-7 + 9q^-6 + 17q^-5 - 6q^-4 - 20q^-3 + 6q^-2 + 18q^-1 - 17q - 3q² + 12q³ + 6q⁴ - 7q⁵ - 6q⁶ + 2q⁷ + 6q⁸ - q⁹ - 2q¹⁰ - 2q¹¹ + 2q¹²
4	q^-60 - 2q^-59 - q^-58 + 2q^-57 + q^-56 + 5q^-55 - 6q^-54 - 5q^-53 + 17q^-50 - 4q^-49 - 9q^-48 - 7q^-47 - 10q^-46 + 24q^-45 + 5q^-44 - 6q^-42 - 25q^-41 + 16q^-40 + 3q^-39 + 8q^-38 + 8q^-37 - 21q^-36 + 9q^-35 - 14q^-34 - 3q^-33 + 18q^-32 - q^-31 + 23q^-30 - 24q^-29 - 30q^-28 + 9q^-27 + 15q^-26 + 51q^-25 - 19q^-24 - 57q^-23 - 12q^-22 + 20q^-21 + 75q^-20 - 4q^-19 - 75q^-18 - 30q^-17 + 21q^-16 + 89q^-15 + 8q^-14 - 87q^-13 - 42q^-12 + 25q^-11 + 100q^-10 + 16q^-9 - 95q^-8 - 54q^-7 + 26q^-6 + 104q^-5 + 28q^-4 - 87q^-3 - 63q^-2 + 7q^-1 + 91 + 43q - 57q² - 56q³ - 16q⁴ + 57q⁵ + 40q⁶ - 20q⁷ - 29q⁸ - 23q⁹ + 21q¹⁰ + 21q¹¹ - q¹² - 7q¹³ - 13q¹⁴ + 5q¹⁵ + 6q¹⁶ + q¹⁷ - q¹⁸ - 4q¹⁹ + q²⁰ + q²¹
5	q^-90 - 2q^-89 - q^-88 + 2q^-87 + q^-86 + 2q^-85 + q^-84 - 4q^-83 - 7q^-82 + 4q^-80 + 7q^-79 + 8q^-78 - 12q^-76 - 15q^-75 - 3q^-74 + 8q^-73 + 17q^-72 + 17q^-71 - q^-70 - 19q^-69 - 21q^-68 - 9q^-67 + 6q^-66 + 21q^-65 + 21q^-64 + 2q^-63 - 11q^-62 - 17q^-61 - 13q^-60 - 4q^-59 + 6q^-58 + 9q^-57 + 16q^-56 + 17q^-55 + 6q^-54 - 14q^-53 - 36q^-52 - 36q^-51 - 2q^-50 + 46q^-49 + 68q^-48 + 31q^-47 - 39q^-46 - 92q^-45 - 78q^-44 + 19q^-43 + 111q^-42 + 112q^-41 + 16q^-40 - 104q^-39 - 154q^-38 - 58q^-37 + 101q^-36 + 176q^-35 + 96q^-34 - 74q^-33 - 195q^-32 - 138q^-31 + 56q^-30 + 205q^-29 + 165q^-28 - 29q^-27 - 208q^-26 - 196q^-25 + 11q^-24 + 214q^-23 + 212q^-22 + 2q^-21 - 214q^-20 - 232q^-19 - 12q^-18 + 226q^-17 + 241q^-16 + 15q^-15 - 227q^-14 - 258q^-13 - 21q^-12 + 234q^-11 + 266q^-10 + 37q^-9 - 228q^-8 - 279q^-7 - 55q^-6 + 214q^-5 + 275q^-4 + 82q^-3 - 178q^-2 - 267q^-1 - 104 + 137q + 234q² + 115q³ - 86q⁴ - 186q⁵ - 113q⁶ + 42q⁷ + 131q⁸ + 97q⁹ - 14q¹⁰ - 81q¹¹ - 64q¹² - 6q¹³ + 40q¹⁴ + 43q¹⁵ + 7q¹⁶ - 22q¹⁷ - 17q¹⁸ - 3q¹⁹ + 5q²⁰ + 11q²¹ + q²² - 8q²³ - 2q²⁴ + 2q²⁵ + 3q²⁶ + 2q²⁷ - 4q²⁹ + q³²
6	q^-126 - 2q^-125 - q^-124 + 2q^-123 + q^-122 + 2q^-121 - 2q^-120 + 3q^-119 - 6q^-118 - 7q^-117 + 3q^-116 + 3q^-115 + 8q^-114 + q^-113 + 13q^-112 - 10q^-111 - 17q^-110 - 8q^-109 - 6q^-108 + 9q^-107 + 4q^-106 + 40q^-105 + 5q^-104 - 13q^-103 - 17q^-102 - 28q^-101 - 15q^-100 - 20q^-99 + 49q^-98 + 27q^-97 + 21q^-96 + 9q^-95 - 15q^-94 - 26q^-93 - 61q^-92 + 13q^-91 - 2q^-90 + 21q^-89 + 29q^-88 + 27q^-87 + 24q^-86 - 32q^-85 + 12q^-84 - 38q^-83 - 36q^-82 - 39q^-81 - 12q^-80 + 43q^-79 + 38q^-78 + 106q^-77 + 42q^-76 - 13q^-75 - 110q^-74 - 145q^-73 - 76q^-72 - 15q^-71 + 159q^-70 + 194q^-69 + 162q^-68 - 20q^-67 - 195q^-66 - 236q^-65 - 221q^-64 + 27q^-63 + 226q^-62 + 345q^-61 + 208q^-60 - 49q^-59 - 247q^-58 - 415q^-57 - 226q^-56 + 65q^-55 + 371q^-54 + 404q^-53 + 202q^-52 - 74q^-51 - 456q^-50 - 444q^-49 - 186q^-48 + 242q^-47 + 467q^-46 + 417q^-45 + 171q^-44 - 366q^-43 - 556q^-42 - 410q^-41 + 60q^-40 + 435q^-39 + 547q^-38 + 387q^-37 - 243q^-36 - 597q^-35 - 561q^-34 - 90q^-33 + 383q^-32 + 617q^-31 + 528q^-30 - 154q^-29 - 617q^-28 - 648q^-27 - 173q^-26 + 360q^-25 + 662q^-24 + 598q^-23 - 123q^-22 - 643q^-21 - 700q^-20 - 207q^-19 + 372q^-18 + 709q^-17 + 645q^-16 - 109q^-15 - 673q^-14 - 760q^-13 - 260q^-12 + 357q^-11 + 748q^-10 + 719q^-9 - 25q^-8 - 632q^-7 - 805q^-6 - 379q^-5 + 224q^-4 + 683q^-3 + 768q^-2 + 151q^-1 - 430 - 711q - 467q² - 13q³ + 435q⁴ + 646q⁵ + 281q⁶ - 127q⁷ - 426q⁸ - 377q⁹ - 177q¹⁰ + 120q¹¹ + 352q¹² + 228q¹³ + 67q¹⁴ - 122q¹⁵ - 159q¹⁶ - 151q¹⁷ - 45q¹⁸ + 92q¹⁹ + 77q²⁰ + 71q²¹ + 11q²² - 5q²³ - 49q²⁴ - 41q²⁵ - q²⁶ - 9q²⁷ + 17q²⁸ + 10q²⁹ + 23q³⁰ + q³¹ - 6q³² - 3q³³ - 14q³⁴ - 2q³⁵ - 3q³⁶ + 9q³⁷ + 4q³⁸ + 2q³⁹ + q⁴⁰ - 4q⁴¹ - q⁴² - 2q⁴³ + q⁴⁴ + q⁴⁵

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	PD[Knot[10, 141]]
Out[2]=	PD[X[1, 4, 2, 5], X[3, 10, 4, 11], X[14, 6, 15, 5], X[16, 8, 17, 7], > X[6, 16, 7, 15], X[17, 20, 18, 1], X[11, 18, 12, 19], X[19, 12, 20, 13], > X[8, 14, 9, 13], X[9, 2, 10, 3]]
In[3]:=	GaussCode[Knot[10, 141]]
Out[3]=	GaussCode[-1, 10, -2, 1, 3, -5, 4, -9, -10, 2, -7, 8, 9, -3, 5, -4, -6, 7, -8, > 6]
In[4]:=	DTCode[Knot[10, 141]]
Out[4]=	DTCode[4, 10, -14, -16, 2, 18, -8, -6, 20, 12]
In[5]:=	br = BR[Knot[10, 141]]
Out[5]=	BR[3, {1, 1, 1, 1, -2, -1, -1, -1, -2, -2}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{3, 10}
In[7]:=	BraidIndex[Knot[10, 141]]
Out[7]=	3
In[8]:=	Show[DrawMorseLink[Knot[10, 141]]]

Out[8]=	-Graphics-
In[9]:=	#[Knot[10, 141]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 1, 3, 3, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 141]][t]
Out[10]=	-3 3 4 2 3 5 - t + -- - - - 4 t + 3 t - t 2 t t
In[11]:=	Conway[Knot[10, 141]][z]
Out[11]=	2 4 6 1 - z - 3 z - z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[8, 5], Knot[10, 141]}
In[13]:=	{KnotDet[Knot[10, 141]], KnotSignature[Knot[10, 141]]}
Out[13]=	{21, 0}
In[14]:=	Jones[Knot[10, 141]][q]
Out[14]=	-6 2 2 3 4 3 2 3 + q - -- + -- - -- + -- - - - 2 q + q 5 4 3 2 q q q q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] \|\| (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[10, 141]}
In[16]:=	A2Invariant[Knot[10, 141]][q]
Out[16]=	-18 -12 -10 -8 -4 2 6 q - q - q + q + q + q + q
In[17]:=	HOMFLYPT[Knot[10, 141]][a, z]
Out[17]=	2 4 2 2 2 4 2 4 2 4 4 4 2 6 2 - 2 a + a + 3 z - 7 a z + 3 a z + z - 5 a z + a z - a z
In[18]:=	Kauffman[Knot[10, 141]][a, z]
Out[18]=	2 2 4 z 3 5 2 z 2 2 4 2 2 + 2 a + a - - - 3 a z - 4 a z - 2 a z - 4 z + -- - 9 a z - a z + a 2 a 3 6 2 2 z 3 3 3 5 3 4 2 4 4 4 > 3 a z + ---- + 5 a z + 13 a z + 10 a z + 3 z + 8 a z + a z - a 6 4 5 3 5 5 5 2 6 4 6 6 6 7 > 4 a z - 3 a z - 12 a z - 9 a z - 4 a z - 3 a z + a z + a z + 3 7 5 7 2 8 4 8 > 3 a z + 2 a z + a z + a z
In[19]:=	{Vassiliev[2][Knot[10, 141]], Vassiliev[3][Knot[10, 141]]}
Out[19]=	{-1, 1}
In[20]:=	Kh[Knot[10, 141]][q, t]
Out[20]=	2 1 1 1 1 1 2 1 2 - + 2 q + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ----- + q 13 6 11 5 9 5 9 4 7 4 7 3 5 3 5 2 q t q t q t q t q t q t q t q t 2 1 2 3 5 2 > ----- + ---- + --- + q t + q t + q t 3 2 3 q t q t q t
In[21]:=	ColouredJones[Knot[10, 141], 2][q]
Out[21]=	-18 2 -16 5 3 4 8 -11 8 7 -8 9 1 + q - --- - q + --- - --- - --- + --- - q - --- + -- + q - -- + 17 15 14 13 12 10 9 7 q q q q q q q q 6 4 8 3 6 6 2 3 5 6 > -- + -- - -- + -- + -- - - + 3 q - 3 q + q - q + q 6 5 4 3 2 q q q q q q

The Non Alternating Knot 10141

The Non Alternating Knot 10₁₄₁