The Alternating Knot 9₁₄

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Acknowledgement

KnotPlot

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

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

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

Minimum Braid Representative:

Length is 10, width is 5
Braid index is 5

A Morse Link Presentation:

3D Invariants:

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

Reversible 1 2 2 / 4--7 1

Alexander Polynomial: 2t^-2 - 9t^-1 + 15 - 9t + 2t²

Conway Polynomial: 1 - z² + 2z⁴

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

Determinant and Signature: {37, 0}

Jones Polynomial: - q^-3 + 3q^-2 - 4q^-1 + 6 - 6q + 6q² - 5q³ + 3q⁴ - 2q⁵ + q⁶

Other knots (up to mirrors) with the same Jones Polynomial: {K11n53, ...}

A2 (sl(3)) Invariant: - q^-10 + q^-8 + q^-6 - q^-4 + 2q^-2 + q² + q⁴ + q⁸ - 2q¹⁰ - q¹² - q¹⁶ + q¹⁸ + q²⁰

HOMFLY-PT Polynomial: a^-6 - 2a^-4 - 2a^-4z² + a^-2 + a^-2z² + a^-2z⁴ + 1 + z² + z⁴ - a²z²

Kauffman Polynomial: - a^-6 + 4a^-6z² - 4a^-6z⁴ + a^-6z⁶ - 3a^-5z + 9a^-5z³ - 8a^-5z⁵ + 2a^-5z⁷ - 2a^-4 + 10a^-4z² - 9a^-4z⁴ + a^-4z⁸ - 5a^-3z + 15a^-3z³ - 16a^-3z⁵ + 5a^-3z⁷ - a^-2 + 8a^-2z² - 12a^-2z⁴ + 3a^-2z⁶ + a^-2z⁸ - 2a^-1z + 2a^-1z³ - 4a^-1z⁵ + 3a^-1z⁷ + 1 - 4z⁴ + 4z⁶ - 3az³ + 4az⁵ - 2a²z² + 3a²z⁴ + a³z³

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=0 is the signature of 9₁₄. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

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

j = 13 1

j = 11 1

j = 9 2 1

j = 7 3 1

j = 5 3 2

j = 3 3 3

j = 1 3 3

j = -1 2 4

j = -3 1 2

j = -5 2

j = -7 1

n Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2 q^-9 - 3q^-8 + q^-7 + 6q^-6 - 11q^-5 + 5q^-4 + 13q^-3 - 23q^-2 + 9q^-1 + 21 - 33q + 9q² + 27q³ - 34q⁴ + 4q⁵ + 28q⁶ - 27q⁷ - 2q⁸ + 23q⁹ - 16q¹⁰ - 5q¹¹ + 15q¹² - 6q¹³ - 5q¹⁴ + 6q¹⁵ - q¹⁶ - 2q¹⁷ + q¹⁸

3 - q^-18 + 3q^-17 - q^-16 - 3q^-15 - q^-14 + 6q^-13 + q^-12 - 11q^-11 + 4q^-10 + 12q^-9 - 5q^-8 - 21q^-7 + 17q^-6 + 23q^-5 - 20q^-4 - 37q^-3 + 34q^-2 + 42q^-1 - 34 - 58q + 41q² + 61q³ - 33q⁴ - 71q⁵ + 29q⁶ + 72q⁷ - 18q⁸ - 73q⁹ + 8q¹⁰ + 70q¹¹ + 3q¹² - 65q¹³ - 14q¹⁴ + 60q¹⁵ + 21q¹⁶ - 49q¹⁷ - 30q¹⁸ + 41q¹⁹ + 32q²⁰ - 27q²¹ - 34q²² + 16q²³ + 30q²⁴ - 7q²⁵ - 23q²⁶ - q²⁷ + 17q²⁸ + 3q²⁹ - 9q³⁰ - 4q³¹ + 5q³² + 2q³³ - q³⁴ - 2q³⁵ + q³⁶

4 q^-30 - 3q^-29 + q^-28 + 3q^-27 - 2q^-26 + 6q^-25 - 12q^-24 + 3q^-23 + 5q^-22 - 8q^-21 + 23q^-20 - 25q^-19 + 8q^-18 + 2q^-17 - 29q^-16 + 48q^-15 - 23q^-14 + 28q^-13 - 14q^-12 - 82q^-11 + 69q^-10 + 3q^-9 + 80q^-8 - 30q^-7 - 170q^-6 + 61q^-5 + 35q^-4 + 164q^-3 - 20q^-2 - 256q^-1 + 23 + 34q + 240q² + 22q³ - 295q⁴ - 15q⁵ - 4q⁶ + 266q⁷ + 67q⁸ - 273q⁹ - 26q¹⁰ - 59q¹¹ + 243q¹² + 98q¹³ - 215q¹⁴ - 20q¹⁵ - 109q¹⁶ + 193q¹⁷ + 115q¹⁸ - 140q¹⁹ - 8q²⁰ - 152q²¹ + 128q²² + 124q²³ - 58q²⁴ + 12q²⁵ - 177q²⁶ + 52q²⁷ + 107q²⁸ + 14q²⁹ + 49q³⁰ - 166q³¹ - 18q³² + 58q³³ + 48q³⁴ + 86q³⁵ - 112q³⁶ - 50q³⁷ + 35q³⁹ + 93q⁴⁰ - 46q⁴¹ - 38q⁴² - 29q⁴³ + 3q⁴⁴ + 63q⁴⁵ - 5q⁴⁶ - 11q⁴⁷ - 23q⁴⁸ - 12q⁴⁹ + 26q⁵⁰ + 3q⁵¹ + 2q⁵² - 7q⁵³ - 8q⁵⁴ + 6q⁵⁵ + q⁵⁶ + 2q⁵⁷ - q⁵⁸ - 2q⁵⁹ + q⁶⁰

5 - q^-45 + 3q^-44 - q^-43 - 3q^-42 + 2q^-41 - 3q^-40 + 8q^-38 + 3q^-37 - 7q^-36 - 2q^-35 - 10q^-34 - 4q^-33 + 16q^-32 + 20q^-31 + 6q^-30 - 18q^-29 - 35q^-28 - 25q^-27 + 14q^-26 + 67q^-25 + 63q^-24 - 22q^-23 - 96q^-22 - 106q^-21 - 18q^-20 + 146q^-19 + 207q^-18 + 34q^-17 - 192q^-16 - 291q^-15 - 130q^-14 + 235q^-13 + 448q^-12 + 207q^-11 - 260q^-10 - 560q^-9 - 368q^-8 + 257q^-7 + 716q^-6 + 491q^-5 - 212q^-4 - 794q^-3 - 674q^-2 + 147q^-1 + 880 + 772q - 49q² - 868q³ - 899q⁴ - 38q⁵ + 866q⁶ + 922q⁷ + 120q⁸ - 787q⁹ - 951q¹⁰ - 183q¹¹ + 737q¹² + 906q¹³ + 225q¹⁴ - 644q¹⁵ - 869q¹⁶ - 251q¹⁷ + 568q¹⁸ + 802q¹⁹ + 273q²⁰ - 477q²¹ - 738q²² - 294q²³ + 386q²⁴ + 661q²⁵ + 314q²⁶ - 283q²⁷ - 575q²⁸ - 335q²⁹ + 175q³⁰ + 481q³¹ + 337q³² - 65q³³ - 360q³⁴ - 334q³⁵ - 39q³⁶ + 248q³⁷ + 290q³⁸ + 115q³⁹ - 108q⁴⁰ - 234q⁴¹ - 174q⁴² + 6q⁴³ + 145q⁴⁴ + 175q⁴⁵ + 101q⁴⁶ - 51q⁴⁷ - 159q⁴⁸ - 152q⁴⁹ - 36q⁵⁰ + 98q⁵¹ + 169q⁵² + 112q⁵³ - 34q⁵⁴ - 150q⁵⁵ - 145q⁵⁶ - 31q⁵⁷ + 99q⁵⁸ + 149q⁵⁹ + 81q⁶⁰ - 49q⁶¹ - 122q⁶² - 97q⁶³ - 4q⁶⁴ + 84q⁶⁵ + 93q⁶⁶ + 30q⁶⁷ - 41q⁶⁸ - 69q⁶⁹ - 46q⁷⁰ + 15q⁷¹ + 46q⁷² + 33q⁷³ + 6q⁷⁴ - 20q⁷⁵ - 29q⁷⁶ - 9q⁷⁷ + 12q⁷⁸ + 11q⁷⁹ + 7q⁸⁰ + 2q⁸¹ - 9q⁸² - 6q⁸³ + 2q⁸⁴ + 2q⁸⁵ + q⁸⁶ + 2q⁸⁷ - q⁸⁸ - 2q⁸⁹ + q⁹⁰

6 q^-63 - 3q^-62 + q^-61 + 3q^-60 - 2q^-59 + 3q^-58 - 3q^-57 + 4q^-56 - 14q^-55 - q^-54 + 17q^-53 - 5q^-52 + 14q^-51 - 3q^-50 + 4q^-49 - 49q^-48 - 11q^-47 + 41q^-46 + 2q^-45 + 50q^-44 + 14q^-43 + q^-42 - 139q^-41 - 58q^-40 + 67q^-39 + 33q^-38 + 155q^-37 + 87q^-36 - 9q^-35 - 323q^-34 - 207q^-33 + 64q^-32 + 117q^-31 + 409q^-30 + 291q^-29 - 20q^-28 - 664q^-27 - 574q^-26 - 49q^-25 + 276q^-24 + 919q^-23 + 765q^-22 + 47q^-21 - 1195q^-20 - 1290q^-19 - 438q^-18 + 415q^-17 + 1707q^-16 + 1636q^-15 + 391q^-14 - 1750q^-13 - 2327q^-12 - 1243q^-11 + 269q^-10 + 2515q^-9 + 2814q^-8 + 1165q^-7 - 1975q^-6 - 3321q^-5 - 2323q^-4 - 334q^-3 + 2909q^-2 + 3857q^-1 + 2192 - 1684q - 3810q² - 3219q³ - 1174q⁴ + 2736q⁵ + 4328q⁶ + 3007q⁷ - 1116q⁸ - 3697q⁹ - 3570q¹⁰ - 1817q¹¹ + 2258q¹² + 4210q¹³ + 3332q¹⁴ - 644q¹⁵ - 3262q¹⁶ - 3443q¹⁷ - 2085q¹⁸ + 1786q¹⁹ + 3795q²⁰ + 3297q²¹ - 352q²² - 2763q²³ - 3128q²⁴ - 2150q²⁵ + 1356q²⁶ + 3299q²⁷ + 3160q²⁸ - 69q²⁹ - 2212q³⁰ - 2784q³¹ - 2217q³² + 832q³³ + 2713q³⁴ + 3007q³⁵ + 326q³⁶ - 1514q³⁷ - 2349q³⁸ - 2283q³⁹ + 167q⁴⁰ + 1943q⁴¹ + 2720q⁴² + 748q⁴³ - 668q⁴⁴ - 1702q⁴⁵ - 2172q⁴⁶ - 494q⁴⁷ + 1001q⁴⁸ + 2146q⁴⁹ + 960q⁵⁰ + 151q⁵¹ - 848q⁵² - 1711q⁵³ - 879q⁵⁴ + 92q⁵⁵ + 1296q⁵⁶ + 765q⁵⁷ + 645q⁵⁸ - 15q⁵⁹ - 930q⁶⁰ - 784q⁶¹ - 453q⁶² + 436q⁶³ + 212q⁶⁴ + 615q⁶⁵ + 447q⁶⁶ - 148q⁶⁷ - 301q⁶⁸ - 447q⁶⁹ - 51q⁷⁰ - 341q⁷¹ + 195q⁷² + 381q⁷³ + 243q⁷⁴ + 170q⁷⁵ - 79q⁷⁶ - 37q⁷⁷ - 516q⁷⁸ - 189q⁷⁹ + 27q⁸⁰ + 170q⁸¹ + 285q⁸² + 214q⁸³ + 201q⁸⁴ - 311q⁸⁵ - 245q⁸⁶ - 200q⁸⁷ - 57q⁸⁸ + 108q⁸⁹ + 209q⁹⁰ + 295q⁹¹ - 49q⁹² - 86q⁹³ - 165q⁹⁴ - 133q⁹⁵ - 60q⁹⁶ + 62q⁹⁷ + 193q⁹⁸ + 43q⁹⁹ + 30q¹⁰⁰ - 48q¹⁰¹ - 68q¹⁰² - 79q¹⁰³ - 21q¹⁰⁴ + 70q¹⁰⁵ + 21q¹⁰⁶ + 36q¹⁰⁷ + 5q¹⁰⁸ - 9q¹⁰⁹ - 35q¹¹⁰ - 22q¹¹¹ + 16q¹¹² + 12q¹¹⁴ + 6q¹¹⁵ + 5q¹¹⁶ - 9q¹¹⁷ - 8q¹¹⁸ + 4q¹¹⁹ - 2q¹²⁰ + 2q¹²¹ + q¹²² + 2q¹²³ - q¹²⁴ - 2q¹²⁵ + q¹²⁶

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, 14]]

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

In[3]:=
GaussCode[Knot[9, 14]]

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

In[4]:=
DTCode[Knot[9, 14]]

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

In[5]:=
br = BR[Knot[9, 14]]

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

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

Out[6]=
{5, 10}

In[7]:=
BraidIndex[Knot[9, 14]]

Out[7]=
5

In[8]:=
Show[DrawMorseLink[Knot[9, 14]]]

Out[8]=
-Graphics-

In[9]:=
#[Knot[9, 14]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}

Out[9]=
{Reversible, 1, 2, 2, {4, 7}, 1}

In[10]:=
alex = Alexander[Knot[9, 14]][t]

Out[10]=
2 9 2 15 + -- - - - 9 t + 2 t 2 t t

In[11]:=
Conway[Knot[9, 14]][z]

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

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

Out[12]=
{Knot[9, 14]}

In[13]:=
{KnotDet[Knot[9, 14]], KnotSignature[Knot[9, 14]]}

Out[13]=
{37, 0}

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

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

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

Out[15]=
{Knot[9, 14], Knot[11, NonAlternating, 53]}

In[16]:=
A2Invariant[Knot[9, 14]][q]

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

In[17]:=
HOMFLYPT[Knot[9, 14]][a, z]

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

In[18]:=
Kauffman[Knot[9, 14]][a, z]

Out[18]=
2 2 2 3 -6 2 -2 3 z 5 z 2 z 4 z 10 z 8 z 2 2 9 z 1 - a - -- - a - --- - --- - --- + ---- + ----- + ---- - 2 a z + ---- + 4 5 3 a 6 4 2 5 a a a a a a a 3 3 4 4 4 15 z 2 z 3 3 3 4 4 z 9 z 12 z 2 4 > ----- + ---- - 3 a z + a z - 4 z - ---- - ---- - ----- + 3 a z - 3 a 6 4 2 a a a a 5 5 5 6 6 7 7 7 8 8 z 16 z 4 z 5 6 z 3 z 2 z 5 z 3 z z > ---- - ----- - ---- + 4 a z + 4 z + -- + ---- + ---- + ---- + ---- + -- + 5 3 a 6 2 5 3 a 4 a a a a a a a 8 z > -- 2 a

In[19]:=
{Vassiliev[2][Knot[9, 14]], Vassiliev[3][Knot[9, 14]]}

Out[19]=
{-1, -2}

In[20]:=
Kh[Knot[9, 14]][q, t]

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

In[21]:=
ColouredJones[Knot[9, 14], 2][q]

Out[21]=
-9 3 -7 6 11 5 13 23 9 2 3 21 + q - -- + q + -- - -- + -- + -- - -- + - - 33 q + 9 q + 27 q - 8 6 5 4 3 2 q q q q q q q 4 5 6 7 8 9 10 11 12 > 34 q + 4 q + 28 q - 27 q - 2 q + 23 q - 16 q - 5 q + 15 q - 13 14 15 16 17 18 > 6 q - 5 q + 6 q - q - 2 q + q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 9₁₄

9₁₃
9₁₅

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q^-9 - 3q^-8 + q^-7 + 6q^-6 - 11q^-5 + 5q^-4 + 13q^-3 - 23q^-2 + 9q^-1 + 21 - 33q + 9q² + 27q³ - 34q⁴ + 4q⁵ + 28q⁶ - 27q⁷ - 2q⁸ + 23q⁹ - 16q¹⁰ - 5q¹¹ + 15q¹² - 6q¹³ - 5q¹⁴ + 6q¹⁵ - q¹⁶ - 2q¹⁷ + q¹⁸
3	- q^-18 + 3q^-17 - q^-16 - 3q^-15 - q^-14 + 6q^-13 + q^-12 - 11q^-11 + 4q^-10 + 12q^-9 - 5q^-8 - 21q^-7 + 17q^-6 + 23q^-5 - 20q^-4 - 37q^-3 + 34q^-2 + 42q^-1 - 34 - 58q + 41q² + 61q³ - 33q⁴ - 71q⁵ + 29q⁶ + 72q⁷ - 18q⁸ - 73q⁹ + 8q¹⁰ + 70q¹¹ + 3q¹² - 65q¹³ - 14q¹⁴ + 60q¹⁵ + 21q¹⁶ - 49q¹⁷ - 30q¹⁸ + 41q¹⁹ + 32q²⁰ - 27q²¹ - 34q²² + 16q²³ + 30q²⁴ - 7q²⁵ - 23q²⁶ - q²⁷ + 17q²⁸ + 3q²⁹ - 9q³⁰ - 4q³¹ + 5q³² + 2q³³ - q³⁴ - 2q³⁵ + q³⁶
4	q^-30 - 3q^-29 + q^-28 + 3q^-27 - 2q^-26 + 6q^-25 - 12q^-24 + 3q^-23 + 5q^-22 - 8q^-21 + 23q^-20 - 25q^-19 + 8q^-18 + 2q^-17 - 29q^-16 + 48q^-15 - 23q^-14 + 28q^-13 - 14q^-12 - 82q^-11 + 69q^-10 + 3q^-9 + 80q^-8 - 30q^-7 - 170q^-6 + 61q^-5 + 35q^-4 + 164q^-3 - 20q^-2 - 256q^-1 + 23 + 34q + 240q² + 22q³ - 295q⁴ - 15q⁵ - 4q⁶ + 266q⁷ + 67q⁸ - 273q⁹ - 26q¹⁰ - 59q¹¹ + 243q¹² + 98q¹³ - 215q¹⁴ - 20q¹⁵ - 109q¹⁶ + 193q¹⁷ + 115q¹⁸ - 140q¹⁹ - 8q²⁰ - 152q²¹ + 128q²² + 124q²³ - 58q²⁴ + 12q²⁵ - 177q²⁶ + 52q²⁷ + 107q²⁸ + 14q²⁹ + 49q³⁰ - 166q³¹ - 18q³² + 58q³³ + 48q³⁴ + 86q³⁵ - 112q³⁶ - 50q³⁷ + 35q³⁹ + 93q⁴⁰ - 46q⁴¹ - 38q⁴² - 29q⁴³ + 3q⁴⁴ + 63q⁴⁵ - 5q⁴⁶ - 11q⁴⁷ - 23q⁴⁸ - 12q⁴⁹ + 26q⁵⁰ + 3q⁵¹ + 2q⁵² - 7q⁵³ - 8q⁵⁴ + 6q⁵⁵ + q⁵⁶ + 2q⁵⁷ - q⁵⁸ - 2q⁵⁹ + q⁶⁰
5	- q^-45 + 3q^-44 - q^-43 - 3q^-42 + 2q^-41 - 3q^-40 + 8q^-38 + 3q^-37 - 7q^-36 - 2q^-35 - 10q^-34 - 4q^-33 + 16q^-32 + 20q^-31 + 6q^-30 - 18q^-29 - 35q^-28 - 25q^-27 + 14q^-26 + 67q^-25 + 63q^-24 - 22q^-23 - 96q^-22 - 106q^-21 - 18q^-20 + 146q^-19 + 207q^-18 + 34q^-17 - 192q^-16 - 291q^-15 - 130q^-14 + 235q^-13 + 448q^-12 + 207q^-11 - 260q^-10 - 560q^-9 - 368q^-8 + 257q^-7 + 716q^-6 + 491q^-5 - 212q^-4 - 794q^-3 - 674q^-2 + 147q^-1 + 880 + 772q - 49q² - 868q³ - 899q⁴ - 38q⁵ + 866q⁶ + 922q⁷ + 120q⁸ - 787q⁹ - 951q¹⁰ - 183q¹¹ + 737q¹² + 906q¹³ + 225q¹⁴ - 644q¹⁵ - 869q¹⁶ - 251q¹⁷ + 568q¹⁸ + 802q¹⁹ + 273q²⁰ - 477q²¹ - 738q²² - 294q²³ + 386q²⁴ + 661q²⁵ + 314q²⁶ - 283q²⁷ - 575q²⁸ - 335q²⁹ + 175q³⁰ + 481q³¹ + 337q³² - 65q³³ - 360q³⁴ - 334q³⁵ - 39q³⁶ + 248q³⁷ + 290q³⁸ + 115q³⁹ - 108q⁴⁰ - 234q⁴¹ - 174q⁴² + 6q⁴³ + 145q⁴⁴ + 175q⁴⁵ + 101q⁴⁶ - 51q⁴⁷ - 159q⁴⁸ - 152q⁴⁹ - 36q⁵⁰ + 98q⁵¹ + 169q⁵² + 112q⁵³ - 34q⁵⁴ - 150q⁵⁵ - 145q⁵⁶ - 31q⁵⁷ + 99q⁵⁸ + 149q⁵⁹ + 81q⁶⁰ - 49q⁶¹ - 122q⁶² - 97q⁶³ - 4q⁶⁴ + 84q⁶⁵ + 93q⁶⁶ + 30q⁶⁷ - 41q⁶⁸ - 69q⁶⁹ - 46q⁷⁰ + 15q⁷¹ + 46q⁷² + 33q⁷³ + 6q⁷⁴ - 20q⁷⁵ - 29q⁷⁶ - 9q⁷⁷ + 12q⁷⁸ + 11q⁷⁹ + 7q⁸⁰ + 2q⁸¹ - 9q⁸² - 6q⁸³ + 2q⁸⁴ + 2q⁸⁵ + q⁸⁶ + 2q⁸⁷ - q⁸⁸ - 2q⁸⁹ + q⁹⁰
6	q^-63 - 3q^-62 + q^-61 + 3q^-60 - 2q^-59 + 3q^-58 - 3q^-57 + 4q^-56 - 14q^-55 - q^-54 + 17q^-53 - 5q^-52 + 14q^-51 - 3q^-50 + 4q^-49 - 49q^-48 - 11q^-47 + 41q^-46 + 2q^-45 + 50q^-44 + 14q^-43 + q^-42 - 139q^-41 - 58q^-40 + 67q^-39 + 33q^-38 + 155q^-37 + 87q^-36 - 9q^-35 - 323q^-34 - 207q^-33 + 64q^-32 + 117q^-31 + 409q^-30 + 291q^-29 - 20q^-28 - 664q^-27 - 574q^-26 - 49q^-25 + 276q^-24 + 919q^-23 + 765q^-22 + 47q^-21 - 1195q^-20 - 1290q^-19 - 438q^-18 + 415q^-17 + 1707q^-16 + 1636q^-15 + 391q^-14 - 1750q^-13 - 2327q^-12 - 1243q^-11 + 269q^-10 + 2515q^-9 + 2814q^-8 + 1165q^-7 - 1975q^-6 - 3321q^-5 - 2323q^-4 - 334q^-3 + 2909q^-2 + 3857q^-1 + 2192 - 1684q - 3810q² - 3219q³ - 1174q⁴ + 2736q⁵ + 4328q⁶ + 3007q⁷ - 1116q⁸ - 3697q⁹ - 3570q¹⁰ - 1817q¹¹ + 2258q¹² + 4210q¹³ + 3332q¹⁴ - 644q¹⁵ - 3262q¹⁶ - 3443q¹⁷ - 2085q¹⁸ + 1786q¹⁹ + 3795q²⁰ + 3297q²¹ - 352q²² - 2763q²³ - 3128q²⁴ - 2150q²⁵ + 1356q²⁶ + 3299q²⁷ + 3160q²⁸ - 69q²⁹ - 2212q³⁰ - 2784q³¹ - 2217q³² + 832q³³ + 2713q³⁴ + 3007q³⁵ + 326q³⁶ - 1514q³⁷ - 2349q³⁸ - 2283q³⁹ + 167q⁴⁰ + 1943q⁴¹ + 2720q⁴² + 748q⁴³ - 668q⁴⁴ - 1702q⁴⁵ - 2172q⁴⁶ - 494q⁴⁷ + 1001q⁴⁸ + 2146q⁴⁹ + 960q⁵⁰ + 151q⁵¹ - 848q⁵² - 1711q⁵³ - 879q⁵⁴ + 92q⁵⁵ + 1296q⁵⁶ + 765q⁵⁷ + 645q⁵⁸ - 15q⁵⁹ - 930q⁶⁰ - 784q⁶¹ - 453q⁶² + 436q⁶³ + 212q⁶⁴ + 615q⁶⁵ + 447q⁶⁶ - 148q⁶⁷ - 301q⁶⁸ - 447q⁶⁹ - 51q⁷⁰ - 341q⁷¹ + 195q⁷² + 381q⁷³ + 243q⁷⁴ + 170q⁷⁵ - 79q⁷⁶ - 37q⁷⁷ - 516q⁷⁸ - 189q⁷⁹ + 27q⁸⁰ + 170q⁸¹ + 285q⁸² + 214q⁸³ + 201q⁸⁴ - 311q⁸⁵ - 245q⁸⁶ - 200q⁸⁷ - 57q⁸⁸ + 108q⁸⁹ + 209q⁹⁰ + 295q⁹¹ - 49q⁹² - 86q⁹³ - 165q⁹⁴ - 133q⁹⁵ - 60q⁹⁶ + 62q⁹⁷ + 193q⁹⁸ + 43q⁹⁹ + 30q¹⁰⁰ - 48q¹⁰¹ - 68q¹⁰² - 79q¹⁰³ - 21q¹⁰⁴ + 70q¹⁰⁵ + 21q¹⁰⁶ + 36q¹⁰⁷ + 5q¹⁰⁸ - 9q¹⁰⁹ - 35q¹¹⁰ - 22q¹¹¹ + 16q¹¹² + 12q¹¹⁴ + 6q¹¹⁵ + 5q¹¹⁶ - 9q¹¹⁷ - 8q¹¹⁸ + 4q¹¹⁹ - 2q¹²⁰ + 2q¹²¹ + q¹²² + 2q¹²³ - q¹²⁴ - 2q¹²⁵ + q¹²⁶

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

Out[8]=	-Graphics-
In[9]:=	#[Knot[9, 14]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 1, 2, 2, {4, 7}, 1}
In[10]:=	alex = Alexander[Knot[9, 14]][t]
Out[10]=	2 9 2 15 + -- - - - 9 t + 2 t 2 t t
In[11]:=	Conway[Knot[9, 14]][z]
Out[11]=	2 4 1 - z + 2 z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[9, 14]}
In[13]:=	{KnotDet[Knot[9, 14]], KnotSignature[Knot[9, 14]]}
Out[13]=	{37, 0}
In[14]:=	Jones[Knot[9, 14]][q]
Out[14]=	-3 3 4 2 3 4 5 6 6 - q + -- - - - 6 q + 6 q - 5 q + 3 q - 2 q + q 2 q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] \|\| (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[9, 14], Knot[11, NonAlternating, 53]}
In[16]:=	A2Invariant[Knot[9, 14]][q]
Out[16]=	-10 -8 -6 -4 2 2 4 8 10 12 16 18 20 -q + q + q - q + -- + q + q + q - 2 q - q - q + q + q 2 q
In[17]:=	HOMFLYPT[Knot[9, 14]][a, z]
Out[17]=	2 2 4 -6 2 -2 2 2 z z 2 2 4 z 1 + a - -- + a + z - ---- + -- - a z + z + -- 4 4 2 2 a a a a
In[18]:=	Kauffman[Knot[9, 14]][a, z]
Out[18]=	2 2 2 3 -6 2 -2 3 z 5 z 2 z 4 z 10 z 8 z 2 2 9 z 1 - a - -- - a - --- - --- - --- + ---- + ----- + ---- - 2 a z + ---- + 4 5 3 a 6 4 2 5 a a a a a a a 3 3 4 4 4 15 z 2 z 3 3 3 4 4 z 9 z 12 z 2 4 > ----- + ---- - 3 a z + a z - 4 z - ---- - ---- - ----- + 3 a z - 3 a 6 4 2 a a a a 5 5 5 6 6 7 7 7 8 8 z 16 z 4 z 5 6 z 3 z 2 z 5 z 3 z z > ---- - ----- - ---- + 4 a z + 4 z + -- + ---- + ---- + ---- + ---- + -- + 5 3 a 6 2 5 3 a 4 a a a a a a a 8 z > -- 2 a
In[19]:=	{Vassiliev[2][Knot[9, 14]], Vassiliev[3][Knot[9, 14]]}
Out[19]=	{-1, -2}
In[20]:=	Kh[Knot[9, 14]][q, t]
Out[20]=	4 1 2 1 2 2 3 3 2 - + 3 q + ----- + ----- + ----- + ---- + --- + 3 q t + 3 q t + 3 q t + q 7 3 5 2 3 2 3 q t q t q t q t q t 5 2 5 3 7 3 7 4 9 4 9 5 11 5 13 6 > 3 q t + 2 q t + 3 q t + q t + 2 q t + q t + q t + q t
In[21]:=	ColouredJones[Knot[9, 14], 2][q]
Out[21]=	-9 3 -7 6 11 5 13 23 9 2 3 21 + q - -- + q + -- - -- + -- + -- - -- + - - 33 q + 9 q + 27 q - 8 6 5 4 3 2 q q q q q q q 4 5 6 7 8 9 10 11 12 > 34 q + 4 q + 28 q - 27 q - 2 q + 23 q - 16 q - 5 q + 15 q - 13 14 15 16 17 18 > 6 q - 5 q + 6 q - q - 2 q + q

The Alternating Knot 914

The Alternating Knot 9₁₄