The Alternating Knot 10₁₃

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Acknowledgement

KnotPlot

PD Presentation: X₁₄₂₅ X_9,12,10,13 X_3,11,4,10 X_11,3,12,2 X_5,18,6,19 X_13,1,14,20 X_19,15,20,14 X_7,16,8,17 X_15,8,16,9 X_17,6,18,7

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

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

Minimum Braid Representative:

Length is 11, width is 6
Braid index is 6

A Morse Link Presentation:

3D Invariants:

Symmetry Type Unknotting Number 3-Genus Bridge/Super Bridge Index Nakanishi Index

Reversible 2 2 2 / NotAvailable 1

Alexander Polynomial: 2t^-2 - 13t^-1 + 23 - 13t + 2t²

Conway Polynomial: 1 - 5z² + 2z⁴

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

Determinant and Signature: {53, 0}

Jones Polynomial: q^-6 - 2q^-5 + 4q^-4 - 6q^-3 + 8q^-2 - 9q^-1 + 8 - 7q + 5q² - 2q³ + q⁴

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

A2 (sl(3)) Invariant: q^-20 + q^-18 - q^-16 + q^-14 - 2q^-10 + 2q^-8 - 2 + q² - 2q⁴ + q⁶ + 2q⁸ - q¹⁰ + q¹² + q¹⁴

HOMFLY-PT Polynomial: a^-4 - 2a^-2z² - 1 - z² + z⁴ + a² + a²z⁴ - a⁴ - 2a⁴z² + a⁶

Kauffman Polynomial: a^-4 - 2a^-4z² + a^-4z⁴ - 2a^-3z³ + 2a^-3z⁵ + a^-2z² - 3a^-2z⁴ + 3a^-2z⁶ - 2a^-1z + 3a^-1z³ - 3a^-1z⁵ + 3a^-1z⁷ - 1 + 4z² - 3z⁴ + 2z⁸ - 2az⁵ + az⁷ + az⁹ - a² - a²z² + 6a²z⁴ - 9a²z⁶ + 4a²z⁸ + a³z + a³z³ - 4a³z⁵ + a³z⁹ - a⁴ + 2a⁴z² + a⁴z⁴ - 5a⁴z⁶ + 2a⁴z⁸ - a⁵z + 6a⁵z³ - 7a⁵z⁵ + 2a⁵z⁷ - a⁶ + 4a⁶z² - 4a⁶z⁴ + a⁶z⁶

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

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 10₁₃. 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 r = 3 r = 4

j = 9 1

j = 7 1

j = 5 4 1

j = 3 3 1

j = 1 5 4

j = -1 5 4

j = -3 3 4

j = -5 3 5

j = -7 1 3

j = -9 1 3

j = -11 1

j = -13 1

n Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2 q^-18 - 2q^-17 + 6q^-15 - 8q^-14 - 4q^-13 + 20q^-12 - 14q^-11 - 17q^-10 + 38q^-9 - 15q^-8 - 36q^-7 + 54q^-6 - 9q^-5 - 53q^-4 + 60q^-3 - 58q^-1 + 52 + 6q - 47q² + 34q³ + 6q⁴ - 27q⁵ + 16q⁶ + 3q⁷ - 10q⁸ + 5q⁹ + q¹⁰ - 2q¹¹ + q¹²

3 q^-36 - 2q^-35 + 2q^-33 + 3q^-32 - 7q^-31 - 5q^-30 + 10q^-29 + 15q^-28 - 17q^-27 - 25q^-26 + 15q^-25 + 47q^-24 - 14q^-23 - 65q^-22 - 2q^-21 + 87q^-20 + 21q^-19 - 100q^-18 - 50q^-17 + 110q^-16 + 79q^-15 - 108q^-14 - 112q^-13 + 104q^-12 + 139q^-11 - 91q^-10 - 165q^-9 + 75q^-8 + 188q^-7 - 61q^-6 - 197q^-5 + 38q^-4 + 204q^-3 - 23q^-2 - 193q^-1 + 2 + 178q + 8q² - 148q³ - 16q⁴ + 116q⁵ + 19q⁶ - 87q⁷ - 12q⁸ + 56q⁹ + 13q¹⁰ - 43q¹¹ + 21q¹³ + 4q¹⁴ - 18q¹⁵ + 2q¹⁶ + 8q¹⁷ + q¹⁸ - 7q¹⁹ + 2q²⁰ + q²¹ + q²² - 2q²³ + q²⁴

4 q^-60 - 2q^-59 + 2q^-57 - q^-56 + 4q^-55 - 9q^-54 - q^-53 + 10q^-52 + 15q^-50 - 31q^-49 - 17q^-48 + 24q^-47 + 15q^-46 + 57q^-45 - 64q^-44 - 68q^-43 + 9q^-42 + 30q^-41 + 165q^-40 - 59q^-39 - 134q^-38 - 74q^-37 - 22q^-36 + 306q^-35 + 30q^-34 - 125q^-33 - 187q^-32 - 196q^-31 + 377q^-30 + 160q^-29 + 18q^-28 - 222q^-27 - 440q^-26 + 310q^-25 + 228q^-24 + 253q^-23 - 127q^-22 - 653q^-21 + 145q^-20 + 189q^-19 + 484q^-18 + 53q^-17 - 781q^-16 - 42q^-15 + 87q^-14 + 664q^-13 + 240q^-12 - 834q^-11 - 206q^-10 - 29q^-9 + 773q^-8 + 401q^-7 - 803q^-6 - 334q^-5 - 156q^-4 + 782q^-3 + 517q^-2 - 665q^-1 - 374 - 283q + 644q² + 541q³ - 434q⁴ - 286q⁵ - 351q⁶ + 399q⁷ + 435q⁸ - 214q⁹ - 114q¹⁰ - 303q¹¹ + 170q¹² + 251q¹³ - 92q¹⁴ + 21q¹⁵ - 179q¹⁶ + 50q¹⁷ + 96q¹⁸ - 56q¹⁹ + 61q²⁰ - 73q²¹ + 17q²² + 25q²³ - 42q²⁴ + 40q²⁵ - 23q²⁶ + 13q²⁷ + 7q²⁸ - 25q²⁹ + 16q³⁰ - 7q³¹ + 6q³² + 3q³³ - 9q³⁴ + 5q³⁵ - 2q³⁶ + q³⁷ + q³⁸ - 2q³⁹ + q⁴⁰

5 q^-90 - 2q^-89 + 2q^-87 - q^-86 + 2q^-84 - 5q^-83 - 2q^-82 + 9q^-81 + 3q^-80 - 3q^-79 - q^-78 - 17q^-77 - 11q^-76 + 21q^-75 + 32q^-74 + 14q^-73 - 12q^-72 - 57q^-71 - 63q^-70 + 15q^-69 + 90q^-68 + 106q^-67 + 37q^-66 - 110q^-65 - 195q^-64 - 104q^-63 + 93q^-62 + 258q^-61 + 246q^-60 - 23q^-59 - 320q^-58 - 366q^-57 - 128q^-56 + 270q^-55 + 518q^-54 + 328q^-53 - 165q^-52 - 550q^-51 - 542q^-50 - 91q^-49 + 505q^-48 + 722q^-47 + 374q^-46 - 285q^-45 - 788q^-44 - 708q^-43 - 46q^-42 + 710q^-41 + 975q^-40 + 491q^-39 - 475q^-38 - 1161q^-37 - 948q^-36 + 89q^-35 + 1197q^-34 + 1420q^-33 + 382q^-32 - 1127q^-31 - 1790q^-30 - 909q^-29 + 921q^-28 + 2109q^-27 + 1431q^-26 - 672q^-25 - 2326q^-24 - 1902q^-23 + 374q^-22 + 2471q^-21 + 2338q^-20 - 85q^-19 - 2585q^-18 - 2696q^-17 - 173q^-16 + 2624q^-15 + 3017q^-14 + 453q^-13 - 2679q^-12 - 3283q^-11 - 682q^-10 + 2629q^-9 + 3493q^-8 + 980q^-7 - 2550q^-6 - 3642q^-5 - 1236q^-4 + 2315q^-3 + 3683q^-2 + 1546q^-1 - 2009 - 3585q - 1786q² + 1559q³ + 3332q⁴ + 1978q⁵ - 1073q⁶ - 2925q⁷ - 2021q⁸ + 568q⁹ + 2388q¹⁰ + 1936q¹¹ - 121q¹² - 1811q¹³ - 1719q¹⁴ - 177q¹⁵ + 1226q¹⁶ + 1378q¹⁷ + 402q¹⁸ - 748q¹⁹ - 1059q²⁰ - 409q²¹ + 369q²² + 676q²³ + 426q²⁴ - 128q²⁵ - 450q²⁶ - 291q²⁷ - q²⁸ + 201q²⁹ + 236q³⁰ + 55q³¹ - 112q³² - 115q³³ - 61q³⁴ + 13q³⁵ + 80q³⁶ + 45q³⁷ - 8q³⁸ - 12q³⁹ - 31q⁴⁰ - 24q⁴¹ + 20q⁴² + 11q⁴³ + 2q⁴⁴ + 11q⁴⁵ - 4q⁴⁶ - 18q⁴⁷ + 7q⁴⁸ + 2q⁴⁹ - 4q⁵⁰ + 7q⁵¹ + q⁵² - 7q⁵³ + 3q⁵⁴ + q⁵⁵ - 2q⁵⁶ + q⁵⁷ + q⁵⁸ - 2q⁵⁹ + q⁶⁰

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, 13]]

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

In[3]:=
GaussCode[Knot[10, 13]]

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

In[4]:=
DTCode[Knot[10, 13]]

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

In[5]:=
br = BR[Knot[10, 13]]

Out[5]=
BR[6, {-1, -1, -2, 1, 3, -2, -4, 3, 5, -4, 5}]

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

Out[6]=
{6, 11}

In[7]:=
BraidIndex[Knot[10, 13]]

Out[7]=
6

In[8]:=
Show[DrawMorseLink[Knot[10, 13]]]

Out[8]=
-Graphics-

In[9]:=
#[Knot[10, 13]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}

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

In[10]:=
alex = Alexander[Knot[10, 13]][t]

Out[10]=
2 13 2 23 + -- - -- - 13 t + 2 t 2 t t

In[11]:=
Conway[Knot[10, 13]][z]

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

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

Out[12]=
{Knot[10, 13]}

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

Out[13]=
{53, 0}

In[14]:=
Jones[Knot[10, 13]][q]

Out[14]=
-6 2 4 6 8 9 2 3 4 8 + q - -- + -- - -- + -- - - - 7 q + 5 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, 13]}

In[16]:=
A2Invariant[Knot[10, 13]][q]

Out[16]=
-20 -18 -16 -14 2 2 2 4 6 8 10 12 -2 + q + q - q + q - --- + -- + q - 2 q + q + 2 q - q + q + 10 8 q q 14 > q

In[17]:=
HOMFLYPT[Knot[10, 13]][a, z]

Out[17]=
2 -4 2 4 6 2 2 z 4 2 4 2 4 -1 + a + a - a + a - z - ---- - 2 a z + z + a z 2 a

In[18]:=
Kauffman[Knot[10, 13]][a, z]

Out[18]=
2 2 -4 2 4 6 2 z 3 5 2 2 z z 2 2 -1 + a - a - a - a - --- + a z - a z + 4 z - ---- + -- - a z + a 4 2 a a 3 3 4 4 4 2 6 2 2 z 3 z 3 3 5 3 4 z 3 z > 2 a z + 4 a z - ---- + ---- + a z + 6 a z - 3 z + -- - ---- + 3 a 4 2 a a a 5 5 2 4 4 4 6 4 2 z 3 z 5 3 5 5 5 > 6 a z + a z - 4 a z + ---- - ---- - 2 a z - 4 a z - 7 a z + 3 a a 6 7 3 z 2 6 4 6 6 6 3 z 7 5 7 8 2 8 > ---- - 9 a z - 5 a z + a z + ---- + a z + 2 a z + 2 z + 4 a z + 2 a a 4 8 9 3 9 > 2 a z + a z + a z

In[19]:=
{Vassiliev[2][Knot[10, 13]], Vassiliev[3][Knot[10, 13]]}

Out[19]=
{-5, 2}

In[20]:=
Kh[Knot[10, 13]][q, t]

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

In[21]:=
ColouredJones[Knot[10, 13], 2][q]

Out[21]=
-18 2 6 8 4 20 14 17 38 15 36 54 9 52 + q - --- + --- - --- - --- + --- - --- - --- + -- - -- - -- + -- - -- - 17 15 14 13 12 11 10 9 8 7 6 5 q q q q q q q q q q q q 53 60 58 2 3 4 5 6 7 8 > -- + -- - -- + 6 q - 47 q + 34 q + 6 q - 27 q + 16 q + 3 q - 10 q + 4 3 q q q 9 10 11 12 > 5 q + q - 2 q + q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 10₁₃

10₁₂
10₁₄

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q^-18 - 2q^-17 + 6q^-15 - 8q^-14 - 4q^-13 + 20q^-12 - 14q^-11 - 17q^-10 + 38q^-9 - 15q^-8 - 36q^-7 + 54q^-6 - 9q^-5 - 53q^-4 + 60q^-3 - 58q^-1 + 52 + 6q - 47q² + 34q³ + 6q⁴ - 27q⁵ + 16q⁶ + 3q⁷ - 10q⁸ + 5q⁹ + q¹⁰ - 2q¹¹ + q¹²
3	q^-36 - 2q^-35 + 2q^-33 + 3q^-32 - 7q^-31 - 5q^-30 + 10q^-29 + 15q^-28 - 17q^-27 - 25q^-26 + 15q^-25 + 47q^-24 - 14q^-23 - 65q^-22 - 2q^-21 + 87q^-20 + 21q^-19 - 100q^-18 - 50q^-17 + 110q^-16 + 79q^-15 - 108q^-14 - 112q^-13 + 104q^-12 + 139q^-11 - 91q^-10 - 165q^-9 + 75q^-8 + 188q^-7 - 61q^-6 - 197q^-5 + 38q^-4 + 204q^-3 - 23q^-2 - 193q^-1 + 2 + 178q + 8q² - 148q³ - 16q⁴ + 116q⁵ + 19q⁶ - 87q⁷ - 12q⁸ + 56q⁹ + 13q¹⁰ - 43q¹¹ + 21q¹³ + 4q¹⁴ - 18q¹⁵ + 2q¹⁶ + 8q¹⁷ + q¹⁸ - 7q¹⁹ + 2q²⁰ + q²¹ + q²² - 2q²³ + q²⁴
4	q^-60 - 2q^-59 + 2q^-57 - q^-56 + 4q^-55 - 9q^-54 - q^-53 + 10q^-52 + 15q^-50 - 31q^-49 - 17q^-48 + 24q^-47 + 15q^-46 + 57q^-45 - 64q^-44 - 68q^-43 + 9q^-42 + 30q^-41 + 165q^-40 - 59q^-39 - 134q^-38 - 74q^-37 - 22q^-36 + 306q^-35 + 30q^-34 - 125q^-33 - 187q^-32 - 196q^-31 + 377q^-30 + 160q^-29 + 18q^-28 - 222q^-27 - 440q^-26 + 310q^-25 + 228q^-24 + 253q^-23 - 127q^-22 - 653q^-21 + 145q^-20 + 189q^-19 + 484q^-18 + 53q^-17 - 781q^-16 - 42q^-15 + 87q^-14 + 664q^-13 + 240q^-12 - 834q^-11 - 206q^-10 - 29q^-9 + 773q^-8 + 401q^-7 - 803q^-6 - 334q^-5 - 156q^-4 + 782q^-3 + 517q^-2 - 665q^-1 - 374 - 283q + 644q² + 541q³ - 434q⁴ - 286q⁵ - 351q⁶ + 399q⁷ + 435q⁸ - 214q⁹ - 114q¹⁰ - 303q¹¹ + 170q¹² + 251q¹³ - 92q¹⁴ + 21q¹⁵ - 179q¹⁶ + 50q¹⁷ + 96q¹⁸ - 56q¹⁹ + 61q²⁰ - 73q²¹ + 17q²² + 25q²³ - 42q²⁴ + 40q²⁵ - 23q²⁶ + 13q²⁷ + 7q²⁸ - 25q²⁹ + 16q³⁰ - 7q³¹ + 6q³² + 3q³³ - 9q³⁴ + 5q³⁵ - 2q³⁶ + q³⁷ + q³⁸ - 2q³⁹ + q⁴⁰
5	q^-90 - 2q^-89 + 2q^-87 - q^-86 + 2q^-84 - 5q^-83 - 2q^-82 + 9q^-81 + 3q^-80 - 3q^-79 - q^-78 - 17q^-77 - 11q^-76 + 21q^-75 + 32q^-74 + 14q^-73 - 12q^-72 - 57q^-71 - 63q^-70 + 15q^-69 + 90q^-68 + 106q^-67 + 37q^-66 - 110q^-65 - 195q^-64 - 104q^-63 + 93q^-62 + 258q^-61 + 246q^-60 - 23q^-59 - 320q^-58 - 366q^-57 - 128q^-56 + 270q^-55 + 518q^-54 + 328q^-53 - 165q^-52 - 550q^-51 - 542q^-50 - 91q^-49 + 505q^-48 + 722q^-47 + 374q^-46 - 285q^-45 - 788q^-44 - 708q^-43 - 46q^-42 + 710q^-41 + 975q^-40 + 491q^-39 - 475q^-38 - 1161q^-37 - 948q^-36 + 89q^-35 + 1197q^-34 + 1420q^-33 + 382q^-32 - 1127q^-31 - 1790q^-30 - 909q^-29 + 921q^-28 + 2109q^-27 + 1431q^-26 - 672q^-25 - 2326q^-24 - 1902q^-23 + 374q^-22 + 2471q^-21 + 2338q^-20 - 85q^-19 - 2585q^-18 - 2696q^-17 - 173q^-16 + 2624q^-15 + 3017q^-14 + 453q^-13 - 2679q^-12 - 3283q^-11 - 682q^-10 + 2629q^-9 + 3493q^-8 + 980q^-7 - 2550q^-6 - 3642q^-5 - 1236q^-4 + 2315q^-3 + 3683q^-2 + 1546q^-1 - 2009 - 3585q - 1786q² + 1559q³ + 3332q⁴ + 1978q⁵ - 1073q⁶ - 2925q⁷ - 2021q⁸ + 568q⁹ + 2388q¹⁰ + 1936q¹¹ - 121q¹² - 1811q¹³ - 1719q¹⁴ - 177q¹⁵ + 1226q¹⁶ + 1378q¹⁷ + 402q¹⁸ - 748q¹⁹ - 1059q²⁰ - 409q²¹ + 369q²² + 676q²³ + 426q²⁴ - 128q²⁵ - 450q²⁶ - 291q²⁷ - q²⁸ + 201q²⁹ + 236q³⁰ + 55q³¹ - 112q³² - 115q³³ - 61q³⁴ + 13q³⁵ + 80q³⁶ + 45q³⁷ - 8q³⁸ - 12q³⁹ - 31q⁴⁰ - 24q⁴¹ + 20q⁴² + 11q⁴³ + 2q⁴⁴ + 11q⁴⁵ - 4q⁴⁶ - 18q⁴⁷ + 7q⁴⁸ + 2q⁴⁹ - 4q⁵⁰ + 7q⁵¹ + q⁵² - 7q⁵³ + 3q⁵⁴ + q⁵⁵ - 2q⁵⁶ + q⁵⁷ + q⁵⁸ - 2q⁵⁹ + q⁶⁰

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

Out[8]=	-Graphics-
In[9]:=	#[Knot[10, 13]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 2, 2, 2, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 13]][t]
Out[10]=	2 13 2 23 + -- - -- - 13 t + 2 t 2 t t
In[11]:=	Conway[Knot[10, 13]][z]
Out[11]=	2 4 1 - 5 z + 2 z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 13]}
In[13]:=	{KnotDet[Knot[10, 13]], KnotSignature[Knot[10, 13]]}
Out[13]=	{53, 0}
In[14]:=	Jones[Knot[10, 13]][q]
Out[14]=	-6 2 4 6 8 9 2 3 4 8 + q - -- + -- - -- + -- - - - 7 q + 5 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, 13]}
In[16]:=	A2Invariant[Knot[10, 13]][q]
Out[16]=	-20 -18 -16 -14 2 2 2 4 6 8 10 12 -2 + q + q - q + q - --- + -- + q - 2 q + q + 2 q - q + q + 10 8 q q 14 > q
In[17]:=	HOMFLYPT[Knot[10, 13]][a, z]
Out[17]=	2 -4 2 4 6 2 2 z 4 2 4 2 4 -1 + a + a - a + a - z - ---- - 2 a z + z + a z 2 a
In[18]:=	Kauffman[Knot[10, 13]][a, z]
Out[18]=	2 2 -4 2 4 6 2 z 3 5 2 2 z z 2 2 -1 + a - a - a - a - --- + a z - a z + 4 z - ---- + -- - a z + a 4 2 a a 3 3 4 4 4 2 6 2 2 z 3 z 3 3 5 3 4 z 3 z > 2 a z + 4 a z - ---- + ---- + a z + 6 a z - 3 z + -- - ---- + 3 a 4 2 a a a 5 5 2 4 4 4 6 4 2 z 3 z 5 3 5 5 5 > 6 a z + a z - 4 a z + ---- - ---- - 2 a z - 4 a z - 7 a z + 3 a a 6 7 3 z 2 6 4 6 6 6 3 z 7 5 7 8 2 8 > ---- - 9 a z - 5 a z + a z + ---- + a z + 2 a z + 2 z + 4 a z + 2 a a 4 8 9 3 9 > 2 a z + a z + a z
In[19]:=	{Vassiliev[2][Knot[10, 13]], Vassiliev[3][Knot[10, 13]]}
Out[19]=	{-5, 2}
In[20]:=	Kh[Knot[10, 13]][q, t]
Out[20]=	4 1 1 1 3 1 3 3 5 - + 5 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 3 4 5 3 3 2 5 2 5 3 7 3 > ----- + ---- + --- + 4 q t + 3 q t + q t + 4 q t + q t + q t + 3 2 3 q t q t q t 9 4 > q t
In[21]:=	ColouredJones[Knot[10, 13], 2][q]
Out[21]=	-18 2 6 8 4 20 14 17 38 15 36 54 9 52 + q - --- + --- - --- - --- + --- - --- - --- + -- - -- - -- + -- - -- - 17 15 14 13 12 11 10 9 8 7 6 5 q q q q q q q q q q q q 53 60 58 2 3 4 5 6 7 8 > -- + -- - -- + 6 q - 47 q + 34 q + 6 q - 27 q + 16 q + 3 q - 10 q + 4 3 q q q 9 10 11 12 > 5 q + q - 2 q + q

The Alternating Knot 1013

The Alternating Knot 10₁₃