The Non Alternating Knot 9₄₇

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Acknowledgement

KnotPlot

PD Presentation: X₆₂₇₁ X_16,8,17,7 X₈₃₉₄ X_2,15,3,16 X_14,9,15,10 X_10,6,11,5 X_4,14,5,13 X_11,1,12,18 X_17,13,18,12

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

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

Alexander Polynomial: t^-3 - 4t^-2 + 6t^-1 - 5 + 6t - 4t² + t³

Conway Polynomial: 1 - z² + 2z⁴ + z⁶

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

Determinant and Signature: {27, 2}

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

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

A2 (sl(3)) Invariant: - q^-6 + q^-4 + q^-2 + 2 + 2q² - q⁴ + q⁶ - 2q⁸ - q¹² - q¹⁴ + q¹⁶ + q²⁰

HOMFLY-PT Polynomial: a^-6 - 2a^-4 - 3a^-4z² - a^-4z⁴ + a^-2 + 4a^-2z² + 4a^-2z⁴ + a^-2z⁶ + 1 - 2z² - z⁴

Kauffman Polynomial: - a^-6 + 3a^-6z² - 3a^-5z + 3a^-5z³ + a^-5z⁵ - 2a^-4 + 9a^-4z² - 7a^-4z⁴ + 3a^-4z⁶ - 5a^-3z + 6a^-3z³ - 4a^-3z⁵ + 2a^-3z⁷ - a^-2 + 11a^-2z² - 16a^-2z⁴ + 6a^-2z⁶ - 2a^-1z + a^-1z³ - 4a^-1z⁵ + 2a^-1z⁷ + 1 + 5z² - 9z⁴ + 3z⁶ - 2az³ + az⁵

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=2 is the signature of 9₄₇. 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

j = 11 2

j = 9 2

j = 7 2 2

j = 5 3 2

j = 3 2 2

j = 1 2 4

j = -1 1 1

j = -3 2

j = -5 1

n Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2 q^-7 - 3q^-6 - q^-5 + 9q^-4 - 6q^-3 - 8q^-2 + 17q^-1 - 3 - 18q + 20q² + 2q³ - 25q⁴ + 19q⁵ + 8q⁶ - 25q⁷ + 14q⁸ + 10q⁹ - 17q¹⁰ + 6q¹¹ + 6q¹² - 6q¹³ + q¹⁵

3 - q^-15 + 3q^-14 + q^-13 - 5q^-12 - 7q^-11 + 6q^-10 + 17q^-9 - 5q^-8 - 23q^-7 - 6q^-6 + 32q^-5 + 17q^-4 - 28q^-3 - 35q^-2 + 26q^-1 + 43 - 10q - 60q² + 3q³ + 60q⁴ + 16q⁵ - 69q⁶ - 25q⁷ + 70q⁸ + 37q⁹ - 70q¹⁰ - 46q¹¹ + 69q¹² + 51q¹³ - 58q¹⁴ - 59q¹⁵ + 50q¹⁶ + 54q¹⁷ - 32q¹⁸ - 51q¹⁹ + 19q²⁰ + 37q²¹ - 4q²² - 25q²³ - 3q²⁴ + 15q²⁵ + q²⁶ - 2q²⁷ - 4q²⁸ + 2q²⁹

4 q^-26 - 3q^-25 - q^-24 + 5q^-23 + 3q^-22 + 7q^-21 - 16q^-20 - 15q^-19 + 3q^-18 + 11q^-17 + 46q^-16 - 12q^-15 - 38q^-14 - 33q^-13 - 21q^-12 + 88q^-11 + 39q^-10 - 3q^-9 - 60q^-8 - 115q^-7 + 56q^-6 + 77q^-5 + 98q^-4 - 7q^-3 - 192q^-2 - 47q^-1 + 32 + 190q + 115q² - 193q³ - 149q⁴ - 72q⁵ + 227q⁶ + 236q⁷ - 145q⁸ - 214q⁹ - 173q¹⁰ + 228q¹¹ + 321q¹² - 92q¹³ - 254q¹⁴ - 245q¹⁵ + 217q¹⁶ + 374q¹⁷ - 38q¹⁸ - 276q¹⁹ - 297q²⁰ + 185q²¹ + 392q²² + 28q²³ - 248q²⁴ - 319q²⁵ + 100q²⁶ + 344q²⁷ + 100q²⁸ - 150q²⁹ - 278q³⁰ - 7q³¹ + 215q³² + 118q³³ - 30q³⁴ - 161q³⁵ - 58q³⁶ + 73q³⁷ + 68q³⁸ + 23q³⁹ - 46q⁴⁰ - 34q⁴¹ + 5q⁴² + 14q⁴³ + 12q⁴⁴ - 3q⁴⁵ - 6q⁴⁶ + q⁴⁸

5 - q^-40 + 3q^-39 + q^-38 - 5q^-37 - 3q^-36 - 3q^-35 + 3q^-34 + 14q^-33 + 18q^-32 - 5q^-31 - 24q^-30 - 33q^-29 - 21q^-28 + 17q^-27 + 61q^-26 + 66q^-25 + 5q^-24 - 61q^-23 - 101q^-22 - 82q^-21 + 17q^-20 + 132q^-19 + 153q^-18 + 71q^-17 - 71q^-16 - 209q^-15 - 213q^-14 - 34q^-13 + 192q^-12 + 318q^-11 + 224q^-10 - 78q^-9 - 392q^-8 - 410q^-7 - 114q^-6 + 336q^-5 + 593q^-4 + 367q^-3 - 224q^-2 - 674q^-1 - 619 - 15q + 714q² + 858q³ + 231q⁴ - 630q⁵ - 1028q⁶ - 516q⁷ + 546q⁸ + 1161q⁹ + 721q¹⁰ - 404q¹¹ - 1226q¹² - 944q¹³ + 290q¹⁴ + 1287q¹⁵ + 1083q¹⁶ - 173q¹⁷ - 1314q¹⁸ - 1221q¹⁹ + 84q²⁰ + 1349q²¹ + 1323q²² - 15q²³ - 1369q²⁴ - 1406q²⁵ - 65q²⁶ + 1373q²⁷ + 1495q²⁸ + 145q²⁹ - 1351q³⁰ - 1542q³¹ - 270q³² + 1268q³³ + 1590q³⁴ + 397q³⁵ - 1124q³⁶ - 1549q³⁷ - 563q³⁸ + 904q³⁹ + 1466q⁴⁰ + 675q⁴¹ - 637q⁴² - 1255q⁴³ - 755q⁴⁴ + 338q⁴⁵ + 1010q⁴⁶ + 727q⁴⁷ - 95q⁴⁸ - 691q⁴⁹ - 626q⁵⁰ - 83q⁵¹ + 404q⁵² + 472q⁵³ + 154q⁵⁴ - 192q⁵⁵ - 283q⁵⁶ - 151q⁵⁷ + 42q⁵⁸ + 154q⁵⁹ + 111q⁶⁰ - 8q⁶¹ - 49q⁶² - 51q⁶³ - 21q⁶⁴ + 19q⁶⁵ + 21q⁶⁶ + 3q⁶⁷ - 2q⁶⁸ - 2q⁶⁹ - 4q⁷⁰ + 2q⁷¹

6 q^-57 - 3q^-56 - q^-55 + 5q^-54 + 3q^-53 + 3q^-52 - 7q^-51 - q^-50 - 17q^-49 - 16q^-48 + 17q^-47 + 25q^-46 + 39q^-45 + 10q^-44 + 12q^-43 - 62q^-42 - 94q^-41 - 48q^-40 - 2q^-39 + 89q^-38 + 103q^-37 + 186q^-36 + 28q^-35 - 118q^-34 - 196q^-33 - 236q^-32 - 124q^-31 - 15q^-30 + 360q^-29 + 368q^-28 + 284q^-27 + 48q^-26 - 284q^-25 - 536q^-24 - 684q^-23 - 126q^-22 + 271q^-21 + 759q^-20 + 881q^-19 + 578q^-18 - 195q^-17 - 1184q^-16 - 1199q^-15 - 917q^-14 + 161q^-13 + 1247q^-12 + 1925q^-11 + 1362q^-10 - 320q^-9 - 1547q^-8 - 2423q^-7 - 1695q^-6 + 68q^-5 + 2319q^-4 + 3078q^-3 + 1775q^-2 - 290q^-1 - 2831 - 3594q - 2236q² + 1147q³ + 3677q⁴ + 3828q⁵ + 1944q⁶ - 1809q⁷ - 4462q⁸ - 4434q⁹ - 844q¹⁰ + 3060q¹¹ + 4979q¹² + 4030q¹³ - 152q¹⁴ - 4346q¹⁵ - 5850q¹⁶ - 2664q¹⁷ + 1991q¹⁸ + 5347q¹⁹ + 5430q²⁰ + 1292q²¹ - 3894q²² - 6593q²³ - 3892q²⁴ + 1127q²⁵ + 5439q²⁶ + 6240q²⁷ + 2217q²⁸ - 3574q²⁹ - 7027q³⁰ - 4638q³¹ + 612q³² + 5549q³³ + 6777q³⁴ + 2818q³⁵ - 3392q³⁶ - 7358q³⁷ - 5253q³⁸ + 139q³⁹ + 5580q⁴⁰ + 7244q⁴¹ + 3523q⁴² - 2940q⁴³ - 7440q⁴⁴ - 5949q⁴⁵ - 767q⁴⁶ + 5032q⁴⁷ + 7418q⁴⁸ + 4511q⁴⁹ - 1701q⁵⁰ - 6680q⁵¹ - 6367q⁵² - 2186q⁵³ + 3386q⁵⁴ + 6574q⁵⁵ + 5213q⁵⁶ + 244q⁵⁷ - 4594q⁵⁸ - 5637q⁵⁹ - 3303q⁶⁰ + 970q⁶¹ + 4325q⁶² + 4634q⁶³ + 1792q⁶⁴ - 1819q⁶⁵ - 3524q⁶⁶ - 3059q⁶⁷ - 829q⁶⁸ + 1631q⁶⁹ + 2748q⁷⁰ + 1873q⁷¹ + 77q⁷² - 1225q⁷³ - 1656q⁷⁴ - 1075q⁷⁵ + 32q⁷⁶ + 897q⁷⁷ + 935q⁷⁸ + 445q⁷⁹ - 65q⁸⁰ - 434q⁸¹ - 477q⁸² - 220q⁸³ + 92q⁸⁴ + 204q⁸⁵ + 162q⁸⁶ + 78q⁸⁷ - 18q⁸⁸ - 83q⁸⁹ - 64q⁹⁰ - 7q⁹¹ + 12q⁹² + 14q⁹³ + 11q⁹⁴ + 6q⁹⁵ - 3q⁹⁶ - 6q⁹⁷ + q⁹⁹

7 - q^-77 + 3q^-76 + q^-75 - 5q^-74 - 3q^-73 - 3q^-72 + 7q^-71 + 5q^-70 + 4q^-69 + 15q^-68 + 4q^-67 - 18q^-66 - 30q^-65 - 42q^-64 - 12q^-63 + 21q^-62 + 32q^-61 + 90q^-60 + 84q^-59 + 42q^-58 - 30q^-57 - 152q^-56 - 175q^-55 - 140q^-54 - 96q^-53 + 88q^-52 + 236q^-51 + 332q^-50 + 360q^-49 + 94q^-48 - 146q^-47 - 368q^-46 - 605q^-45 - 529q^-44 - 303q^-43 + 138q^-42 + 734q^-41 + 926q^-40 + 907q^-39 + 579q^-38 - 256q^-37 - 970q^-36 - 1589q^-35 - 1648q^-34 - 772q^-33 + 318q^-32 + 1629q^-31 + 2579q^-30 + 2355q^-29 + 1299q^-28 - 721q^-27 - 2888q^-26 - 3781q^-25 - 3472q^-24 - 1381q^-23 + 1803q^-22 + 4371q^-21 + 5691q^-20 + 4387q^-19 + 670q^-18 - 3494q^-17 - 6944q^-16 - 7487q^-15 - 4385q^-14 + 794q^-13 + 6668q^-12 + 9890q^-11 + 8477q^-10 + 3309q^-9 - 4404q^-8 - 10696q^-7 - 12190q^-6 - 8266q^-5 + 496q^-4 + 9688q^-3 + 14552q^-2 + 13133q^-1 + 4661 - 6824q - 15407q² - 17269q³ - 10035q⁴ + 2675q⁵ + 14458q⁶ + 20080q⁷ + 15299q⁸ + 2229q⁹ - 12361q¹⁰ - 21648q¹¹ - 19589q¹² - 7100q¹³ + 9245q¹⁴ + 21943q¹⁵ + 23066q¹⁶ + 11708q¹⁷ - 6000q¹⁸ - 21539q¹⁹ - 25433q²⁰ - 15509q²¹ + 2746q²² + 20527q²³ + 27138q²⁴ + 18678q²⁵ + 19q²⁶ - 19533q²⁷ - 28182q²⁸ - 20988q²⁹ - 2320q³⁰ + 18523q³¹ + 28917q³² + 22811q³³ + 4011q³⁴ - 17828q³⁵ - 29454q³⁶ - 24091q³⁷ - 5240q³⁸ + 17342q³⁹ + 29948q⁴⁰ + 25133q⁴¹ + 6142q⁴² - 17103q⁴³ - 30510q⁴⁴ - 26075q⁴⁵ - 6870q⁴⁶ + 16987q⁴⁷ + 31078q⁴⁸ + 27068q⁴⁹ + 7756q⁵⁰ - 16723q⁵¹ - 31718q⁵² - 28287q⁵³ - 8934q⁵⁴ + 16131q⁵⁵ + 32091q⁵⁶ + 29642q⁵⁷ + 10730q⁵⁸ - 14775q⁵⁹ - 32019q⁶⁰ - 31045q⁶¹ - 13064q⁶² + 12461q⁶³ + 30941q⁶⁴ + 32023q⁶⁵ + 15921q⁶⁶ - 8988q⁶⁷ - 28605q⁶⁸ - 32143q⁶⁹ - 18666q⁷⁰ + 4578q⁷¹ + 24610q⁷² + 30785q⁷³ + 20885q⁷⁴ + 351q⁷⁵ - 19347q⁷⁶ - 27727q⁷⁷ - 21648q⁷⁸ - 4888q⁷⁹ + 13091q⁸⁰ + 22927q⁸¹ + 20776q⁸² + 8363q⁸³ - 6983q⁸⁴ - 17161q⁸⁵ - 17999q⁸⁶ - 9990q⁸⁷ + 1803q⁸⁸ + 11087q⁸⁹ + 13991q⁹⁰ + 9855q⁹¹ + 1670q⁹² - 5881q⁹³ - 9574q⁹⁴ - 8154q⁹⁵ - 3244q⁹⁶ + 2098q⁹⁷ + 5505q⁹⁸ + 5762q⁹⁹ + 3396q¹⁰⁰ + 49q¹⁰¹ - 2611q¹⁰² - 3461q¹⁰³ - 2516q¹⁰⁴ - 812q¹⁰⁵ + 791q¹⁰⁶ + 1653q¹⁰⁷ + 1567q¹⁰⁸ + 846q¹⁰⁹ - 75q¹¹⁰ - 649q¹¹¹ - 725q¹¹² - 489q¹¹³ - 160q¹¹⁴ + 144q¹¹⁵ + 278q¹¹⁶ + 246q¹¹⁷ + 109q¹¹⁸ - 28q¹¹⁹ - 68q¹²⁰ - 70q¹²¹ - 47q¹²² - 13q¹²³ + 15q¹²⁴ + 19q¹²⁵ + 15q¹²⁶ - 2q¹²⁸ - 2q¹²⁹ - 4q¹³⁰ + 2q¹³¹

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, 47]]

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

In[3]:=
GaussCode[Knot[9, 47]]

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

In[4]:=
DTCode[Knot[9, 47]]

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

In[5]:=
br = BR[Knot[9, 47]]

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

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

Out[6]=
{4, 9}

In[7]:=
BraidIndex[Knot[9, 47]]

Out[7]=
4

In[8]:=
Show[DrawMorseLink[Knot[9, 47]]]

Out[8]=
-Graphics-

In[9]:=
#[Knot[9, 47]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}

Out[9]=
{Reversible, 2, 3, 3, {4, 6}, 2}

In[10]:=
alex = Alexander[Knot[9, 47]][t]

Out[10]=
-3 4 6 2 3 -5 + t - -- + - + 6 t - 4 t + t 2 t t

In[11]:=
Conway[Knot[9, 47]][z]

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

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

Out[12]=
{Knot[9, 47]}

In[13]:=
{KnotDet[Knot[9, 47]], KnotSignature[Knot[9, 47]]}

Out[13]=
{27, 2}

In[14]:=
Jones[Knot[9, 47]][q]

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

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

Out[15]=
{Knot[9, 47]}

In[16]:=
A2Invariant[Knot[9, 47]][q]

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

In[17]:=
HOMFLYPT[Knot[9, 47]][a, z]

Out[17]=
2 2 4 4 6 -6 2 -2 2 3 z 4 z 4 z 4 z z 1 + a - -- + a - 2 z - ---- + ---- - z - -- + ---- + -- 4 4 2 4 2 2 a a a a a a

In[18]:=
Kauffman[Knot[9, 47]][a, z]

Out[18]=
2 2 2 3 -6 2 -2 3 z 5 z 2 z 2 3 z 9 z 11 z 3 z 1 - a - -- - a - --- - --- - --- + 5 z + ---- + ---- + ----- + ---- + 4 5 3 a 6 4 2 5 a a a a a a a 3 3 4 4 5 5 5 6 z z 3 4 7 z 16 z z 4 z 4 z 5 6 > ---- + -- - 2 a z - 9 z - ---- - ----- + -- - ---- - ---- + a z + 3 z + 3 a 4 2 5 3 a a a a a a 6 6 7 7 3 z 6 z 2 z 2 z > ---- + ---- + ---- + ---- 4 2 3 a a a a

In[19]:=
{Vassiliev[2][Knot[9, 47]], Vassiliev[3][Knot[9, 47]]}

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

In[20]:=
Kh[Knot[9, 47]][q, t]

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

In[21]:=
ColouredJones[Knot[9, 47], 2][q]

Out[21]=
-7 3 -5 9 6 8 17 2 3 4 5 -3 + q - -- - q + -- - -- - -- + -- - 18 q + 20 q + 2 q - 25 q + 19 q + 6 4 3 2 q q q q q 6 7 8 9 10 11 12 13 15 > 8 q - 25 q + 14 q + 10 q - 17 q + 6 q + 6 q - 6 q + q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 9₄₇

9₄₆
9₄₈

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q^-7 - 3q^-6 - q^-5 + 9q^-4 - 6q^-3 - 8q^-2 + 17q^-1 - 3 - 18q + 20q² + 2q³ - 25q⁴ + 19q⁵ + 8q⁶ - 25q⁷ + 14q⁸ + 10q⁹ - 17q¹⁰ + 6q¹¹ + 6q¹² - 6q¹³ + q¹⁵
3	- q^-15 + 3q^-14 + q^-13 - 5q^-12 - 7q^-11 + 6q^-10 + 17q^-9 - 5q^-8 - 23q^-7 - 6q^-6 + 32q^-5 + 17q^-4 - 28q^-3 - 35q^-2 + 26q^-1 + 43 - 10q - 60q² + 3q³ + 60q⁴ + 16q⁵ - 69q⁶ - 25q⁷ + 70q⁸ + 37q⁹ - 70q¹⁰ - 46q¹¹ + 69q¹² + 51q¹³ - 58q¹⁴ - 59q¹⁵ + 50q¹⁶ + 54q¹⁷ - 32q¹⁸ - 51q¹⁹ + 19q²⁰ + 37q²¹ - 4q²² - 25q²³ - 3q²⁴ + 15q²⁵ + q²⁶ - 2q²⁷ - 4q²⁸ + 2q²⁹
4	q^-26 - 3q^-25 - q^-24 + 5q^-23 + 3q^-22 + 7q^-21 - 16q^-20 - 15q^-19 + 3q^-18 + 11q^-17 + 46q^-16 - 12q^-15 - 38q^-14 - 33q^-13 - 21q^-12 + 88q^-11 + 39q^-10 - 3q^-9 - 60q^-8 - 115q^-7 + 56q^-6 + 77q^-5 + 98q^-4 - 7q^-3 - 192q^-2 - 47q^-1 + 32 + 190q + 115q² - 193q³ - 149q⁴ - 72q⁵ + 227q⁶ + 236q⁷ - 145q⁸ - 214q⁹ - 173q¹⁰ + 228q¹¹ + 321q¹² - 92q¹³ - 254q¹⁴ - 245q¹⁵ + 217q¹⁶ + 374q¹⁷ - 38q¹⁸ - 276q¹⁹ - 297q²⁰ + 185q²¹ + 392q²² + 28q²³ - 248q²⁴ - 319q²⁵ + 100q²⁶ + 344q²⁷ + 100q²⁸ - 150q²⁹ - 278q³⁰ - 7q³¹ + 215q³² + 118q³³ - 30q³⁴ - 161q³⁵ - 58q³⁶ + 73q³⁷ + 68q³⁸ + 23q³⁹ - 46q⁴⁰ - 34q⁴¹ + 5q⁴² + 14q⁴³ + 12q⁴⁴ - 3q⁴⁵ - 6q⁴⁶ + q⁴⁸
5	- q^-40 + 3q^-39 + q^-38 - 5q^-37 - 3q^-36 - 3q^-35 + 3q^-34 + 14q^-33 + 18q^-32 - 5q^-31 - 24q^-30 - 33q^-29 - 21q^-28 + 17q^-27 + 61q^-26 + 66q^-25 + 5q^-24 - 61q^-23 - 101q^-22 - 82q^-21 + 17q^-20 + 132q^-19 + 153q^-18 + 71q^-17 - 71q^-16 - 209q^-15 - 213q^-14 - 34q^-13 + 192q^-12 + 318q^-11 + 224q^-10 - 78q^-9 - 392q^-8 - 410q^-7 - 114q^-6 + 336q^-5 + 593q^-4 + 367q^-3 - 224q^-2 - 674q^-1 - 619 - 15q + 714q² + 858q³ + 231q⁴ - 630q⁵ - 1028q⁶ - 516q⁷ + 546q⁸ + 1161q⁹ + 721q¹⁰ - 404q¹¹ - 1226q¹² - 944q¹³ + 290q¹⁴ + 1287q¹⁵ + 1083q¹⁶ - 173q¹⁷ - 1314q¹⁸ - 1221q¹⁹ + 84q²⁰ + 1349q²¹ + 1323q²² - 15q²³ - 1369q²⁴ - 1406q²⁵ - 65q²⁶ + 1373q²⁷ + 1495q²⁸ + 145q²⁹ - 1351q³⁰ - 1542q³¹ - 270q³² + 1268q³³ + 1590q³⁴ + 397q³⁵ - 1124q³⁶ - 1549q³⁷ - 563q³⁸ + 904q³⁹ + 1466q⁴⁰ + 675q⁴¹ - 637q⁴² - 1255q⁴³ - 755q⁴⁴ + 338q⁴⁵ + 1010q⁴⁶ + 727q⁴⁷ - 95q⁴⁸ - 691q⁴⁹ - 626q⁵⁰ - 83q⁵¹ + 404q⁵² + 472q⁵³ + 154q⁵⁴ - 192q⁵⁵ - 283q⁵⁶ - 151q⁵⁷ + 42q⁵⁸ + 154q⁵⁹ + 111q⁶⁰ - 8q⁶¹ - 49q⁶² - 51q⁶³ - 21q⁶⁴ + 19q⁶⁵ + 21q⁶⁶ + 3q⁶⁷ - 2q⁶⁸ - 2q⁶⁹ - 4q⁷⁰ + 2q⁷¹
6	q^-57 - 3q^-56 - q^-55 + 5q^-54 + 3q^-53 + 3q^-52 - 7q^-51 - q^-50 - 17q^-49 - 16q^-48 + 17q^-47 + 25q^-46 + 39q^-45 + 10q^-44 + 12q^-43 - 62q^-42 - 94q^-41 - 48q^-40 - 2q^-39 + 89q^-38 + 103q^-37 + 186q^-36 + 28q^-35 - 118q^-34 - 196q^-33 - 236q^-32 - 124q^-31 - 15q^-30 + 360q^-29 + 368q^-28 + 284q^-27 + 48q^-26 - 284q^-25 - 536q^-24 - 684q^-23 - 126q^-22 + 271q^-21 + 759q^-20 + 881q^-19 + 578q^-18 - 195q^-17 - 1184q^-16 - 1199q^-15 - 917q^-14 + 161q^-13 + 1247q^-12 + 1925q^-11 + 1362q^-10 - 320q^-9 - 1547q^-8 - 2423q^-7 - 1695q^-6 + 68q^-5 + 2319q^-4 + 3078q^-3 + 1775q^-2 - 290q^-1 - 2831 - 3594q - 2236q² + 1147q³ + 3677q⁴ + 3828q⁵ + 1944q⁶ - 1809q⁷ - 4462q⁸ - 4434q⁹ - 844q¹⁰ + 3060q¹¹ + 4979q¹² + 4030q¹³ - 152q¹⁴ - 4346q¹⁵ - 5850q¹⁶ - 2664q¹⁷ + 1991q¹⁸ + 5347q¹⁹ + 5430q²⁰ + 1292q²¹ - 3894q²² - 6593q²³ - 3892q²⁴ + 1127q²⁵ + 5439q²⁶ + 6240q²⁷ + 2217q²⁸ - 3574q²⁹ - 7027q³⁰ - 4638q³¹ + 612q³² + 5549q³³ + 6777q³⁴ + 2818q³⁵ - 3392q³⁶ - 7358q³⁷ - 5253q³⁸ + 139q³⁹ + 5580q⁴⁰ + 7244q⁴¹ + 3523q⁴² - 2940q⁴³ - 7440q⁴⁴ - 5949q⁴⁵ - 767q⁴⁶ + 5032q⁴⁷ + 7418q⁴⁸ + 4511q⁴⁹ - 1701q⁵⁰ - 6680q⁵¹ - 6367q⁵² - 2186q⁵³ + 3386q⁵⁴ + 6574q⁵⁵ + 5213q⁵⁶ + 244q⁵⁷ - 4594q⁵⁸ - 5637q⁵⁹ - 3303q⁶⁰ + 970q⁶¹ + 4325q⁶² + 4634q⁶³ + 1792q⁶⁴ - 1819q⁶⁵ - 3524q⁶⁶ - 3059q⁶⁷ - 829q⁶⁸ + 1631q⁶⁹ + 2748q⁷⁰ + 1873q⁷¹ + 77q⁷² - 1225q⁷³ - 1656q⁷⁴ - 1075q⁷⁵ + 32q⁷⁶ + 897q⁷⁷ + 935q⁷⁸ + 445q⁷⁹ - 65q⁸⁰ - 434q⁸¹ - 477q⁸² - 220q⁸³ + 92q⁸⁴ + 204q⁸⁵ + 162q⁸⁶ + 78q⁸⁷ - 18q⁸⁸ - 83q⁸⁹ - 64q⁹⁰ - 7q⁹¹ + 12q⁹² + 14q⁹³ + 11q⁹⁴ + 6q⁹⁵ - 3q⁹⁶ - 6q⁹⁷ + q⁹⁹
7	- q^-77 + 3q^-76 + q^-75 - 5q^-74 - 3q^-73 - 3q^-72 + 7q^-71 + 5q^-70 + 4q^-69 + 15q^-68 + 4q^-67 - 18q^-66 - 30q^-65 - 42q^-64 - 12q^-63 + 21q^-62 + 32q^-61 + 90q^-60 + 84q^-59 + 42q^-58 - 30q^-57 - 152q^-56 - 175q^-55 - 140q^-54 - 96q^-53 + 88q^-52 + 236q^-51 + 332q^-50 + 360q^-49 + 94q^-48 - 146q^-47 - 368q^-46 - 605q^-45 - 529q^-44 - 303q^-43 + 138q^-42 + 734q^-41 + 926q^-40 + 907q^-39 + 579q^-38 - 256q^-37 - 970q^-36 - 1589q^-35 - 1648q^-34 - 772q^-33 + 318q^-32 + 1629q^-31 + 2579q^-30 + 2355q^-29 + 1299q^-28 - 721q^-27 - 2888q^-26 - 3781q^-25 - 3472q^-24 - 1381q^-23 + 1803q^-22 + 4371q^-21 + 5691q^-20 + 4387q^-19 + 670q^-18 - 3494q^-17 - 6944q^-16 - 7487q^-15 - 4385q^-14 + 794q^-13 + 6668q^-12 + 9890q^-11 + 8477q^-10 + 3309q^-9 - 4404q^-8 - 10696q^-7 - 12190q^-6 - 8266q^-5 + 496q^-4 + 9688q^-3 + 14552q^-2 + 13133q^-1 + 4661 - 6824q - 15407q² - 17269q³ - 10035q⁴ + 2675q⁵ + 14458q⁶ + 20080q⁷ + 15299q⁸ + 2229q⁹ - 12361q¹⁰ - 21648q¹¹ - 19589q¹² - 7100q¹³ + 9245q¹⁴ + 21943q¹⁵ + 23066q¹⁶ + 11708q¹⁷ - 6000q¹⁸ - 21539q¹⁹ - 25433q²⁰ - 15509q²¹ + 2746q²² + 20527q²³ + 27138q²⁴ + 18678q²⁵ + 19q²⁶ - 19533q²⁷ - 28182q²⁸ - 20988q²⁹ - 2320q³⁰ + 18523q³¹ + 28917q³² + 22811q³³ + 4011q³⁴ - 17828q³⁵ - 29454q³⁶ - 24091q³⁷ - 5240q³⁸ + 17342q³⁹ + 29948q⁴⁰ + 25133q⁴¹ + 6142q⁴² - 17103q⁴³ - 30510q⁴⁴ - 26075q⁴⁵ - 6870q⁴⁶ + 16987q⁴⁷ + 31078q⁴⁸ + 27068q⁴⁹ + 7756q⁵⁰ - 16723q⁵¹ - 31718q⁵² - 28287q⁵³ - 8934q⁵⁴ + 16131q⁵⁵ + 32091q⁵⁶ + 29642q⁵⁷ + 10730q⁵⁸ - 14775q⁵⁹ - 32019q⁶⁰ - 31045q⁶¹ - 13064q⁶² + 12461q⁶³ + 30941q⁶⁴ + 32023q⁶⁵ + 15921q⁶⁶ - 8988q⁶⁷ - 28605q⁶⁸ - 32143q⁶⁹ - 18666q⁷⁰ + 4578q⁷¹ + 24610q⁷² + 30785q⁷³ + 20885q⁷⁴ + 351q⁷⁵ - 19347q⁷⁶ - 27727q⁷⁷ - 21648q⁷⁸ - 4888q⁷⁹ + 13091q⁸⁰ + 22927q⁸¹ + 20776q⁸² + 8363q⁸³ - 6983q⁸⁴ - 17161q⁸⁵ - 17999q⁸⁶ - 9990q⁸⁷ + 1803q⁸⁸ + 11087q⁸⁹ + 13991q⁹⁰ + 9855q⁹¹ + 1670q⁹² - 5881q⁹³ - 9574q⁹⁴ - 8154q⁹⁵ - 3244q⁹⁶ + 2098q⁹⁷ + 5505q⁹⁸ + 5762q⁹⁹ + 3396q¹⁰⁰ + 49q¹⁰¹ - 2611q¹⁰² - 3461q¹⁰³ - 2516q¹⁰⁴ - 812q¹⁰⁵ + 791q¹⁰⁶ + 1653q¹⁰⁷ + 1567q¹⁰⁸ + 846q¹⁰⁹ - 75q¹¹⁰ - 649q¹¹¹ - 725q¹¹² - 489q¹¹³ - 160q¹¹⁴ + 144q¹¹⁵ + 278q¹¹⁶ + 246q¹¹⁷ + 109q¹¹⁸ - 28q¹¹⁹ - 68q¹²⁰ - 70q¹²¹ - 47q¹²² - 13q¹²³ + 15q¹²⁴ + 19q¹²⁵ + 15q¹²⁶ - 2q¹²⁸ - 2q¹²⁹ - 4q¹³⁰ + 2q¹³¹

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	PD[Knot[9, 47]]
Out[2]=	PD[X[6, 2, 7, 1], X[16, 8, 17, 7], X[8, 3, 9, 4], X[2, 15, 3, 16], > X[14, 9, 15, 10], X[10, 6, 11, 5], X[4, 14, 5, 13], X[11, 1, 12, 18], > X[17, 13, 18, 12]]
In[3]:=	GaussCode[Knot[9, 47]]
Out[3]=	GaussCode[1, -4, 3, -7, 6, -1, 2, -3, 5, -6, -8, 9, 7, -5, 4, -2, -9, 8]
In[4]:=	DTCode[Knot[9, 47]]
Out[4]=	DTCode[6, 8, 10, 16, 14, -18, 4, 2, -12]
In[5]:=	br = BR[Knot[9, 47]]
Out[5]=	BR[4, {-1, 2, -1, 2, 3, 2, -1, 2, 3}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{4, 9}
In[7]:=	BraidIndex[Knot[9, 47]]
Out[7]=	4
In[8]:=	Show[DrawMorseLink[Knot[9, 47]]]

Out[8]=	-Graphics-
In[9]:=	#[Knot[9, 47]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 2, 3, 3, {4, 6}, 2}
In[10]:=	alex = Alexander[Knot[9, 47]][t]
Out[10]=	-3 4 6 2 3 -5 + t - -- + - + 6 t - 4 t + t 2 t t
In[11]:=	Conway[Knot[9, 47]][z]
Out[11]=	2 4 6 1 - z + 2 z + z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[9, 47]}
In[13]:=	{KnotDet[Knot[9, 47]], KnotSignature[Knot[9, 47]]}
Out[13]=	{27, 2}
In[14]:=	Jones[Knot[9, 47]][q]
Out[14]=	-2 3 2 3 4 5 -3 - q + - + 5 q - 5 q + 4 q - 4 q + 2 q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] \|\| (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[9, 47]}
In[16]:=	A2Invariant[Knot[9, 47]][q]
Out[16]=	-6 -4 -2 2 4 6 8 12 14 16 20 2 - q + q + q + 2 q - q + q - 2 q - q - q + q + q
In[17]:=	HOMFLYPT[Knot[9, 47]][a, z]
Out[17]=	2 2 4 4 6 -6 2 -2 2 3 z 4 z 4 z 4 z z 1 + a - -- + a - 2 z - ---- + ---- - z - -- + ---- + -- 4 4 2 4 2 2 a a a a a a
In[18]:=	Kauffman[Knot[9, 47]][a, z]
Out[18]=	2 2 2 3 -6 2 -2 3 z 5 z 2 z 2 3 z 9 z 11 z 3 z 1 - a - -- - a - --- - --- - --- + 5 z + ---- + ---- + ----- + ---- + 4 5 3 a 6 4 2 5 a a a a a a a 3 3 4 4 5 5 5 6 z z 3 4 7 z 16 z z 4 z 4 z 5 6 > ---- + -- - 2 a z - 9 z - ---- - ----- + -- - ---- - ---- + a z + 3 z + 3 a 4 2 5 3 a a a a a a 6 6 7 7 3 z 6 z 2 z 2 z > ---- + ---- + ---- + ---- 4 2 3 a a a a
In[19]:=	{Vassiliev[2][Knot[9, 47]], Vassiliev[3][Knot[9, 47]]}
Out[19]=	{-1, -2}
In[20]:=	Kh[Knot[9, 47]][q, t]
Out[20]=	3 1 2 1 1 2 q 3 5 5 2 4 q + 2 q + ----- + ----- + ---- + --- + --- + 2 q t + 3 q t + 2 q t + 5 3 3 2 2 q t t q t q t q t 7 2 7 3 9 3 11 4 > 2 q t + 2 q t + 2 q t + 2 q t
In[21]:=	ColouredJones[Knot[9, 47], 2][q]
Out[21]=	-7 3 -5 9 6 8 17 2 3 4 5 -3 + q - -- - q + -- - -- - -- + -- - 18 q + 20 q + 2 q - 25 q + 19 q + 6 4 3 2 q q q q q 6 7 8 9 10 11 12 13 15 > 8 q - 25 q + 14 q + 10 q - 17 q + 6 q + 6 q - 6 q + q

The Non Alternating Knot 947

The Non Alternating Knot 9₄₇