The Non Alternating Knot 10₁₂₅

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Acknowledgement

KnotPlot

PD Presentation: X₁₄₂₅ X₃₈₄₉ X_5,14,6,15 X_20,16,1,15 X_16,10,17,9 X_18,12,19,11 X_10,18,11,17 X_12,20,13,19 X_13,6,14,7 X₇₂₈₃

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 + 2t^-1 - 1 + 2t - 2t² + t³

Conway Polynomial: 1 + 3z² + 4z⁴ + z⁶

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

Determinant and Signature: {11, 2}

Jones Polynomial: - q^-4 + q^-3 - q^-2 + 2q^-1 - 1 + 2q - q² + q³ - q⁴

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

A2 (sl(3)) Invariant: - q^-12 - q^-10 - q^-8 + q^-4 + 2q^-2 + 3 + 2q² + q⁴ - q⁸ - q¹⁰ - q¹²

HOMFLY-PT Polynomial: - 3a^-2 - 4a^-2z² - a^-2z⁴ + 7 + 11z² + 6z⁴ + z⁶ - 3a² - 4a²z² - a²z⁴

Kauffman Polynomial: a^-5z + a^-4z² - a^-3z + a^-3z³ + 3a^-2 - 6a^-2z² + 2a^-2z⁴ - 6a^-1z + 8a^-1z³ - 5a^-1z⁵ + a^-1z⁷ + 7 - 15z² + 13z⁴ - 6z⁶ + z⁸ - 8az + 17az³ - 11az⁵ + 2az⁷ + 3a² - 8a²z² + 11a²z⁴ - 6a²z⁶ + a²z⁸ - 4a³z + 10a³z³ - 6a³z⁵ + a³z⁷

V₂ and V₃, the type 2 and 3 Vassiliev invariants: {3, 0}

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

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

j = 9 1

j = 7

j = 5 1 1

j = 3 1

j = 1 1 2

j = -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^-13 - q^-12 - q^-11 + 2q^-10 - q^-9 - 2q^-8 + 2q^-7 - 2q^-5 + 2q^-4 - q^-2 + 2q^-1 + q² - q³ + q⁴ - q⁶ + q⁸ - q⁹ - q¹⁰ + q¹¹

3 - q^-27 + q^-26 + q^-25 - 2q^-23 + 2q^-21 + q^-20 - 2q^-19 - q^-18 + q^-17 + q^-16 - q^-15 - q^-14 + q^-13 - q^-11 - q^-10 + 2q^-9 - 2q^-7 - q^-6 + 4q^-5 + q^-4 - 2q^-3 - 3q^-2 + 5q^-1 + 2 - 3q - 4q² + 6q³ + 2q⁴ - 4q⁵ - 4q⁶ + 6q⁷ + 3q⁸ - 5q⁹ - 5q¹⁰ + 5q¹¹ + 4q¹² - 3q¹³ - 4q¹⁴ + 2q¹⁵ + 3q¹⁶ - 2q¹⁷ - q¹⁸ + 2q²⁰ - q²¹

4 q^-46 - q^-45 - q^-44 + 3q^-41 - q^-40 - q^-39 - q^-38 - 2q^-37 + 4q^-36 - 3q^-32 + 3q^-31 - q^-30 + q^-28 - 2q^-27 + 3q^-26 - q^-25 - q^-24 - 2q^-22 + 3q^-21 + q^-20 - q^-19 - q^-18 - 4q^-17 + q^-16 + 3q^-15 + q^-14 + q^-13 - 5q^-12 - 2q^-11 + 3q^-10 + 3q^-9 + 4q^-8 - 5q^-7 - 5q^-6 + 2q^-5 + 4q^-4 + 6q^-3 - 4q^-2 - 7q^-1 + 1 + 3q + 8q² - 3q³ - 8q⁴ + 3q⁶ + 9q⁷ - 3q⁸ - 10q⁹ - q¹⁰ + 4q¹¹ + 11q¹² - 3q¹³ - 11q¹⁴ - 2q¹⁵ + 4q¹⁶ + 10q¹⁷ - q¹⁸ - 8q¹⁹ - 3q²⁰ + 2q²¹ + 7q²² - q²³ - 4q²⁴ - q²⁵ + q²⁶ + 3q²⁷ - 2q²⁸ - q²⁹ + q³¹ + q³² - q³³

5 - q^-70 + q^-69 + q^-68 - q^-65 - 2q^-64 + 2q^-62 + q^-61 + q^-60 - 2q^-58 - 2q^-57 + q^-55 + q^-54 + q^-53 - q^-51 - q^-47 + q^-44 + q^-43 + q^-42 - q^-41 - q^-40 - q^-39 - q^-38 + q^-37 + 2q^-36 + q^-35 + 2q^-34 - q^-33 - 3q^-32 - 2q^-31 - q^-30 + 5q^-28 + 3q^-27 - q^-26 - 3q^-25 - 6q^-24 - 3q^-23 + 4q^-22 + 6q^-21 + 4q^-20 - q^-19 - 7q^-18 - 6q^-17 + 6q^-15 + 6q^-14 + 3q^-13 - 5q^-12 - 6q^-11 - 3q^-10 + 3q^-9 + 5q^-8 + 6q^-7 - 2q^-6 - 4q^-5 - 5q^-4 - q^-3 + 3q^-2 + 7q^-1 + 2 - 2q - 6q² - 4q³ + 8q⁵ + 5q⁶ - 7q⁸ - 7q⁹ + 9q¹¹ + 6q¹² - q¹³ - 9q¹⁴ - 8q¹⁵ + 2q¹⁶ + 11q¹⁷ + 8q¹⁸ - 2q¹⁹ - 10q²⁰ - 10q²¹ + 11q²³ + 10q²⁴ + q²⁵ - 7q²⁶ - 10q²⁷ - 4q²⁸ + 5q²⁹ + 8q³⁰ + 3q³¹ - q³² - 5q³³ - 2q³⁴ - q³⁵ + q³⁶ + q³⁷ + q³⁸ + q³⁹ + q⁴⁰ - 2q⁴¹ - q⁴² - q⁴³ + q⁴⁴ + 2q⁴⁵ + q⁴⁶ - q⁴⁷ - q⁴⁸ - q⁵⁰ + q⁵¹

6 q^-99 - q^-98 - q^-97 + q^-94 + 3q^-92 - q^-91 - 2q^-90 - q^-89 - q^-88 - q^-86 + 5q^-85 - q^-81 - q^-80 - 4q^-79 + 4q^-78 - q^-77 + q^-75 + q^-74 + q^-73 - 3q^-72 + 4q^-71 - 2q^-70 - 2q^-69 - q^-68 + q^-66 - 2q^-65 + 5q^-64 - q^-62 + q^-61 - q^-60 - q^-59 - 4q^-58 + 2q^-57 - q^-56 - q^-55 + 5q^-54 + 3q^-53 + 2q^-52 - 3q^-51 + q^-50 - 6q^-49 - 7q^-48 + 3q^-47 + 3q^-46 + 7q^-45 + 3q^-44 + 6q^-43 - 6q^-42 - 11q^-41 - 3q^-40 - 4q^-39 + 5q^-38 + 6q^-37 + 12q^-36 - q^-35 - 7q^-34 - 3q^-33 - 10q^-32 - q^-31 + 2q^-30 + 11q^-29 + q^-28 + 3q^-26 - 8q^-25 - 4q^-24 - 3q^-23 + 5q^-22 - 3q^-21 + 3q^-20 + 9q^-19 - q^-17 - 4q^-16 - 2q^-15 - 11q^-14 + 12q^-12 + 7q^-11 + 5q^-10 - 7q^-8 - 19q^-7 - 5q^-6 + 11q^-5 + 13q^-4 + 11q^-3 + 5q^-2 - 10q^-1 - 26 - 9q + 10q² + 18q³ + 16q⁴ + 8q⁵ - 12q⁶ - 32q⁷ - 12q⁸ + 9q⁹ + 21q¹⁰ + 19q¹¹ + 11q¹² - 14q¹³ - 35q¹⁴ - 13q¹⁵ + 10q¹⁶ + 24q¹⁷ + 18q¹⁸ + 10q¹⁹ - 16q²⁰ - 36q²¹ - 11q²² + 12q²³ + 28q²⁴ + 18q²⁵ + 8q²⁶ - 20q²⁷ - 39q²⁸ - 12q²⁹ + 11q³⁰ + 31q³¹ + 23q³² + 13q³³ - 18q³⁴ - 40q³⁵ - 19q³⁶ + 3q³⁷ + 24q³⁸ + 24q³⁹ + 20q⁴⁰ - 7q⁴¹ - 29q⁴² - 20q⁴³ - 4q⁴⁴ + 11q⁴⁵ + 14q⁴⁶ + 17q⁴⁷ + 2q⁴⁸ - 14q⁴⁹ - 12q⁵⁰ - 4q⁵¹ + 2q⁵² + 5q⁵³ + 9q⁵⁴ + 5q⁵⁵ - 5q⁵⁶ - 6q⁵⁷ - 2q⁵⁸ - q⁵⁹ + q⁶⁰ + 4q⁶¹ + 4q⁶² - 2q⁶³ - 3q⁶⁴ - q⁶⁵ - q⁶⁶ + q⁶⁸ + 2q⁶⁹ - q⁷¹

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, 125]]

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

In[3]:=
GaussCode[Knot[10, 125]]

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, 125]]

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

In[5]:=
br = BR[Knot[10, 125]]

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, 125]]

Out[7]=
3

In[8]:=
Show[DrawMorseLink[Knot[10, 125]]]

Out[8]=
-Graphics-

In[9]:=
#[Knot[10, 125]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}

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

In[10]:=
alex = Alexander[Knot[10, 125]][t]

Out[10]=
-3 2 2 2 3 -1 + t - -- + - + 2 t - 2 t + t 2 t t

In[11]:=
Conway[Knot[10, 125]][z]

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

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

Out[12]=
{Knot[10, 125]}

In[13]:=
{KnotDet[Knot[10, 125]], KnotSignature[Knot[10, 125]]}

Out[13]=
{11, 2}

In[14]:=
Jones[Knot[10, 125]][q]

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

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

Out[15]=
{Knot[10, 125]}

In[16]:=
A2Invariant[Knot[10, 125]][q]

Out[16]=
-12 -10 -8 -4 2 2 4 8 10 12 3 - q - q - q + q + -- + 2 q + q - q - q - q 2 q

In[17]:=
HOMFLYPT[Knot[10, 125]][a, z]

Out[17]=
2 4 3 2 2 4 z 2 2 4 z 2 4 6 7 - -- - 3 a + 11 z - ---- - 4 a z + 6 z - -- - a z + z 2 2 2 a a a

In[18]:=
Kauffman[Knot[10, 125]][a, z]

Out[18]=
2 2 3 2 z z 6 z 3 2 z 6 z 2 2 7 + -- + 3 a + -- - -- - --- - 8 a z - 4 a z - 15 z + -- - ---- - 8 a z + 2 5 3 a 4 2 a a a a a 3 3 4 5 z 8 z 3 3 3 4 2 z 2 4 5 z 5 > -- + ---- + 17 a z + 10 a z + 13 z + ---- + 11 a z - ---- - 11 a z - 3 a 2 a a a 7 3 5 6 2 6 z 7 3 7 8 2 8 > 6 a z - 6 z - 6 a z + -- + 2 a z + a z + z + a z a

In[19]:=
{Vassiliev[2][Knot[10, 125]], Vassiliev[3][Knot[10, 125]]}

Out[19]=
{3, 0}

In[20]:=
Kh[Knot[10, 125]][q, t]

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

In[21]:=
ColouredJones[Knot[10, 125], 2][q]

Out[21]=
-13 -12 -11 2 -9 2 2 2 2 -2 2 2 3 4 q - q - q + --- - q - -- + -- - -- + -- - q + - + q - q + q - 10 8 7 5 4 q q q q q q 6 8 9 10 11 > q + q - q - q + q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 10₁₂₅

10₁₂₄
10₁₂₆

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q^-13 - q^-12 - q^-11 + 2q^-10 - q^-9 - 2q^-8 + 2q^-7 - 2q^-5 + 2q^-4 - q^-2 + 2q^-1 + q² - q³ + q⁴ - q⁶ + q⁸ - q⁹ - q¹⁰ + q¹¹
3	- q^-27 + q^-26 + q^-25 - 2q^-23 + 2q^-21 + q^-20 - 2q^-19 - q^-18 + q^-17 + q^-16 - q^-15 - q^-14 + q^-13 - q^-11 - q^-10 + 2q^-9 - 2q^-7 - q^-6 + 4q^-5 + q^-4 - 2q^-3 - 3q^-2 + 5q^-1 + 2 - 3q - 4q² + 6q³ + 2q⁴ - 4q⁵ - 4q⁶ + 6q⁷ + 3q⁸ - 5q⁹ - 5q¹⁰ + 5q¹¹ + 4q¹² - 3q¹³ - 4q¹⁴ + 2q¹⁵ + 3q¹⁶ - 2q¹⁷ - q¹⁸ + 2q²⁰ - q²¹
4	q^-46 - q^-45 - q^-44 + 3q^-41 - q^-40 - q^-39 - q^-38 - 2q^-37 + 4q^-36 - 3q^-32 + 3q^-31 - q^-30 + q^-28 - 2q^-27 + 3q^-26 - q^-25 - q^-24 - 2q^-22 + 3q^-21 + q^-20 - q^-19 - q^-18 - 4q^-17 + q^-16 + 3q^-15 + q^-14 + q^-13 - 5q^-12 - 2q^-11 + 3q^-10 + 3q^-9 + 4q^-8 - 5q^-7 - 5q^-6 + 2q^-5 + 4q^-4 + 6q^-3 - 4q^-2 - 7q^-1 + 1 + 3q + 8q² - 3q³ - 8q⁴ + 3q⁶ + 9q⁷ - 3q⁸ - 10q⁹ - q¹⁰ + 4q¹¹ + 11q¹² - 3q¹³ - 11q¹⁴ - 2q¹⁵ + 4q¹⁶ + 10q¹⁷ - q¹⁸ - 8q¹⁹ - 3q²⁰ + 2q²¹ + 7q²² - q²³ - 4q²⁴ - q²⁵ + q²⁶ + 3q²⁷ - 2q²⁸ - q²⁹ + q³¹ + q³² - q³³
5	- q^-70 + q^-69 + q^-68 - q^-65 - 2q^-64 + 2q^-62 + q^-61 + q^-60 - 2q^-58 - 2q^-57 + q^-55 + q^-54 + q^-53 - q^-51 - q^-47 + q^-44 + q^-43 + q^-42 - q^-41 - q^-40 - q^-39 - q^-38 + q^-37 + 2q^-36 + q^-35 + 2q^-34 - q^-33 - 3q^-32 - 2q^-31 - q^-30 + 5q^-28 + 3q^-27 - q^-26 - 3q^-25 - 6q^-24 - 3q^-23 + 4q^-22 + 6q^-21 + 4q^-20 - q^-19 - 7q^-18 - 6q^-17 + 6q^-15 + 6q^-14 + 3q^-13 - 5q^-12 - 6q^-11 - 3q^-10 + 3q^-9 + 5q^-8 + 6q^-7 - 2q^-6 - 4q^-5 - 5q^-4 - q^-3 + 3q^-2 + 7q^-1 + 2 - 2q - 6q² - 4q³ + 8q⁵ + 5q⁶ - 7q⁸ - 7q⁹ + 9q¹¹ + 6q¹² - q¹³ - 9q¹⁴ - 8q¹⁵ + 2q¹⁶ + 11q¹⁷ + 8q¹⁸ - 2q¹⁹ - 10q²⁰ - 10q²¹ + 11q²³ + 10q²⁴ + q²⁵ - 7q²⁶ - 10q²⁷ - 4q²⁸ + 5q²⁹ + 8q³⁰ + 3q³¹ - q³² - 5q³³ - 2q³⁴ - q³⁵ + q³⁶ + q³⁷ + q³⁸ + q³⁹ + q⁴⁰ - 2q⁴¹ - q⁴² - q⁴³ + q⁴⁴ + 2q⁴⁵ + q⁴⁶ - q⁴⁷ - q⁴⁸ - q⁵⁰ + q⁵¹
6	q^-99 - q^-98 - q^-97 + q^-94 + 3q^-92 - q^-91 - 2q^-90 - q^-89 - q^-88 - q^-86 + 5q^-85 - q^-81 - q^-80 - 4q^-79 + 4q^-78 - q^-77 + q^-75 + q^-74 + q^-73 - 3q^-72 + 4q^-71 - 2q^-70 - 2q^-69 - q^-68 + q^-66 - 2q^-65 + 5q^-64 - q^-62 + q^-61 - q^-60 - q^-59 - 4q^-58 + 2q^-57 - q^-56 - q^-55 + 5q^-54 + 3q^-53 + 2q^-52 - 3q^-51 + q^-50 - 6q^-49 - 7q^-48 + 3q^-47 + 3q^-46 + 7q^-45 + 3q^-44 + 6q^-43 - 6q^-42 - 11q^-41 - 3q^-40 - 4q^-39 + 5q^-38 + 6q^-37 + 12q^-36 - q^-35 - 7q^-34 - 3q^-33 - 10q^-32 - q^-31 + 2q^-30 + 11q^-29 + q^-28 + 3q^-26 - 8q^-25 - 4q^-24 - 3q^-23 + 5q^-22 - 3q^-21 + 3q^-20 + 9q^-19 - q^-17 - 4q^-16 - 2q^-15 - 11q^-14 + 12q^-12 + 7q^-11 + 5q^-10 - 7q^-8 - 19q^-7 - 5q^-6 + 11q^-5 + 13q^-4 + 11q^-3 + 5q^-2 - 10q^-1 - 26 - 9q + 10q² + 18q³ + 16q⁴ + 8q⁵ - 12q⁶ - 32q⁷ - 12q⁸ + 9q⁹ + 21q¹⁰ + 19q¹¹ + 11q¹² - 14q¹³ - 35q¹⁴ - 13q¹⁵ + 10q¹⁶ + 24q¹⁷ + 18q¹⁸ + 10q¹⁹ - 16q²⁰ - 36q²¹ - 11q²² + 12q²³ + 28q²⁴ + 18q²⁵ + 8q²⁶ - 20q²⁷ - 39q²⁸ - 12q²⁹ + 11q³⁰ + 31q³¹ + 23q³² + 13q³³ - 18q³⁴ - 40q³⁵ - 19q³⁶ + 3q³⁷ + 24q³⁸ + 24q³⁹ + 20q⁴⁰ - 7q⁴¹ - 29q⁴² - 20q⁴³ - 4q⁴⁴ + 11q⁴⁵ + 14q⁴⁶ + 17q⁴⁷ + 2q⁴⁸ - 14q⁴⁹ - 12q⁵⁰ - 4q⁵¹ + 2q⁵² + 5q⁵³ + 9q⁵⁴ + 5q⁵⁵ - 5q⁵⁶ - 6q⁵⁷ - 2q⁵⁸ - q⁵⁹ + q⁶⁰ + 4q⁶¹ + 4q⁶² - 2q⁶³ - 3q⁶⁴ - q⁶⁵ - q⁶⁶ + q⁶⁸ + 2q⁶⁹ - q⁷¹

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	PD[Knot[10, 125]]
Out[2]=	PD[X[1, 4, 2, 5], X[3, 8, 4, 9], X[5, 14, 6, 15], X[20, 16, 1, 15], > X[16, 10, 17, 9], X[18, 12, 19, 11], X[10, 18, 11, 17], X[12, 20, 13, 19], > X[13, 6, 14, 7], X[7, 2, 8, 3]]
In[3]:=	GaussCode[Knot[10, 125]]
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, 125]]
Out[4]=	DTCode[4, 8, 14, 2, -16, -18, 6, -20, -10, -12]
In[5]:=	br = BR[Knot[10, 125]]
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, 125]]
Out[7]=	3
In[8]:=	Show[DrawMorseLink[Knot[10, 125]]]

Out[8]=	-Graphics-
In[9]:=	#[Knot[10, 125]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 2, 3, 3, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 125]][t]
Out[10]=	-3 2 2 2 3 -1 + t - -- + - + 2 t - 2 t + t 2 t t
In[11]:=	Conway[Knot[10, 125]][z]
Out[11]=	2 4 6 1 + 3 z + 4 z + z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 125]}
In[13]:=	{KnotDet[Knot[10, 125]], KnotSignature[Knot[10, 125]]}
Out[13]=	{11, 2}
In[14]:=	Jones[Knot[10, 125]][q]
Out[14]=	-4 -3 -2 2 2 3 4 -1 - q + q - q + - + 2 q - q + q - q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] \|\| (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[10, 125]}
In[16]:=	A2Invariant[Knot[10, 125]][q]
Out[16]=	-12 -10 -8 -4 2 2 4 8 10 12 3 - q - q - q + q + -- + 2 q + q - q - q - q 2 q
In[17]:=	HOMFLYPT[Knot[10, 125]][a, z]
Out[17]=	2 4 3 2 2 4 z 2 2 4 z 2 4 6 7 - -- - 3 a + 11 z - ---- - 4 a z + 6 z - -- - a z + z 2 2 2 a a a
In[18]:=	Kauffman[Knot[10, 125]][a, z]
Out[18]=	2 2 3 2 z z 6 z 3 2 z 6 z 2 2 7 + -- + 3 a + -- - -- - --- - 8 a z - 4 a z - 15 z + -- - ---- - 8 a z + 2 5 3 a 4 2 a a a a a 3 3 4 5 z 8 z 3 3 3 4 2 z 2 4 5 z 5 > -- + ---- + 17 a z + 10 a z + 13 z + ---- + 11 a z - ---- - 11 a z - 3 a 2 a a a 7 3 5 6 2 6 z 7 3 7 8 2 8 > 6 a z - 6 z - 6 a z + -- + 2 a z + a z + z + a z a
In[19]:=	{Vassiliev[2][Knot[10, 125]], Vassiliev[3][Knot[10, 125]]}
Out[19]=	{3, 0}
In[20]:=	Kh[Knot[10, 125]][q, t]
Out[20]=	3 1 1 1 1 1 q 5 5 2 9 3 2 q + q + ----- + ----- + ----- + ----- + ---- + - + q t + q t + q t 9 5 5 4 5 3 3 2 2 t q t q t q t q t q t
In[21]:=	ColouredJones[Knot[10, 125], 2][q]
Out[21]=	-13 -12 -11 2 -9 2 2 2 2 -2 2 2 3 4 q - q - q + --- - q - -- + -- - -- + -- - q + - + q - q + q - 10 8 7 5 4 q q q q q q 6 8 9 10 11 > q + q - q - q + q

The Non Alternating Knot 10125

The Non Alternating Knot 10₁₂₅