The Non Alternating Knot 9₄₃

Visit 9₄₃'s page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 9₄₃'s page at Knotilus!

Acknowledgement

KnotPlot

PD Presentation: X₄₂₅₁ X_10,6,11,5 X₈₃₉₄ X_2,9,3,10 X_14,8,15,7 X_15,1,16,18 X_11,17,12,16 X_17,13,18,12 X_6,14,7,13

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

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

Minimum Braid Representative:

Length is 9, width is 4
Braid index is 4

A Morse Link Presentation:

3D Invariants:

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

Reversible 2 3 3 / 4--5 1

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

Conway Polynomial: 1 + z² - 3z⁴ - z⁶

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

Determinant and Signature: {13, 4}

Jones Polynomial: 1 - q + 2q² - 2q³ + 2q⁴ - 2q⁵ + 2q⁶ - q⁷

Other knots (up to mirrors) with the same Jones Polynomial: {K11n12, ...}

A2 (sl(3)) Invariant: 1 + q² + q⁴ + q⁶ - 2q¹² + q¹⁸ + q²⁰ - q²⁶

HOMFLY-PT Polynomial: - a^-8 + 3a^-6 + 4a^-6z² + a^-6z⁴ - 4a^-4 - 7a^-4z² - 5a^-4z⁴ - a^-4z⁶ + 3a^-2 + 4a^-2z² + a^-2z⁴

Kauffman Polynomial: a^-9z - a^-8 + 2a^-8z² + a^-7z - 2a^-7z³ + a^-7z⁵ - 3a^-6 + 9a^-6z² - 8a^-6z⁴ + 2a^-6z⁶ + a^-5z³ - 3a^-5z⁵ + a^-5z⁷ - 4a^-4 + 14a^-4z² - 13a^-4z⁴ + 3a^-4z⁶ + 3a^-3z³ - 4a^-3z⁵ + a^-3z⁷ - 3a^-2 + 7a^-2z² - 5a^-2z⁴ + a^-2z⁶

V₂ and V₃, the type 2 and 3 Vassiliev invariants: {1, 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=4 is the signature of 9₄₃. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

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

j = 15 1

j = 13 1

j = 11 1 1

j = 9 1 1

j = 7 1 1

j = 5 1 1

j = 3 1 2

j = 1

j = -1 1

n Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2 q^-2 - q^-1 - 1 + 3q - q² - 3q³ + 4q⁴ - 3q⁶ + 3q⁷ + q⁸ - 3q⁹ + q¹⁰ + 2q¹¹ - 2q¹² - q¹³ + 2q¹⁴ - q¹⁵ - q¹⁶ + q¹⁷

3 q^-6 - q^-5 - q^-4 + 3q^-2 - 3 - 2q + 4q² + 2q³ - 2q⁴ - 3q⁵ + 3q⁶ + q⁷ - q⁹ + q¹⁰ - 2q¹¹ + 3q¹³ + q¹⁴ - 5q¹⁵ - q¹⁶ + 6q¹⁷ + 2q¹⁸ - 8q¹⁹ - 2q²⁰ + 9q²¹ + 2q²² - 11q²³ - 2q²⁴ + 11q²⁵ + 3q²⁶ - 11q²⁷ - 3q²⁸ + 9q²⁹ + 4q³⁰ - 7q³¹ - 3q³² + 4q³³ + 2q³⁴ - q³⁵ - 2q³⁶ + q³⁷

4 q^-12 - q^-11 - q^-10 + 4q^-7 - q^-6 - 2q^-5 - 2q^-4 - 3q^-3 + 7q^-2 + q^-1 - 2q - 6q² + 6q³ - q⁴ + q⁵ + 2q⁶ - 4q⁷ + 6q⁸ - 5q⁹ - 4q¹⁰ + 3q¹¹ + q¹² + 11q¹³ - 5q¹⁴ - 12q¹⁵ + 2q¹⁷ + 17q¹⁸ + q¹⁹ - 17q²⁰ - 6q²¹ - q²² + 20q²³ + 9q²⁴ - 18q²⁵ - 11q²⁶ - 5q²⁷ + 21q²⁸ + 16q²⁹ - 18q³⁰ - 15q³¹ - 7q³² + 22q³³ + 21q³⁴ - 19q³⁵ - 18q³⁶ - 9q³⁷ + 23q³⁸ + 26q³⁹ - 19q⁴⁰ - 20q⁴¹ - 11q⁴² + 20q⁴³ + 27q⁴⁴ - 13q⁴⁵ - 16q⁴⁶ - 13q⁴⁷ + 11q⁴⁸ + 21q⁴⁹ - 6q⁵⁰ - 7q⁵¹ - 9q⁵² + 3q⁵³ + 9q⁵⁴ - 2q⁵⁵ - q⁵⁶ - 3q⁵⁷ + q⁵⁸ + 2q⁵⁹ - q⁶⁰

5 q^-20 - q^-19 - q^-18 + q^-15 + 3q^-14 - 3q^-12 - 2q^-11 - 2q^-10 + 5q^-8 + 4q^-7 - 3q^-5 - 4q^-4 - 3q^-3 + 2q^-2 + 3q^-1 + 2 + q⁴ - 2q⁵ - 5q⁶ - 3q⁷ + 2q⁸ + 7q⁹ + 10q¹⁰ + 2q¹¹ - 10q¹² - 14q¹³ - 6q¹⁴ + 6q¹⁵ + 18q¹⁶ + 14q¹⁷ - 2q¹⁸ - 18q¹⁹ - 18q²⁰ - 6q²¹ + 14q²² + 22q²³ + 12q²⁴ - 8q²⁵ - 21q²⁶ - 19q²⁷ + q²⁸ + 19q²⁹ + 23q³⁰ + 6q³¹ - 14q³² - 26q³³ - 14q³⁴ + 10q³⁵ + 26q³⁶ + 20q³⁷ - 4q³⁸ - 27q³⁹ - 25q⁴⁰ - q⁴¹ + 26q⁴² + 30q⁴³ + 5q⁴⁴ - 27q⁴⁵ - 33q⁴⁶ - 8q⁴⁷ + 27q⁴⁸ + 39q⁴⁹ + 9q⁵⁰ - 30q⁵¹ - 41q⁵² - 11q⁵³ + 32q⁵⁴ + 46q⁵⁵ + 12q⁵⁶ - 34q⁵⁷ - 48q⁵⁸ - 15q⁵⁹ + 32q⁶⁰ + 51q⁶¹ + 20q⁶² - 29q⁶³ - 50q⁶⁴ - 23q⁶⁵ + 21q⁶⁶ + 44q⁶⁷ + 27q⁶⁸ - 13q⁶⁹ - 38q⁷⁰ - 24q⁷¹ + 6q⁷² + 25q⁷³ + 20q⁷⁴ - 15q⁷⁶ - 14q⁷⁷ + 7q⁷⁹ + 6q⁸⁰ + q⁸¹ - 2q⁸² - 3q⁸³ + q⁸⁵

6 q^-30 - q^-29 - q^-28 + q^-25 + 4q^-23 - q^-22 - 3q^-21 - 2q^-20 - 2q^-19 - q^-17 + 9q^-16 + 2q^-15 - 2q^-13 - 4q^-12 - 4q^-11 - 7q^-10 + 9q^-9 + q^-8 + 3q^-7 + 2q^-6 + q^-5 - 7q^-3 + 7q^-2 - 6q^-1 - 4 - 4q + 6q³ + 4q⁴ + 18q⁵ - q⁶ - 6q⁷ - 15q⁸ - 17q⁹ - 4q¹⁰ + 2q¹¹ + 28q¹² + 16q¹³ + 12q¹⁴ - 6q¹⁵ - 27q¹⁶ - 19q¹⁷ - 20q¹⁸ + 12q¹⁹ + 17q²⁰ + 30q²¹ + 20q²² - 11q²³ - 10q²⁴ - 35q²⁵ - 17q²⁶ - 8q²⁷ + 22q²⁸ + 32q²⁹ + 14q³⁰ + 24q³¹ - 23q³² - 30q³³ - 38q³⁴ - 7q³⁵ + 17q³⁶ + 21q³⁷ + 59q³⁸ + 8q³⁹ - 18q⁴⁰ - 51q⁴¹ - 36q⁴² - 15q⁴³ + 8q⁴⁴ + 79q⁴⁵ + 40q⁴⁶ + 8q⁴⁷ - 47q⁴⁸ - 54q⁴⁹ - 48q⁵⁰ - 15q⁵¹ + 86q⁵² + 64q⁵³ + 35q⁵⁴ - 36q⁵⁵ - 64q⁵⁶ - 76q⁵⁷ - 36q⁵⁸ + 89q⁵⁹ + 81q⁶⁰ + 55q⁶¹ - 27q⁶² - 71q⁶³ - 95q⁶⁴ - 48q⁶⁵ + 93q⁶⁶ + 93q⁶⁷ + 65q⁶⁸ - 26q⁶⁹ - 80q⁷⁰ - 106q⁷¹ - 49q⁷² + 103q⁷³ + 103q⁷⁴ + 68q⁷⁵ - 32q⁷⁶ - 91q⁷⁷ - 116q⁷⁸ - 50q⁷⁹ + 110q⁸⁰ + 115q⁸¹ + 77q⁸² - 28q⁸³ - 95q⁸⁴ - 128q⁸⁵ - 64q⁸⁶ + 96q⁸⁷ + 117q⁸⁸ + 92q⁸⁹ - 3q⁹⁰ - 74q⁹¹ - 124q⁹² - 81q⁹³ + 55q⁹⁴ + 87q⁹⁵ + 86q⁹⁶ + 24q⁹⁷ - 31q⁹⁸ - 85q⁹⁹ - 70q¹⁰⁰ + 16q¹⁰¹ + 38q¹⁰² + 49q¹⁰³ + 25q¹⁰⁴ + q¹⁰⁵ - 34q¹⁰⁶ - 35q¹⁰⁷ + q¹⁰⁸ + 6q¹⁰⁹ + 14q¹¹⁰ + 9q¹¹¹ + 6q¹¹² - 8q¹¹³ - 9q¹¹⁴ + 2q¹¹⁵ - q¹¹⁶ + 2q¹¹⁷ + q¹¹⁸ + 2q¹¹⁹ - q¹²⁰ - 2q¹²¹ + q¹²²

7 q^-42 - q^-41 - q^-40 + q^-37 + q^-35 + 3q^-34 - q^-33 - 3q^-32 - 2q^-31 - 3q^-30 + q^-29 + q^-27 + 8q^-26 + 3q^-25 - 2q^-23 - 7q^-22 - 3q^-21 - 4q^-20 - 4q^-19 + 7q^-18 + 4q^-17 + 5q^-16 + 6q^-15 - 4q^-14 - q^-12 - 6q^-11 + q^-10 - 5q^-9 - 4q^-8 + 3q^-7 - 6q^-6 + 3q^-5 + 10q^-4 + 8q^-3 + 15q^-2 - 10 - 8q - 23q² - 18q³ - q⁴ + 8q⁵ + 31q⁶ + 25q⁷ + 13q⁸ + 9q⁹ - 19q¹⁰ - 33q¹¹ - 29q¹² - 25q¹³ + 6q¹⁴ + 21q¹⁵ + 27q¹⁶ + 41q¹⁷ + 21q¹⁸ - 3q¹⁹ - 17q²⁰ - 39q²¹ - 35q²² - 26q²³ - 9q²⁴ + 27q²⁵ + 40q²⁶ + 39q²⁷ + 40q²⁸ + 7q²⁹ - 25q³⁰ - 52q³¹ - 62q³² - 36q³³ - 4q³⁴ + 32q³⁵ + 73q³⁶ + 75q³⁷ + 40q³⁸ - 12q³⁹ - 67q⁴⁰ - 88q⁴¹ - 74q⁴² - 33q⁴³ + 43q⁴⁴ + 99q⁴⁵ + 105q⁴⁶ + 66q⁴⁷ - 10q⁴⁸ - 85q⁴⁹ - 118q⁵⁰ - 108q⁵¹ - 33q⁵² + 67q⁵³ + 125q⁵⁴ + 134q⁵⁵ + 71q⁵⁶ - 34q⁵⁷ - 117q⁵⁸ - 158q⁵⁹ - 112q⁶⁰ + 5q⁶¹ + 105q⁶² + 168q⁶³ + 146q⁶⁴ + 29q⁶⁵ - 87q⁶⁶ - 179q⁶⁷ - 174q⁶⁸ - 57q⁶⁹ + 68q⁷⁰ + 182q⁷¹ + 199q⁷² + 86q⁷³ - 52q⁷⁴ - 184q⁷⁵ - 220q⁷⁶ - 109q⁷⁷ + 37q⁷⁸ + 187q⁷⁹ + 238q⁸⁰ + 127q⁸¹ - 25q⁸² - 189q⁸³ - 255q⁸⁴ - 141q⁸⁵ + 20q⁸⁶ + 194q⁸⁷ + 264q⁸⁸ + 150q⁸⁹ - 17q⁹⁰ - 198q⁹¹ - 277q⁹² - 153q⁹³ + 22q⁹⁴ + 206q⁹⁵ + 283q⁹⁶ + 154q⁹⁷ - 27q⁹⁸ - 215q⁹⁹ - 294q¹⁰⁰ - 156q¹⁰¹ + 32q¹⁰² + 226q¹⁰³ + 305q¹⁰⁴ + 164q¹⁰⁵ - 31q¹⁰⁶ - 232q¹⁰⁷ - 318q¹⁰⁸ - 181q¹⁰⁹ + 18q¹¹⁰ + 228q¹¹¹ + 329q¹¹² + 204q¹¹³ + 6q¹¹⁴ - 212q¹¹⁵ - 325q¹¹⁶ - 221q¹¹⁷ - 43q¹¹⁸ + 172q¹¹⁹ + 307q¹²⁰ + 236q¹²¹ + 77q¹²² - 129q¹²³ - 264q¹²⁴ - 220q¹²⁵ - 101q¹²⁶ + 68q¹²⁷ + 206q¹²⁸ + 194q¹²⁹ + 111q¹³⁰ - 30q¹³¹ - 145q¹³² - 142q¹³³ - 97q¹³⁴ - 3q¹³⁵ + 85q¹³⁶ + 96q¹³⁷ + 74q¹³⁸ + 16q¹³⁹ - 46q¹⁴⁰ - 57q¹⁴¹ - 45q¹⁴² - 12q¹⁴³ + 22q¹⁴⁴ + 21q¹⁴⁵ + 24q¹⁴⁶ + 13q¹⁴⁷ - 9q¹⁴⁸ - 13q¹⁴⁹ - 11q¹⁵⁰ - q¹⁵¹ + 5q¹⁵² + q¹⁵³ + 2q¹⁵⁴ + 4q¹⁵⁵ - 2q¹⁵⁶ - 2q¹⁵⁷ - 2q¹⁵⁸ + q¹⁵⁹ + 2q¹⁶⁰ - q¹⁶¹

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[9, 43]]

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

In[3]:=
GaussCode[Knot[9, 43]]

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

In[4]:=
DTCode[Knot[9, 43]]

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

In[5]:=
br = BR[Knot[9, 43]]

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

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

Out[6]=
{4, 9}

In[7]:=
BraidIndex[Knot[9, 43]]

Out[7]=
4

In[8]:=
Show[DrawMorseLink[Knot[9, 43]]]

Out[8]=
-Graphics-

In[9]:=
#[Knot[9, 43]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}

Out[9]=
{Reversible, 2, 3, 3, {4, 5}, 1}

In[10]:=
alex = Alexander[Knot[9, 43]][t]

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

In[11]:=
Conway[Knot[9, 43]][z]

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

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

Out[12]=
{Knot[9, 43]}

In[13]:=
{KnotDet[Knot[9, 43]], KnotSignature[Knot[9, 43]]}

Out[13]=
{13, 4}

In[14]:=
Jones[Knot[9, 43]][q]

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

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

Out[15]=
{Knot[9, 43], Knot[11, NonAlternating, 12]}

In[16]:=
A2Invariant[Knot[9, 43]][q]

Out[16]=
2 4 6 12 18 20 26 1 + q + q + q - 2 q + q + q - q

In[17]:=
HOMFLYPT[Knot[9, 43]][a, z]

Out[17]=
2 2 2 4 4 4 6 -8 3 4 3 4 z 7 z 4 z z 5 z z z -a + -- - -- + -- + ---- - ---- + ---- + -- - ---- + -- - -- 6 4 2 6 4 2 6 4 2 4 a a a a a a a a a a

In[18]:=
Kauffman[Knot[9, 43]][a, z]

Out[18]=
2 2 2 2 3 3 3 -8 3 4 3 z z 2 z 9 z 14 z 7 z 2 z z 3 z -a - -- - -- - -- + -- + -- + ---- + ---- + ----- + ---- - ---- + -- + ---- - 6 4 2 9 7 8 6 4 2 7 5 3 a a a a a a a a a a a a 4 4 4 5 5 5 6 6 6 7 7 8 z 13 z 5 z z 3 z 4 z 2 z 3 z z z z > ---- - ----- - ---- + -- - ---- - ---- + ---- + ---- + -- + -- + -- 6 4 2 7 5 3 6 4 2 5 3 a a a a a a a a a a a

In[19]:=
{Vassiliev[2][Knot[9, 43]], Vassiliev[3][Knot[9, 43]]}

Out[19]=
{1, 2}

In[20]:=
Kh[Knot[9, 43]][q, t]

Out[20]=
3 3 5 1 q 5 7 7 2 9 2 9 3 11 3 11 4 2 q + q + ---- + -- + q t + q t + q t + q t + q t + q t + q t + 2 t q t 13 4 15 5 > q t + q t

In[21]:=
ColouredJones[Knot[9, 43], 2][q]

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

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 9₄₃

9₄₂
9₄₄

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q^-2 - q^-1 - 1 + 3q - q² - 3q³ + 4q⁴ - 3q⁶ + 3q⁷ + q⁸ - 3q⁹ + q¹⁰ + 2q¹¹ - 2q¹² - q¹³ + 2q¹⁴ - q¹⁵ - q¹⁶ + q¹⁷
3	q^-6 - q^-5 - q^-4 + 3q^-2 - 3 - 2q + 4q² + 2q³ - 2q⁴ - 3q⁵ + 3q⁶ + q⁷ - q⁹ + q¹⁰ - 2q¹¹ + 3q¹³ + q¹⁴ - 5q¹⁵ - q¹⁶ + 6q¹⁷ + 2q¹⁸ - 8q¹⁹ - 2q²⁰ + 9q²¹ + 2q²² - 11q²³ - 2q²⁴ + 11q²⁵ + 3q²⁶ - 11q²⁷ - 3q²⁸ + 9q²⁹ + 4q³⁰ - 7q³¹ - 3q³² + 4q³³ + 2q³⁴ - q³⁵ - 2q³⁶ + q³⁷
4	q^-12 - q^-11 - q^-10 + 4q^-7 - q^-6 - 2q^-5 - 2q^-4 - 3q^-3 + 7q^-2 + q^-1 - 2q - 6q² + 6q³ - q⁴ + q⁵ + 2q⁶ - 4q⁷ + 6q⁸ - 5q⁹ - 4q¹⁰ + 3q¹¹ + q¹² + 11q¹³ - 5q¹⁴ - 12q¹⁵ + 2q¹⁷ + 17q¹⁸ + q¹⁹ - 17q²⁰ - 6q²¹ - q²² + 20q²³ + 9q²⁴ - 18q²⁵ - 11q²⁶ - 5q²⁷ + 21q²⁸ + 16q²⁹ - 18q³⁰ - 15q³¹ - 7q³² + 22q³³ + 21q³⁴ - 19q³⁵ - 18q³⁶ - 9q³⁷ + 23q³⁸ + 26q³⁹ - 19q⁴⁰ - 20q⁴¹ - 11q⁴² + 20q⁴³ + 27q⁴⁴ - 13q⁴⁵ - 16q⁴⁶ - 13q⁴⁷ + 11q⁴⁸ + 21q⁴⁹ - 6q⁵⁰ - 7q⁵¹ - 9q⁵² + 3q⁵³ + 9q⁵⁴ - 2q⁵⁵ - q⁵⁶ - 3q⁵⁷ + q⁵⁸ + 2q⁵⁹ - q⁶⁰
5	q^-20 - q^-19 - q^-18 + q^-15 + 3q^-14 - 3q^-12 - 2q^-11 - 2q^-10 + 5q^-8 + 4q^-7 - 3q^-5 - 4q^-4 - 3q^-3 + 2q^-2 + 3q^-1 + 2 + q⁴ - 2q⁵ - 5q⁶ - 3q⁷ + 2q⁸ + 7q⁹ + 10q¹⁰ + 2q¹¹ - 10q¹² - 14q¹³ - 6q¹⁴ + 6q¹⁵ + 18q¹⁶ + 14q¹⁷ - 2q¹⁸ - 18q¹⁹ - 18q²⁰ - 6q²¹ + 14q²² + 22q²³ + 12q²⁴ - 8q²⁵ - 21q²⁶ - 19q²⁷ + q²⁸ + 19q²⁹ + 23q³⁰ + 6q³¹ - 14q³² - 26q³³ - 14q³⁴ + 10q³⁵ + 26q³⁶ + 20q³⁷ - 4q³⁸ - 27q³⁹ - 25q⁴⁰ - q⁴¹ + 26q⁴² + 30q⁴³ + 5q⁴⁴ - 27q⁴⁵ - 33q⁴⁶ - 8q⁴⁷ + 27q⁴⁸ + 39q⁴⁹ + 9q⁵⁰ - 30q⁵¹ - 41q⁵² - 11q⁵³ + 32q⁵⁴ + 46q⁵⁵ + 12q⁵⁶ - 34q⁵⁷ - 48q⁵⁸ - 15q⁵⁹ + 32q⁶⁰ + 51q⁶¹ + 20q⁶² - 29q⁶³ - 50q⁶⁴ - 23q⁶⁵ + 21q⁶⁶ + 44q⁶⁷ + 27q⁶⁸ - 13q⁶⁹ - 38q⁷⁰ - 24q⁷¹ + 6q⁷² + 25q⁷³ + 20q⁷⁴ - 15q⁷⁶ - 14q⁷⁷ + 7q⁷⁹ + 6q⁸⁰ + q⁸¹ - 2q⁸² - 3q⁸³ + q⁸⁵
6	q^-30 - q^-29 - q^-28 + q^-25 + 4q^-23 - q^-22 - 3q^-21 - 2q^-20 - 2q^-19 - q^-17 + 9q^-16 + 2q^-15 - 2q^-13 - 4q^-12 - 4q^-11 - 7q^-10 + 9q^-9 + q^-8 + 3q^-7 + 2q^-6 + q^-5 - 7q^-3 + 7q^-2 - 6q^-1 - 4 - 4q + 6q³ + 4q⁴ + 18q⁵ - q⁶ - 6q⁷ - 15q⁸ - 17q⁹ - 4q¹⁰ + 2q¹¹ + 28q¹² + 16q¹³ + 12q¹⁴ - 6q¹⁵ - 27q¹⁶ - 19q¹⁷ - 20q¹⁸ + 12q¹⁹ + 17q²⁰ + 30q²¹ + 20q²² - 11q²³ - 10q²⁴ - 35q²⁵ - 17q²⁶ - 8q²⁷ + 22q²⁸ + 32q²⁹ + 14q³⁰ + 24q³¹ - 23q³² - 30q³³ - 38q³⁴ - 7q³⁵ + 17q³⁶ + 21q³⁷ + 59q³⁸ + 8q³⁹ - 18q⁴⁰ - 51q⁴¹ - 36q⁴² - 15q⁴³ + 8q⁴⁴ + 79q⁴⁵ + 40q⁴⁶ + 8q⁴⁷ - 47q⁴⁸ - 54q⁴⁹ - 48q⁵⁰ - 15q⁵¹ + 86q⁵² + 64q⁵³ + 35q⁵⁴ - 36q⁵⁵ - 64q⁵⁶ - 76q⁵⁷ - 36q⁵⁸ + 89q⁵⁹ + 81q⁶⁰ + 55q⁶¹ - 27q⁶² - 71q⁶³ - 95q⁶⁴ - 48q⁶⁵ + 93q⁶⁶ + 93q⁶⁷ + 65q⁶⁸ - 26q⁶⁹ - 80q⁷⁰ - 106q⁷¹ - 49q⁷² + 103q⁷³ + 103q⁷⁴ + 68q⁷⁵ - 32q⁷⁶ - 91q⁷⁷ - 116q⁷⁸ - 50q⁷⁹ + 110q⁸⁰ + 115q⁸¹ + 77q⁸² - 28q⁸³ - 95q⁸⁴ - 128q⁸⁵ - 64q⁸⁶ + 96q⁸⁷ + 117q⁸⁸ + 92q⁸⁹ - 3q⁹⁰ - 74q⁹¹ - 124q⁹² - 81q⁹³ + 55q⁹⁴ + 87q⁹⁵ + 86q⁹⁶ + 24q⁹⁷ - 31q⁹⁸ - 85q⁹⁹ - 70q¹⁰⁰ + 16q¹⁰¹ + 38q¹⁰² + 49q¹⁰³ + 25q¹⁰⁴ + q¹⁰⁵ - 34q¹⁰⁶ - 35q¹⁰⁷ + q¹⁰⁸ + 6q¹⁰⁹ + 14q¹¹⁰ + 9q¹¹¹ + 6q¹¹² - 8q¹¹³ - 9q¹¹⁴ + 2q¹¹⁵ - q¹¹⁶ + 2q¹¹⁷ + q¹¹⁸ + 2q¹¹⁹ - q¹²⁰ - 2q¹²¹ + q¹²²
7	q^-42 - q^-41 - q^-40 + q^-37 + q^-35 + 3q^-34 - q^-33 - 3q^-32 - 2q^-31 - 3q^-30 + q^-29 + q^-27 + 8q^-26 + 3q^-25 - 2q^-23 - 7q^-22 - 3q^-21 - 4q^-20 - 4q^-19 + 7q^-18 + 4q^-17 + 5q^-16 + 6q^-15 - 4q^-14 - q^-12 - 6q^-11 + q^-10 - 5q^-9 - 4q^-8 + 3q^-7 - 6q^-6 + 3q^-5 + 10q^-4 + 8q^-3 + 15q^-2 - 10 - 8q - 23q² - 18q³ - q⁴ + 8q⁵ + 31q⁶ + 25q⁷ + 13q⁸ + 9q⁹ - 19q¹⁰ - 33q¹¹ - 29q¹² - 25q¹³ + 6q¹⁴ + 21q¹⁵ + 27q¹⁶ + 41q¹⁷ + 21q¹⁸ - 3q¹⁹ - 17q²⁰ - 39q²¹ - 35q²² - 26q²³ - 9q²⁴ + 27q²⁵ + 40q²⁶ + 39q²⁷ + 40q²⁸ + 7q²⁹ - 25q³⁰ - 52q³¹ - 62q³² - 36q³³ - 4q³⁴ + 32q³⁵ + 73q³⁶ + 75q³⁷ + 40q³⁸ - 12q³⁹ - 67q⁴⁰ - 88q⁴¹ - 74q⁴² - 33q⁴³ + 43q⁴⁴ + 99q⁴⁵ + 105q⁴⁶ + 66q⁴⁷ - 10q⁴⁸ - 85q⁴⁹ - 118q⁵⁰ - 108q⁵¹ - 33q⁵² + 67q⁵³ + 125q⁵⁴ + 134q⁵⁵ + 71q⁵⁶ - 34q⁵⁷ - 117q⁵⁸ - 158q⁵⁹ - 112q⁶⁰ + 5q⁶¹ + 105q⁶² + 168q⁶³ + 146q⁶⁴ + 29q⁶⁵ - 87q⁶⁶ - 179q⁶⁷ - 174q⁶⁸ - 57q⁶⁹ + 68q⁷⁰ + 182q⁷¹ + 199q⁷² + 86q⁷³ - 52q⁷⁴ - 184q⁷⁵ - 220q⁷⁶ - 109q⁷⁷ + 37q⁷⁸ + 187q⁷⁹ + 238q⁸⁰ + 127q⁸¹ - 25q⁸² - 189q⁸³ - 255q⁸⁴ - 141q⁸⁵ + 20q⁸⁶ + 194q⁸⁷ + 264q⁸⁸ + 150q⁸⁹ - 17q⁹⁰ - 198q⁹¹ - 277q⁹² - 153q⁹³ + 22q⁹⁴ + 206q⁹⁵ + 283q⁹⁶ + 154q⁹⁷ - 27q⁹⁸ - 215q⁹⁹ - 294q¹⁰⁰ - 156q¹⁰¹ + 32q¹⁰² + 226q¹⁰³ + 305q¹⁰⁴ + 164q¹⁰⁵ - 31q¹⁰⁶ - 232q¹⁰⁷ - 318q¹⁰⁸ - 181q¹⁰⁹ + 18q¹¹⁰ + 228q¹¹¹ + 329q¹¹² + 204q¹¹³ + 6q¹¹⁴ - 212q¹¹⁵ - 325q¹¹⁶ - 221q¹¹⁷ - 43q¹¹⁸ + 172q¹¹⁹ + 307q¹²⁰ + 236q¹²¹ + 77q¹²² - 129q¹²³ - 264q¹²⁴ - 220q¹²⁵ - 101q¹²⁶ + 68q¹²⁷ + 206q¹²⁸ + 194q¹²⁹ + 111q¹³⁰ - 30q¹³¹ - 145q¹³² - 142q¹³³ - 97q¹³⁴ - 3q¹³⁵ + 85q¹³⁶ + 96q¹³⁷ + 74q¹³⁸ + 16q¹³⁹ - 46q¹⁴⁰ - 57q¹⁴¹ - 45q¹⁴² - 12q¹⁴³ + 22q¹⁴⁴ + 21q¹⁴⁵ + 24q¹⁴⁶ + 13q¹⁴⁷ - 9q¹⁴⁸ - 13q¹⁴⁹ - 11q¹⁵⁰ - q¹⁵¹ + 5q¹⁵² + q¹⁵³ + 2q¹⁵⁴ + 4q¹⁵⁵ - 2q¹⁵⁶ - 2q¹⁵⁷ - 2q¹⁵⁸ + q¹⁵⁹ + 2q¹⁶⁰ - q¹⁶¹

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	PD[Knot[9, 43]]
Out[2]=	PD[X[4, 2, 5, 1], X[10, 6, 11, 5], X[8, 3, 9, 4], X[2, 9, 3, 10], > X[14, 8, 15, 7], X[15, 1, 16, 18], X[11, 17, 12, 16], X[17, 13, 18, 12], > X[6, 14, 7, 13]]
In[3]:=	GaussCode[Knot[9, 43]]
Out[3]=	GaussCode[1, -4, 3, -1, 2, -9, 5, -3, 4, -2, -7, 8, 9, -5, -6, 7, -8, 6]
In[4]:=	DTCode[Knot[9, 43]]
Out[4]=	DTCode[4, 8, 10, 14, 2, -16, 6, -18, -12]
In[5]:=	br = BR[Knot[9, 43]]
Out[5]=	BR[4, {1, 1, 1, 2, 1, 1, -3, 2, -3}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{4, 9}
In[7]:=	BraidIndex[Knot[9, 43]]
Out[7]=	4
In[8]:=	Show[DrawMorseLink[Knot[9, 43]]]

Out[8]=	-Graphics-
In[9]:=	#[Knot[9, 43]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 2, 3, 3, {4, 5}, 1}
In[10]:=	alex = Alexander[Knot[9, 43]][t]
Out[10]=	-3 3 2 2 3 1 - t + -- - - - 2 t + 3 t - t 2 t t
In[11]:=	Conway[Knot[9, 43]][z]
Out[11]=	2 4 6 1 + z - 3 z - z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[9, 43]}
In[13]:=	{KnotDet[Knot[9, 43]], KnotSignature[Knot[9, 43]]}
Out[13]=	{13, 4}
In[14]:=	Jones[Knot[9, 43]][q]
Out[14]=	2 3 4 5 6 7 1 - q + 2 q - 2 q + 2 q - 2 q + 2 q - q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] \|\| (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[9, 43], Knot[11, NonAlternating, 12]}
In[16]:=	A2Invariant[Knot[9, 43]][q]
Out[16]=	2 4 6 12 18 20 26 1 + q + q + q - 2 q + q + q - q
In[17]:=	HOMFLYPT[Knot[9, 43]][a, z]
Out[17]=	2 2 2 4 4 4 6 -8 3 4 3 4 z 7 z 4 z z 5 z z z -a + -- - -- + -- + ---- - ---- + ---- + -- - ---- + -- - -- 6 4 2 6 4 2 6 4 2 4 a a a a a a a a a a
In[18]:=	Kauffman[Knot[9, 43]][a, z]
Out[18]=	2 2 2 2 3 3 3 -8 3 4 3 z z 2 z 9 z 14 z 7 z 2 z z 3 z -a - -- - -- - -- + -- + -- + ---- + ---- + ----- + ---- - ---- + -- + ---- - 6 4 2 9 7 8 6 4 2 7 5 3 a a a a a a a a a a a a 4 4 4 5 5 5 6 6 6 7 7 8 z 13 z 5 z z 3 z 4 z 2 z 3 z z z z > ---- - ----- - ---- + -- - ---- - ---- + ---- + ---- + -- + -- + -- 6 4 2 7 5 3 6 4 2 5 3 a a a a a a a a a a a
In[19]:=	{Vassiliev[2][Knot[9, 43]], Vassiliev[3][Knot[9, 43]]}
Out[19]=	{1, 2}
In[20]:=	Kh[Knot[9, 43]][q, t]
Out[20]=	3 3 5 1 q 5 7 7 2 9 2 9 3 11 3 11 4 2 q + q + ---- + -- + q t + q t + q t + q t + q t + q t + q t + 2 t q t 13 4 15 5 > q t + q t
In[21]:=	ColouredJones[Knot[9, 43], 2][q]
Out[21]=	-2 1 2 3 4 6 7 8 9 10 11 -1 + q - - + 3 q - q - 3 q + 4 q - 3 q + 3 q + q - 3 q + q + 2 q - q 12 13 14 15 16 17 > 2 q - q + 2 q - q - q + q

The Non Alternating Knot 943

The Non Alternating Knot 9₄₃