The Alternating Knot 10₉₅

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Acknowledgement

KnotPlot

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

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

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

Minimum Braid Representative:

Length is 11, 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

Chiral 1 3 3 / NotAvailable 1

Alexander Polynomial: 2t^-3 - 9t^-2 + 21t^-1 - 27 + 21t - 9t² + 2t³

Conway Polynomial: 1 + 3z² + 3z⁴ + 2z⁶

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

Determinant and Signature: {91, 2}

Jones Polynomial: - q^-2 + 4q^-1 - 8 + 12q - 14q² + 16q³ - 14q⁴ + 11q⁵ - 7q⁶ + 3q⁷ - q⁸

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

A2 (sl(3)) Invariant: - q^-6 + 2q^-4 - q^-2 - 1 + 3q² - 3q⁴ + 3q⁶ + q¹⁰ + 3q¹² - 2q¹⁴ + 3q¹⁶ - 2q¹⁸ - 2q²⁰ + q²² - q²⁴

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

Kauffman Polynomial: a^-9z - 2a^-9z³ + a^-9z⁵ + 2a^-8z² - 5a^-8z⁴ + 3a^-8z⁶ - 2a^-7z + 5a^-7z³ - 8a^-7z⁵ + 5a^-7z⁷ + 2a^-6 - 4a^-6z² + 4a^-6z⁴ - 6a^-6z⁶ + 5a^-6z⁸ - 5a^-5z + 17a^-5z³ - 21a^-5z⁵ + 8a^-5z⁷ + 2a^-5z⁹ + 3a^-4 - 7a^-4z² + 13a^-4z⁴ - 19a^-4z⁶ + 11a^-4z⁸ - 3a^-3z + 16a^-3z³ - 25a^-3z⁵ + 10a^-3z⁷ + 2a^-3z⁹ + a^-2z² - 2a^-2z⁴ - 6a^-2z⁶ + 6a^-2z⁸ - a^-1z + 5a^-1z³ - 12a^-1z⁵ + 7a^-1z⁷ + 2z² - 6z⁴ + 4z⁶ - az³ + az⁵

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

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 10₉₅. 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 r = 7

j = 17 1

j = 15 2

j = 13 5 1

j = 11 6 2

j = 9 8 5

j = 7 8 6

j = 5 6 8

j = 3 6 8

j = 1 3 7

j = -1 1 5

j = -3 3

j = -5 1

n Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2 q^-7 - 4q^-6 + 3q^-5 + 12q^-4 - 28q^-3 + 4q^-2 + 57q^-1 - 71 - 20q + 132q² - 105q³ - 70q⁴ + 198q⁵ - 107q⁶ - 117q⁷ + 219q⁸ - 79q⁹ - 135q¹⁰ + 181q¹¹ - 34q¹² - 113q¹³ + 106q¹⁴ - 63q¹⁶ + 39q¹⁷ + 7q¹⁸ - 20q¹⁹ + 8q²⁰ + 2q²¹ - 3q²² + q²³

3 - q^-15 + 4q^-14 - 3q^-13 - 7q^-12 + 4q^-11 + 23q^-10 - 5q^-9 - 61q^-8 + 2q^-7 + 117q^-6 + 35q^-5 - 201q^-4 - 117q^-3 + 296q^-2 + 256q^-1 - 375 - 452q + 406q² + 700q³ - 392q⁴ - 942q⁵ + 303q⁶ + 1181q⁷ - 184q⁸ - 1352q⁹ + 13q¹⁰ + 1490q¹¹ + 137q¹² - 1528q¹³ - 315q¹⁴ + 1528q¹⁵ + 446q¹⁶ - 1425q¹⁷ - 586q¹⁸ + 1283q¹⁹ + 669q²⁰ - 1068q²¹ - 716q²² + 822q²³ + 703q²⁴ - 567q²⁵ - 634q²⁶ + 340q²⁷ + 516q²⁸ - 165q²⁹ - 373q³⁰ + 47q³¹ + 247q³² - 134q³⁴ - 21q³⁵ + 69q³⁶ + 14q³⁷ - 29q³⁸ - 8q³⁹ + 13q⁴⁰ + q⁴¹ - 3q⁴² - 2q⁴³ + 3q⁴⁴ - q⁴⁵

4 q^-26 - 4q^-25 + 3q^-24 + 7q^-23 - 9q^-22 + q^-21 - 22q^-20 + 25q^-19 + 54q^-18 - 41q^-17 - 31q^-16 - 136q^-15 + 85q^-14 + 293q^-13 + 6q^-12 - 136q^-11 - 637q^-10 - 31q^-9 + 873q^-8 + 591q^-7 + 59q^-6 - 1804q^-5 - 1059q^-4 + 1316q^-3 + 2116q^-2 + 1564q^-1 - 3024 - 3510q + 341q² + 3854q³ + 4956q⁴ - 2838q⁵ - 6508q⁶ - 2672q⁷ + 4292q⁸ + 9161q⁹ - 639q¹⁰ - 8392q¹¹ - 6640q¹² + 2870q¹³ + 12405q¹⁴ + 2564q¹⁵ - 8474q¹⁶ - 9920q¹⁷ + 421q¹⁸ + 13826q¹⁹ + 5450q²⁰ - 7196q²¹ - 11750q²² - 2121q²³ + 13506q²⁴ + 7490q²⁵ - 5030q²⁶ - 12064q²⁷ - 4456q²⁸ + 11535q²⁹ + 8565q³⁰ - 2104q³¹ - 10733q³² - 6313q³³ + 7998q³⁴ + 8230q³⁵ + 1022q³⁶ - 7673q³⁷ - 6871q³⁸ + 3738q³⁹ + 6139q⁴⁰ + 3048q⁴¹ - 3778q⁴² - 5528q⁴³ + 470q⁴⁴ + 3097q⁴⁵ + 3054q⁴⁶ - 792q⁴⁷ - 3057q⁴⁸ - 730q⁴⁹ + 753q⁵⁰ + 1747q⁵¹ + 341q⁵² - 1064q⁵³ - 508q⁵⁴ - 123q⁵⁵ + 585q⁵⁶ + 296q⁵⁷ - 220q⁵⁸ - 118q⁵⁹ - 140q⁶⁰ + 117q⁶¹ + 85q⁶² - 40q⁶³ + 4q⁶⁴ - 40q⁶⁵ + 18q⁶⁶ + 14q⁶⁷ - 12q⁶⁸ + 6q⁶⁹ - 6q⁷⁰ + 3q⁷¹ + 2q⁷² - 3q⁷³ + q⁷⁴

5 - q^-40 + 4q^-39 - 3q^-38 - 7q^-37 + 9q^-36 + 4q^-35 - 2q^-34 + 2q^-33 - 18q^-32 - 31q^-31 + 31q^-30 + 75q^-29 + 34q^-28 - 40q^-27 - 159q^-26 - 181q^-25 + 37q^-24 + 395q^-23 + 489q^-22 + 57q^-21 - 668q^-20 - 1134q^-19 - 601q^-18 + 929q^-17 + 2299q^-16 + 1832q^-15 - 768q^-14 - 3747q^-13 - 4271q^-12 - 608q^-11 + 5212q^-10 + 8061q^-9 + 3888q^-8 - 5589q^-7 - 12822q^-6 - 9922q^-5 + 3612q^-4 + 17664q^-3 + 18681q^-2 + 1998q^-1 - 20924 - 29433q - 11936q² + 20776q³ + 40652q⁴ + 26013q⁵ - 16066q⁶ - 50283q⁷ - 42717q⁸ + 6160q⁹ + 56432q¹⁰ + 60438q¹¹ + 7918q¹² - 58163q¹³ - 76576q¹⁴ - 24834q¹⁵ + 55132q¹⁶ + 89989q¹⁷ + 42392q¹⁸ - 48503q¹⁹ - 99208q²⁰ - 59043q²¹ + 39228q²² + 104837q²³ + 73258q²⁴ - 29009q²⁵ - 106690q²⁶ - 84914q²⁷ + 18484q²⁸ + 106299q²⁹ + 93644q³⁰ - 8536q³¹ - 103365q³² - 100296q³³ - 1357q³⁴ + 99127q³⁵ + 104784q³⁶ + 10949q³⁷ - 92468q³⁸ - 107606q³⁹ - 21123q⁴⁰ + 83882q⁴¹ + 108228q⁴² + 31335q⁴³ - 72282q⁴⁴ - 106151q⁴⁵ - 41592q⁴⁶ + 58118q⁴⁷ + 100573q⁴⁸ + 50558q⁴⁹ - 41593q⁵⁰ - 91035q⁵¹ - 57034q⁵² + 24096q⁵³ + 77568q⁵⁴ + 59653q⁵⁵ - 7311q⁵⁶ - 61166q⁵⁷ - 57668q⁵⁸ - 6648q⁵⁹ + 43414q⁶⁰ + 51184q⁶¹ + 16293q⁶² - 26486q⁶³ - 41368q⁶⁴ - 20922q⁶⁵ + 12476q⁶⁶ + 29986q⁶⁷ + 20846q⁶⁸ - 2456q⁶⁹ - 19075q⁷⁰ - 17644q⁷¹ - 3107q⁷² + 10303q⁷³ + 12706q⁷⁴ + 5160q⁷⁵ - 4205q⁷⁶ - 8013q⁷⁷ - 4843q⁷⁸ + 917q⁷⁹ + 4216q⁸⁰ + 3506q⁸¹ + 512q⁸² - 1870q⁸³ - 2082q⁸⁴ - 748q⁸⁵ + 609q⁸⁶ + 1046q⁸⁷ + 564q⁸⁸ - 137q⁸⁹ - 413q⁹⁰ - 304q⁹¹ - 42q⁹² + 163q⁹³ + 143q⁹⁴ + 10q⁹⁵ - 35q⁹⁶ - 33q⁹⁷ - 36q⁹⁸ + 15q⁹⁹ + 27q¹⁰⁰ - 8q¹⁰¹ - 3q¹⁰² + 6q¹⁰³ - 7q¹⁰⁴ - q¹⁰⁵ + 6q¹⁰⁶ - 3q¹⁰⁷ - 2q¹⁰⁸ + 3q¹⁰⁹ - q¹¹⁰

6 q^-57 - 4q^-56 + 3q^-55 + 7q^-54 - 9q^-53 - 4q^-52 - 3q^-51 + 22q^-50 - 9q^-49 - 5q^-48 + 41q^-47 - 59q^-46 - 49q^-45 - 22q^-44 + 123q^-43 + 77q^-42 + 34q^-41 + 130q^-40 - 306q^-39 - 420q^-38 - 322q^-37 + 403q^-36 + 660q^-35 + 804q^-34 + 954q^-33 - 841q^-32 - 2155q^-31 - 2746q^-30 - 515q^-29 + 1774q^-28 + 4433q^-27 + 6364q^-26 + 1677q^-25 - 4917q^-24 - 11502q^-23 - 9968q^-22 - 3587q^-21 + 9420q^-20 + 23192q^-19 + 20705q^-18 + 4571q^-17 - 22488q^-16 - 37935q^-15 - 37682q^-14 - 7378q^-13 + 41846q^-12 + 69773q^-11 + 59930q^-10 + 1410q^-9 - 66011q^-8 - 116340q^-7 - 92700q^-6 + 7180q^-5 + 117936q^-4 + 176693q^-3 + 120351q^-2 - 18973q^-1 - 192118 - 258020q - 151988q² + 69876q³ + 287984q⁴ + 337353q⁵ + 180052q⁶ - 152038q⁷ - 415900q⁸ - 425011q⁹ - 150757q¹⁰ + 268108q¹¹ + 545481q¹² + 502265q¹³ + 73152q¹⁴ - 433994q¹⁵ - 687484q¹⁶ - 495888q¹⁷ + 60074q¹⁸ + 611935q¹⁹ + 809982q²⁰ + 417483q²¹ - 269684q²² - 810590q²³ - 821648q²⁴ - 256910q²⁵ + 507688q²⁶ + 982646q²⁷ + 738632q²⁸ - 7921q²⁹ - 775839q³⁰ - 1020617q³¹ - 550237q³² + 315581q³³ + 1010043q³⁴ + 946196q³⁵ + 236093q³⁶ - 657877q³⁷ - 1090647q³⁸ - 749709q³⁹ + 128777q⁴⁰ + 956458q⁴¹ + 1046366q⁴² + 414389q⁴³ - 526984q⁴⁴ - 1086917q⁴⁵ - 870403q⁴⁶ - 26273q⁴⁷ + 871806q⁴⁸ + 1087006q⁴⁹ + 551136q⁵⁰ - 390720q⁵¹ - 1042736q⁵² - 954361q⁵³ - 181552q⁵⁴ + 748016q⁵⁵ + 1086903q⁵⁶ + 682942q⁵⁷ - 211047q⁵⁸ - 936295q⁵⁹ - 1006059q⁶⁰ - 367069q⁶¹ + 542427q⁶² + 1009815q⁶³ + 798173q⁶⁴ + 33503q⁶⁵ - 719978q⁶⁶ - 972467q⁶⁷ - 552123q⁶⁸ + 243889q⁶⁹ + 801510q⁷⁰ + 823715q⁷¹ + 290137q⁷² - 392694q⁷³ - 791032q⁷⁴ - 641626q⁷⁵ - 71701q⁷⁶ + 470233q⁷⁷ + 688154q⁷⁸ + 442241q⁷⁹ - 50818q⁸⁰ - 478976q⁸¹ - 557893q⁸² - 270830q⁸³ + 129028q⁸⁴ + 418167q⁸⁵ + 411083q⁸⁶ + 163851q⁸⁷ - 161767q⁸⁸ - 338990q⁸⁹ - 281319q⁹⁰ - 80527q⁹¹ + 146781q⁹² + 247689q⁹³ + 193589q⁹⁴ + 26742q⁹⁵ - 120746q⁹⁶ - 167599q⁹⁷ - 119162q⁹⁸ - 6338q⁹⁹ + 85257q¹⁰⁰ + 114244q¹⁰¹ + 66186q¹⁰² - 4579q¹⁰³ - 54595q¹⁰⁴ - 68232q¹⁰⁵ - 37805q¹⁰⁶ + 4935q¹⁰⁷ + 37034q¹⁰⁸ + 36503q¹⁰⁹ + 18604q¹¹⁰ - 3339q¹¹¹ - 20277q¹¹² - 20080q¹¹³ - 9565q¹¹⁴ + 4481q¹¹⁵ + 9556q¹¹⁶ + 9229q¹¹⁷ + 4727q¹¹⁸ - 2180q¹¹⁹ - 5093q¹²⁰ - 4471q¹²¹ - 921q¹²² + 721q¹²³ + 1952q¹²⁴ + 2048q¹²⁵ + 437q¹²⁶ - 604q¹²⁷ - 974q¹²⁸ - 330q¹²⁹ - 246q¹³⁰ + 119q¹³¹ + 443q¹³² + 176q¹³³ - 22q¹³⁴ - 145q¹³⁵ + 8q¹³⁶ - 79q¹³⁷ - 39q¹³⁸ + 75q¹³⁹ + 26q¹⁴⁰ + q¹⁴¹ - 30q¹⁴² + 21q¹⁴³ - 7q¹⁴⁴ - 17q¹⁴⁵ + 13q¹⁴⁶ + 2q¹⁴⁷ + q¹⁴⁸ - 6q¹⁴⁹ + 3q¹⁵⁰ + 2q¹⁵¹ - 3q¹⁵² + q¹⁵³

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, 95]]

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

In[3]:=
GaussCode[Knot[10, 95]]

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

In[4]:=
DTCode[Knot[10, 95]]

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

In[5]:=
br = BR[Knot[10, 95]]

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

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

Out[6]=
{4, 11}

In[7]:=
BraidIndex[Knot[10, 95]]

Out[7]=
4

In[8]:=
Show[DrawMorseLink[Knot[10, 95]]]

Out[8]=
-Graphics-

In[9]:=
#[Knot[10, 95]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}

Out[9]=
{Chiral, 1, 3, 3, NotAvailable, 1}

In[10]:=
alex = Alexander[Knot[10, 95]][t]

Out[10]=
2 9 21 2 3 -27 + -- - -- + -- + 21 t - 9 t + 2 t 3 2 t t t

In[11]:=
Conway[Knot[10, 95]][z]

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

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

Out[12]=
{Knot[10, 95]}

In[13]:=
{KnotDet[Knot[10, 95]], KnotSignature[Knot[10, 95]]}

Out[13]=
{91, 2}

In[14]:=
Jones[Knot[10, 95]][q]

Out[14]=
-2 4 2 3 4 5 6 7 8 -8 - q + - + 12 q - 14 q + 16 q - 14 q + 11 q - 7 q + 3 q - q q

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

Out[15]=
{Knot[10, 95]}

In[16]:=
A2Invariant[Knot[10, 95]][q]

Out[16]=
-6 2 -2 2 4 6 10 12 14 16 -1 - q + -- - q + 3 q - 3 q + 3 q + q + 3 q - 2 q + 3 q - 4 q 18 20 22 24 > 2 q - 2 q + q - q

In[17]:=
HOMFLYPT[Knot[10, 95]][a, z]

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

In[18]:=
Kauffman[Knot[10, 95]][a, z]

Out[18]=
2 2 2 2 3 2 3 z 2 z 5 z 3 z z 2 2 z 4 z 7 z z 2 z -- + -- + -- - --- - --- - --- - - + 2 z + ---- - ---- - ---- + -- - ---- + 6 4 9 7 5 3 a 8 6 4 2 9 a a a a a a a a a a a 3 3 3 3 4 4 4 4 5 z 17 z 16 z 5 z 3 4 5 z 4 z 13 z 2 z > ---- + ----- + ----- + ---- - a z - 6 z - ---- + ---- + ----- - ---- + 7 5 3 a 8 6 4 2 a a a a a a a 5 5 5 5 5 6 6 6 z 8 z 21 z 25 z 12 z 5 6 3 z 6 z 19 z > -- - ---- - ----- - ----- - ----- + a z + 4 z + ---- - ---- - ----- - 9 7 5 3 a 8 6 4 a a a a a a a 6 7 7 7 7 8 8 8 9 9 6 z 5 z 8 z 10 z 7 z 5 z 11 z 6 z 2 z 2 z > ---- + ---- + ---- + ----- + ---- + ---- + ----- + ---- + ---- + ---- 2 7 5 3 a 6 4 2 5 3 a a a a a a a a a

In[19]:=
{Vassiliev[2][Knot[10, 95]], Vassiliev[3][Knot[10, 95]]}

Out[19]=
{3, 5}

In[20]:=
Kh[Knot[10, 95]][q, t]

Out[20]=
3 1 3 1 5 3 q 3 5 5 2 7 q + 6 q + ----- + ----- + ---- + --- + --- + 8 q t + 6 q t + 8 q t + 5 3 3 2 2 q t t q t q t q t 7 2 7 3 9 3 9 4 11 4 11 5 13 5 > 8 q t + 6 q t + 8 q t + 5 q t + 6 q t + 2 q t + 5 q t + 13 6 15 6 17 7 > q t + 2 q t + q t

In[21]:=
ColouredJones[Knot[10, 95], 2][q]

Out[21]=
-7 4 3 12 28 4 57 2 3 4 -71 + q - -- + -- + -- - -- + -- + -- - 20 q + 132 q - 105 q - 70 q + 6 5 4 3 2 q q q q q q 5 6 7 8 9 10 11 12 > 198 q - 107 q - 117 q + 219 q - 79 q - 135 q + 181 q - 34 q - 13 14 16 17 18 19 20 21 > 113 q + 106 q - 63 q + 39 q + 7 q - 20 q + 8 q + 2 q - 22 23 > 3 q + q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 10₉₅

10₉₄
10₉₆

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q^-7 - 4q^-6 + 3q^-5 + 12q^-4 - 28q^-3 + 4q^-2 + 57q^-1 - 71 - 20q + 132q² - 105q³ - 70q⁴ + 198q⁵ - 107q⁶ - 117q⁷ + 219q⁸ - 79q⁹ - 135q¹⁰ + 181q¹¹ - 34q¹² - 113q¹³ + 106q¹⁴ - 63q¹⁶ + 39q¹⁷ + 7q¹⁸ - 20q¹⁹ + 8q²⁰ + 2q²¹ - 3q²² + q²³
3	- q^-15 + 4q^-14 - 3q^-13 - 7q^-12 + 4q^-11 + 23q^-10 - 5q^-9 - 61q^-8 + 2q^-7 + 117q^-6 + 35q^-5 - 201q^-4 - 117q^-3 + 296q^-2 + 256q^-1 - 375 - 452q + 406q² + 700q³ - 392q⁴ - 942q⁵ + 303q⁶ + 1181q⁷ - 184q⁸ - 1352q⁹ + 13q¹⁰ + 1490q¹¹ + 137q¹² - 1528q¹³ - 315q¹⁴ + 1528q¹⁵ + 446q¹⁶ - 1425q¹⁷ - 586q¹⁸ + 1283q¹⁹ + 669q²⁰ - 1068q²¹ - 716q²² + 822q²³ + 703q²⁴ - 567q²⁵ - 634q²⁶ + 340q²⁷ + 516q²⁸ - 165q²⁹ - 373q³⁰ + 47q³¹ + 247q³² - 134q³⁴ - 21q³⁵ + 69q³⁶ + 14q³⁷ - 29q³⁸ - 8q³⁹ + 13q⁴⁰ + q⁴¹ - 3q⁴² - 2q⁴³ + 3q⁴⁴ - q⁴⁵
4	q^-26 - 4q^-25 + 3q^-24 + 7q^-23 - 9q^-22 + q^-21 - 22q^-20 + 25q^-19 + 54q^-18 - 41q^-17 - 31q^-16 - 136q^-15 + 85q^-14 + 293q^-13 + 6q^-12 - 136q^-11 - 637q^-10 - 31q^-9 + 873q^-8 + 591q^-7 + 59q^-6 - 1804q^-5 - 1059q^-4 + 1316q^-3 + 2116q^-2 + 1564q^-1 - 3024 - 3510q + 341q² + 3854q³ + 4956q⁴ - 2838q⁵ - 6508q⁶ - 2672q⁷ + 4292q⁸ + 9161q⁹ - 639q¹⁰ - 8392q¹¹ - 6640q¹² + 2870q¹³ + 12405q¹⁴ + 2564q¹⁵ - 8474q¹⁶ - 9920q¹⁷ + 421q¹⁸ + 13826q¹⁹ + 5450q²⁰ - 7196q²¹ - 11750q²² - 2121q²³ + 13506q²⁴ + 7490q²⁵ - 5030q²⁶ - 12064q²⁷ - 4456q²⁸ + 11535q²⁹ + 8565q³⁰ - 2104q³¹ - 10733q³² - 6313q³³ + 7998q³⁴ + 8230q³⁵ + 1022q³⁶ - 7673q³⁷ - 6871q³⁸ + 3738q³⁹ + 6139q⁴⁰ + 3048q⁴¹ - 3778q⁴² - 5528q⁴³ + 470q⁴⁴ + 3097q⁴⁵ + 3054q⁴⁶ - 792q⁴⁷ - 3057q⁴⁸ - 730q⁴⁹ + 753q⁵⁰ + 1747q⁵¹ + 341q⁵² - 1064q⁵³ - 508q⁵⁴ - 123q⁵⁵ + 585q⁵⁶ + 296q⁵⁷ - 220q⁵⁸ - 118q⁵⁹ - 140q⁶⁰ + 117q⁶¹ + 85q⁶² - 40q⁶³ + 4q⁶⁴ - 40q⁶⁵ + 18q⁶⁶ + 14q⁶⁷ - 12q⁶⁸ + 6q⁶⁹ - 6q⁷⁰ + 3q⁷¹ + 2q⁷² - 3q⁷³ + q⁷⁴
5	- q^-40 + 4q^-39 - 3q^-38 - 7q^-37 + 9q^-36 + 4q^-35 - 2q^-34 + 2q^-33 - 18q^-32 - 31q^-31 + 31q^-30 + 75q^-29 + 34q^-28 - 40q^-27 - 159q^-26 - 181q^-25 + 37q^-24 + 395q^-23 + 489q^-22 + 57q^-21 - 668q^-20 - 1134q^-19 - 601q^-18 + 929q^-17 + 2299q^-16 + 1832q^-15 - 768q^-14 - 3747q^-13 - 4271q^-12 - 608q^-11 + 5212q^-10 + 8061q^-9 + 3888q^-8 - 5589q^-7 - 12822q^-6 - 9922q^-5 + 3612q^-4 + 17664q^-3 + 18681q^-2 + 1998q^-1 - 20924 - 29433q - 11936q² + 20776q³ + 40652q⁴ + 26013q⁵ - 16066q⁶ - 50283q⁷ - 42717q⁸ + 6160q⁹ + 56432q¹⁰ + 60438q¹¹ + 7918q¹² - 58163q¹³ - 76576q¹⁴ - 24834q¹⁵ + 55132q¹⁶ + 89989q¹⁷ + 42392q¹⁸ - 48503q¹⁹ - 99208q²⁰ - 59043q²¹ + 39228q²² + 104837q²³ + 73258q²⁴ - 29009q²⁵ - 106690q²⁶ - 84914q²⁷ + 18484q²⁸ + 106299q²⁹ + 93644q³⁰ - 8536q³¹ - 103365q³² - 100296q³³ - 1357q³⁴ + 99127q³⁵ + 104784q³⁶ + 10949q³⁷ - 92468q³⁸ - 107606q³⁹ - 21123q⁴⁰ + 83882q⁴¹ + 108228q⁴² + 31335q⁴³ - 72282q⁴⁴ - 106151q⁴⁵ - 41592q⁴⁶ + 58118q⁴⁷ + 100573q⁴⁸ + 50558q⁴⁹ - 41593q⁵⁰ - 91035q⁵¹ - 57034q⁵² + 24096q⁵³ + 77568q⁵⁴ + 59653q⁵⁵ - 7311q⁵⁶ - 61166q⁵⁷ - 57668q⁵⁸ - 6648q⁵⁹ + 43414q⁶⁰ + 51184q⁶¹ + 16293q⁶² - 26486q⁶³ - 41368q⁶⁴ - 20922q⁶⁵ + 12476q⁶⁶ + 29986q⁶⁷ + 20846q⁶⁸ - 2456q⁶⁹ - 19075q⁷⁰ - 17644q⁷¹ - 3107q⁷² + 10303q⁷³ + 12706q⁷⁴ + 5160q⁷⁵ - 4205q⁷⁶ - 8013q⁷⁷ - 4843q⁷⁸ + 917q⁷⁹ + 4216q⁸⁰ + 3506q⁸¹ + 512q⁸² - 1870q⁸³ - 2082q⁸⁴ - 748q⁸⁵ + 609q⁸⁶ + 1046q⁸⁷ + 564q⁸⁸ - 137q⁸⁹ - 413q⁹⁰ - 304q⁹¹ - 42q⁹² + 163q⁹³ + 143q⁹⁴ + 10q⁹⁵ - 35q⁹⁶ - 33q⁹⁷ - 36q⁹⁸ + 15q⁹⁹ + 27q¹⁰⁰ - 8q¹⁰¹ - 3q¹⁰² + 6q¹⁰³ - 7q¹⁰⁴ - q¹⁰⁵ + 6q¹⁰⁶ - 3q¹⁰⁷ - 2q¹⁰⁸ + 3q¹⁰⁹ - q¹¹⁰
6	q^-57 - 4q^-56 + 3q^-55 + 7q^-54 - 9q^-53 - 4q^-52 - 3q^-51 + 22q^-50 - 9q^-49 - 5q^-48 + 41q^-47 - 59q^-46 - 49q^-45 - 22q^-44 + 123q^-43 + 77q^-42 + 34q^-41 + 130q^-40 - 306q^-39 - 420q^-38 - 322q^-37 + 403q^-36 + 660q^-35 + 804q^-34 + 954q^-33 - 841q^-32 - 2155q^-31 - 2746q^-30 - 515q^-29 + 1774q^-28 + 4433q^-27 + 6364q^-26 + 1677q^-25 - 4917q^-24 - 11502q^-23 - 9968q^-22 - 3587q^-21 + 9420q^-20 + 23192q^-19 + 20705q^-18 + 4571q^-17 - 22488q^-16 - 37935q^-15 - 37682q^-14 - 7378q^-13 + 41846q^-12 + 69773q^-11 + 59930q^-10 + 1410q^-9 - 66011q^-8 - 116340q^-7 - 92700q^-6 + 7180q^-5 + 117936q^-4 + 176693q^-3 + 120351q^-2 - 18973q^-1 - 192118 - 258020q - 151988q² + 69876q³ + 287984q⁴ + 337353q⁵ + 180052q⁶ - 152038q⁷ - 415900q⁸ - 425011q⁹ - 150757q¹⁰ + 268108q¹¹ + 545481q¹² + 502265q¹³ + 73152q¹⁴ - 433994q¹⁵ - 687484q¹⁶ - 495888q¹⁷ + 60074q¹⁸ + 611935q¹⁹ + 809982q²⁰ + 417483q²¹ - 269684q²² - 810590q²³ - 821648q²⁴ - 256910q²⁵ + 507688q²⁶ + 982646q²⁷ + 738632q²⁸ - 7921q²⁹ - 775839q³⁰ - 1020617q³¹ - 550237q³² + 315581q³³ + 1010043q³⁴ + 946196q³⁵ + 236093q³⁶ - 657877q³⁷ - 1090647q³⁸ - 749709q³⁹ + 128777q⁴⁰ + 956458q⁴¹ + 1046366q⁴² + 414389q⁴³ - 526984q⁴⁴ - 1086917q⁴⁵ - 870403q⁴⁶ - 26273q⁴⁷ + 871806q⁴⁸ + 1087006q⁴⁹ + 551136q⁵⁰ - 390720q⁵¹ - 1042736q⁵² - 954361q⁵³ - 181552q⁵⁴ + 748016q⁵⁵ + 1086903q⁵⁶ + 682942q⁵⁷ - 211047q⁵⁸ - 936295q⁵⁹ - 1006059q⁶⁰ - 367069q⁶¹ + 542427q⁶² + 1009815q⁶³ + 798173q⁶⁴ + 33503q⁶⁵ - 719978q⁶⁶ - 972467q⁶⁷ - 552123q⁶⁸ + 243889q⁶⁹ + 801510q⁷⁰ + 823715q⁷¹ + 290137q⁷² - 392694q⁷³ - 791032q⁷⁴ - 641626q⁷⁵ - 71701q⁷⁶ + 470233q⁷⁷ + 688154q⁷⁸ + 442241q⁷⁹ - 50818q⁸⁰ - 478976q⁸¹ - 557893q⁸² - 270830q⁸³ + 129028q⁸⁴ + 418167q⁸⁵ + 411083q⁸⁶ + 163851q⁸⁷ - 161767q⁸⁸ - 338990q⁸⁹ - 281319q⁹⁰ - 80527q⁹¹ + 146781q⁹² + 247689q⁹³ + 193589q⁹⁴ + 26742q⁹⁵ - 120746q⁹⁶ - 167599q⁹⁷ - 119162q⁹⁸ - 6338q⁹⁹ + 85257q¹⁰⁰ + 114244q¹⁰¹ + 66186q¹⁰² - 4579q¹⁰³ - 54595q¹⁰⁴ - 68232q¹⁰⁵ - 37805q¹⁰⁶ + 4935q¹⁰⁷ + 37034q¹⁰⁸ + 36503q¹⁰⁹ + 18604q¹¹⁰ - 3339q¹¹¹ - 20277q¹¹² - 20080q¹¹³ - 9565q¹¹⁴ + 4481q¹¹⁵ + 9556q¹¹⁶ + 9229q¹¹⁷ + 4727q¹¹⁸ - 2180q¹¹⁹ - 5093q¹²⁰ - 4471q¹²¹ - 921q¹²² + 721q¹²³ + 1952q¹²⁴ + 2048q¹²⁵ + 437q¹²⁶ - 604q¹²⁷ - 974q¹²⁸ - 330q¹²⁹ - 246q¹³⁰ + 119q¹³¹ + 443q¹³² + 176q¹³³ - 22q¹³⁴ - 145q¹³⁵ + 8q¹³⁶ - 79q¹³⁷ - 39q¹³⁸ + 75q¹³⁹ + 26q¹⁴⁰ + q¹⁴¹ - 30q¹⁴² + 21q¹⁴³ - 7q¹⁴⁴ - 17q¹⁴⁵ + 13q¹⁴⁶ + 2q¹⁴⁷ + q¹⁴⁸ - 6q¹⁴⁹ + 3q¹⁵⁰ + 2q¹⁵¹ - 3q¹⁵² + q¹⁵³

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	PD[Knot[10, 95]]
Out[2]=	PD[X[1, 4, 2, 5], X[3, 10, 4, 11], X[11, 17, 12, 16], X[15, 9, 16, 8], > X[19, 7, 20, 6], X[5, 15, 6, 14], X[7, 19, 8, 18], X[13, 1, 14, 20], > X[17, 13, 18, 12], X[9, 2, 10, 3]]
In[3]:=	GaussCode[Knot[10, 95]]
Out[3]=	GaussCode[-1, 10, -2, 1, -6, 5, -7, 4, -10, 2, -3, 9, -8, 6, -4, 3, -9, 7, -5, > 8]
In[4]:=	DTCode[Knot[10, 95]]
Out[4]=	DTCode[4, 10, 14, 18, 2, 16, 20, 8, 12, 6]
In[5]:=	br = BR[Knot[10, 95]]
Out[5]=	BR[4, {-1, -1, 2, 2, -3, 2, -1, 2, 3, 3, 2}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{4, 11}
In[7]:=	BraidIndex[Knot[10, 95]]
Out[7]=	4
In[8]:=	Show[DrawMorseLink[Knot[10, 95]]]

Out[8]=	-Graphics-
In[9]:=	#[Knot[10, 95]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Chiral, 1, 3, 3, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 95]][t]
Out[10]=	2 9 21 2 3 -27 + -- - -- + -- + 21 t - 9 t + 2 t 3 2 t t t
In[11]:=	Conway[Knot[10, 95]][z]
Out[11]=	2 4 6 1 + 3 z + 3 z + 2 z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 95]}
In[13]:=	{KnotDet[Knot[10, 95]], KnotSignature[Knot[10, 95]]}
Out[13]=	{91, 2}
In[14]:=	Jones[Knot[10, 95]][q]
Out[14]=	-2 4 2 3 4 5 6 7 8 -8 - q + - + 12 q - 14 q + 16 q - 14 q + 11 q - 7 q + 3 q - q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] \|\| (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[10, 95]}
In[16]:=	A2Invariant[Knot[10, 95]][q]
Out[16]=	-6 2 -2 2 4 6 10 12 14 16 -1 - q + -- - q + 3 q - 3 q + 3 q + q + 3 q - 2 q + 3 q - 4 q 18 20 22 24 > 2 q - 2 q + q - q
In[17]:=	HOMFLYPT[Knot[10, 95]][a, z]
Out[17]=	2 2 2 4 4 4 6 6 -2 3 2 2 z 5 z z 4 z 3 z 2 z z z -- + -- - z - ---- + ---- + -- - z - -- + ---- + ---- + -- + -- 6 4 6 4 2 6 4 2 4 2 a a a a a a a a a a
In[18]:=	Kauffman[Knot[10, 95]][a, z]
Out[18]=	2 2 2 2 3 2 3 z 2 z 5 z 3 z z 2 2 z 4 z 7 z z 2 z -- + -- + -- - --- - --- - --- - - + 2 z + ---- - ---- - ---- + -- - ---- + 6 4 9 7 5 3 a 8 6 4 2 9 a a a a a a a a a a a 3 3 3 3 4 4 4 4 5 z 17 z 16 z 5 z 3 4 5 z 4 z 13 z 2 z > ---- + ----- + ----- + ---- - a z - 6 z - ---- + ---- + ----- - ---- + 7 5 3 a 8 6 4 2 a a a a a a a 5 5 5 5 5 6 6 6 z 8 z 21 z 25 z 12 z 5 6 3 z 6 z 19 z > -- - ---- - ----- - ----- - ----- + a z + 4 z + ---- - ---- - ----- - 9 7 5 3 a 8 6 4 a a a a a a a 6 7 7 7 7 8 8 8 9 9 6 z 5 z 8 z 10 z 7 z 5 z 11 z 6 z 2 z 2 z > ---- + ---- + ---- + ----- + ---- + ---- + ----- + ---- + ---- + ---- 2 7 5 3 a 6 4 2 5 3 a a a a a a a a a
In[19]:=	{Vassiliev[2][Knot[10, 95]], Vassiliev[3][Knot[10, 95]]}
Out[19]=	{3, 5}
In[20]:=	Kh[Knot[10, 95]][q, t]
Out[20]=	3 1 3 1 5 3 q 3 5 5 2 7 q + 6 q + ----- + ----- + ---- + --- + --- + 8 q t + 6 q t + 8 q t + 5 3 3 2 2 q t t q t q t q t 7 2 7 3 9 3 9 4 11 4 11 5 13 5 > 8 q t + 6 q t + 8 q t + 5 q t + 6 q t + 2 q t + 5 q t + 13 6 15 6 17 7 > q t + 2 q t + q t
In[21]:=	ColouredJones[Knot[10, 95], 2][q]
Out[21]=	-7 4 3 12 28 4 57 2 3 4 -71 + q - -- + -- + -- - -- + -- + -- - 20 q + 132 q - 105 q - 70 q + 6 5 4 3 2 q q q q q q 5 6 7 8 9 10 11 12 > 198 q - 107 q - 117 q + 219 q - 79 q - 135 q + 181 q - 34 q - 13 14 16 17 18 19 20 21 > 113 q + 106 q - 63 q + 39 q + 7 q - 20 q + 8 q + 2 q - 22 23 > 3 q + q

The Alternating Knot 1095

The Alternating Knot 10₉₅