The Alternating Knot 10₉₃

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Acknowledgement

KnotPlot

PD Presentation: X₆₂₇₁ X_16,6,17,5 X_20,8,1,7 X_18,13,19,14 X_14,9,15,10 X_10,3,11,4 X_4,11,5,12 X_12,17,13,18 X_8,20,9,19 X_2,16,3,15

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

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

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 2 3 3 / NotAvailable 1

Alexander Polynomial: 2t^-3 - 8t^-2 + 15t^-1 - 17 + 15t - 8t² + 2t³

Conway Polynomial: 1 + z² + 4z⁴ + 2z⁶

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

Determinant and Signature: {67, -2}

Jones Polynomial: - q^-6 + 3q^-5 - 6q^-4 + 9q^-3 - 10q^-2 + 11q^-1 - 10 + 8q - 5q² + 3q³ - q⁴

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

A2 (sl(3)) Invariant: - q^-18 + q^-16 - q^-14 - q^-12 + 2q^-10 - q^-8 + 3q^-6 + 1 - 2q² + 2q⁴ + q¹⁰ - q¹²

HOMFLY-PT Polynomial: - 2a^-2z² - a^-2z⁴ + 2z² + 3z⁴ + z⁶ + 2a² + 3a²z² + 3a²z⁴ + a²z⁶ - a⁴ - 2a⁴z² - a⁴z⁴

Kauffman Polynomial: - 2a^-3z + 5a^-3z³ - 4a^-3z⁵ + a^-3z⁷ - 6a^-2z² + 17a^-2z⁴ - 13a^-2z⁶ + 3a^-2z⁸ - 6a^-1z + 18a^-1z³ - 10a^-1z⁵ - 3a^-1z⁷ + 2a^-1z⁹ - 6z² + 28z⁴ - 31z⁶ + 9z⁸ - 6az + 25az³ - 29az⁵ + 5az⁷ + 2az⁹ - 2a² + 7a²z² - 6a²z⁴ - 9a²z⁶ + 6a²z⁸ - a³z + 7a³z³ - 17a³z⁵ + 9a³z⁷ - a⁴ + 7a⁴z² - 14a⁴z⁴ + 9a⁴z⁶ + a⁵z - 4a⁵z³ + 6a⁵z⁵ + 3a⁶z⁴ + a⁷z³

V₂ and V₃, the type 2 and 3 Vassiliev invariants: {1, -1}

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 10₉₃. 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 r = 4 r = 5

j = 9 1

j = 7 2

j = 5 3 1

j = 3 5 2

j = 1 5 3

j = -1 6 5

j = -3 5 6

j = -5 4 5

j = -7 2 5

j = -9 1 4

j = -11 2

j = -13 1

n Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2 q^-17 - 3q^-16 + 3q^-15 + 3q^-14 - 14q^-13 + 16q^-12 + 4q^-11 - 35q^-10 + 39q^-9 + 7q^-8 - 63q^-7 + 57q^-6 + 21q^-5 - 85q^-4 + 54q^-3 + 41q^-2 - 88q^-1 + 35 + 52q - 71q² + 11q³ + 48q⁴ - 42q⁵ - 6q⁶ + 31q⁷ - 14q⁸ - 9q⁹ + 11q¹⁰ - q¹¹ - 3q¹² + q¹³

3 - q^-33 + 3q^-32 - 3q^-31 + 2q^-29 + 2q^-28 - 7q^-27 + q^-26 + 13q^-25 - 10q^-24 - 21q^-23 + 29q^-22 + 32q^-21 - 58q^-20 - 55q^-19 + 97q^-18 + 93q^-17 - 139q^-16 - 140q^-15 + 161q^-14 + 205q^-13 - 172q^-12 - 253q^-11 + 142q^-10 + 303q^-9 - 106q^-8 - 318q^-7 + 44q^-6 + 326q^-5 + 9q^-4 - 304q^-3 - 72q^-2 + 282q^-1 + 117 - 240q - 162q² + 196q³ + 190q⁴ - 137q⁵ - 209q⁶ + 80q⁷ + 202q⁸ - 15q⁹ - 183q¹⁰ - 31q¹¹ + 138q¹² + 66q¹³ - 91q¹⁴ - 74q¹⁵ + 44q¹⁶ + 64q¹⁷ - 11q¹⁸ - 43q¹⁹ - 6q²⁰ + 23q²¹ + 9q²² - 9q²³ - 5q²⁴ + q²⁵ + 3q²⁶ - q²⁷

4 q^-54 - 3q^-53 + 3q^-52 - 5q^-50 + 10q^-49 - 11q^-48 + 8q^-47 - 3q^-46 - 12q^-45 + 39q^-44 - 40q^-43 + q^-42 - q^-41 + 8q^-40 + 115q^-39 - 127q^-38 - 95q^-37 - 10q^-36 + 156q^-35 + 353q^-34 - 293q^-33 - 439q^-32 - 164q^-31 + 469q^-30 + 940q^-29 - 348q^-28 - 1042q^-27 - 697q^-26 + 673q^-25 + 1824q^-24 + 10q^-23 - 1501q^-22 - 1515q^-21 + 389q^-20 + 2489q^-19 + 688q^-18 - 1383q^-17 - 2092q^-16 - 277q^-15 + 2511q^-14 + 1195q^-13 - 796q^-12 - 2102q^-11 - 885q^-10 + 2042q^-9 + 1310q^-8 - 149q^-7 - 1742q^-6 - 1241q^-5 + 1432q^-4 + 1197q^-3 + 403q^-2 - 1264q^-1 - 1447 + 776q + 990q² + 889q³ - 682q⁴ - 1488q⁵ + 70q⁶ + 584q⁷ + 1191q⁸ + 26q⁹ - 1177q¹⁰ - 489q¹¹ - 49q¹² + 1055q¹³ + 601q¹⁴ - 505q¹⁵ - 576q¹⁶ - 593q¹⁷ + 489q¹⁸ + 681q¹⁹ + 124q²⁰ - 203q²¹ - 655q²² - 59q²³ + 316q²⁴ + 297q²⁵ + 164q²⁶ - 323q²⁷ - 199q²⁸ - 20q²⁹ + 121q³⁰ + 200q³¹ - 40q³² - 75q³³ - 77q³⁴ - 15q³⁵ + 74q³⁶ + 18q³⁷ + 4q³⁸ - 21q³⁹ - 19q⁴⁰ + 9q⁴¹ + 3q⁴² + 5q⁴³ - q⁴⁴ - 3q⁴⁵ + q⁴⁶

5 - q^-80 + 3q^-79 - 3q^-78 + 5q^-76 - 7q^-75 - q^-74 + 10q^-73 - 6q^-72 - 4q^-71 + 7q^-70 - 8q^-69 + 5q^-68 + 23q^-67 - 19q^-66 - 46q^-65 - 6q^-64 + 54q^-63 + 101q^-62 + 38q^-61 - 152q^-60 - 278q^-59 - 76q^-58 + 363q^-57 + 596q^-56 + 197q^-55 - 687q^-54 - 1202q^-53 - 505q^-52 + 1149q^-51 + 2213q^-50 + 1134q^-49 - 1678q^-48 - 3654q^-47 - 2296q^-46 + 2039q^-45 + 5542q^-44 + 4140q^-43 - 2009q^-42 - 7605q^-41 - 6671q^-40 + 1260q^-39 + 9464q^-38 + 9698q^-37 + 387q^-36 - 10704q^-35 - 12830q^-34 - 2759q^-33 + 10948q^-32 + 15431q^-31 + 5669q^-30 - 10111q^-29 - 17228q^-28 - 8438q^-27 + 8442q^-26 + 17753q^-25 + 10751q^-24 - 6227q^-23 - 17348q^-22 - 12189q^-21 + 4053q^-20 + 16041q^-19 + 12845q^-18 - 2098q^-17 - 14452q^-16 - 12805q^-15 + 580q^-14 + 12692q^-13 + 12452q^-12 + 683q^-11 - 11120q^-10 - 11949q^-9 - 1745q^-8 + 9512q^-7 + 11522q^-6 + 2905q^-5 - 7972q^-4 - 11067q^-3 - 4104q^-2 + 6142q^-1 + 10507 + 5447q - 4124q² - 9612q³ - 6628q⁴ + 1777q⁵ + 8244q⁶ + 7503q⁷ + 628q⁸ - 6314q⁹ - 7739q¹⁰ - 2894q¹¹ + 3939q¹² + 7185q¹³ + 4603q¹⁴ - 1309q¹⁵ - 5808q¹⁶ - 5542q¹⁷ - 1078q¹⁸ + 3786q¹⁹ + 5416q²⁰ + 2941q²¹ - 1490q²² - 4437q²³ - 3846q²⁴ - 556q²⁵ + 2756q²⁶ + 3782q²⁷ + 2015q²⁸ - 978q²⁹ - 2908q³⁰ - 2568q³¹ - 514q³² + 1614q³³ + 2340q³⁴ + 1365q³⁵ - 376q³⁶ - 1586q³⁷ - 1538q³⁸ - 465q³⁹ + 708q⁴⁰ + 1197q⁴¹ + 812q⁴² - 23q⁴³ - 677q⁴⁴ - 740q⁴⁵ - 307q⁴⁶ + 202q⁴⁷ + 463q⁴⁸ + 371q⁴⁹ + 62q⁵⁰ - 207q⁵¹ - 253q⁵² - 129q⁵³ + 22q⁵⁴ + 123q⁵⁵ + 115q⁵⁶ + 27q⁵⁷ - 41q⁵⁸ - 48q⁵⁹ - 31q⁶⁰ - 6q⁶¹ + 24q⁶² + 17q⁶³ + q⁶⁴ - 3q⁶⁵ - 3q⁶⁶ - 5q⁶⁷ + q⁶⁸ + 3q⁶⁹ - q⁷⁰

6 q^-111 - 3q^-110 + 3q^-109 - 5q^-107 + 7q^-106 - 2q^-105 + 2q^-104 - 12q^-103 + 13q^-102 + 9q^-101 - 32q^-100 + 19q^-99 + q^-98 + 2q^-97 - 8q^-96 + 47q^-95 + 8q^-94 - 140q^-93 - 4q^-92 + 41q^-91 + 98q^-90 + 128q^-89 + 145q^-88 - 161q^-87 - 638q^-86 - 251q^-85 + 256q^-84 + 782q^-83 + 964q^-82 + 450q^-81 - 1076q^-80 - 2623q^-79 - 1556q^-78 + 881q^-77 + 3493q^-76 + 4399q^-75 + 1900q^-74 - 3789q^-73 - 8844q^-72 - 6742q^-71 + 1149q^-70 + 10516q^-69 + 14801q^-68 + 8304q^-67 - 7771q^-66 - 22953q^-65 - 21967q^-64 - 4029q^-63 + 21177q^-62 + 36934q^-61 + 27722q^-60 - 5939q^-59 - 43133q^-58 - 52471q^-57 - 24938q^-56 + 25304q^-55 + 66556q^-54 + 65121q^-53 + 14490q^-52 - 55019q^-51 - 90391q^-50 - 65639q^-49 + 7736q^-48 + 84731q^-47 + 108329q^-46 + 55848q^-45 - 41784q^-44 - 112822q^-43 - 109773q^-42 - 31220q^-41 + 74597q^-40 + 132025q^-39 + 98093q^-38 - 7096q^-37 - 104791q^-36 - 131746q^-35 - 68995q^-34 + 43660q^-33 + 124960q^-32 + 117454q^-31 + 25482q^-30 - 77564q^-29 - 125207q^-28 - 85458q^-27 + 15029q^-26 + 101514q^-25 + 112681q^-24 + 40469q^-23 - 52531q^-22 - 106456q^-21 - 83866q^-20 - 484q^-19 + 80423q^-18 + 100459q^-17 + 44212q^-16 - 36351q^-15 - 90771q^-14 - 79734q^-13 - 10744q^-12 + 64708q^-11 + 92073q^-10 + 49855q^-9 - 20665q^-8 - 77915q^-7 - 80175q^-6 - 26053q^-5 + 45773q^-4 + 84289q^-3 + 60590q^-2 + 2410q^-1 - 58840 - 79226q - 46396q² + 17310q³ + 67282q⁴ + 68012q⁵ + 30543q⁶ - 28022q⁷ - 66026q⁸ - 61270q⁹ - 16628q¹⁰ + 35865q¹¹ + 60034q¹² + 51329q¹³ + 8980q¹⁴ - 35657q¹⁵ - 57393q¹⁶ - 41619q¹⁷ - 2674q¹⁸ + 32157q¹⁹ + 50409q²⁰ + 35598q²¹ + 2436q²² - 31230q²³ - 42448q²⁴ - 29444q²⁵ - 3605q²⁶ + 25837q²⁷ + 36305q²⁸ + 27183q²⁹ + 2206q³⁰ - 19470q³¹ - 29339q³² - 24547q³³ - 4302q³⁴ + 14471q³⁵ + 24979q³⁶ + 19415q³⁷ + 6499q³⁸ - 9175q³⁹ - 19757q⁴⁰ - 17038q⁴¹ - 7167q⁴² + 6295q⁴³ + 13180q⁴⁴ + 14542q⁴⁵ + 7650q⁴⁶ - 2996q⁴⁷ - 9559q⁴⁸ - 11243q⁴⁹ - 6301q⁵⁰ - 429q⁵¹ + 6419q⁵² + 8418q⁵³ + 5626q⁵⁴ + 1116q⁵⁵ - 3550q⁵⁶ - 5186q⁵⁷ - 5153q⁵⁸ - 1471q⁵⁹ + 1731q⁶⁰ + 3403q⁶¹ + 3332q⁶² + 1619q⁶³ - 210q⁶⁴ - 2284q⁶⁵ - 2108q⁶⁶ - 1313q⁶⁷ - 75q⁶⁸ + 906q⁶⁹ + 1287q⁷⁰ + 1123q⁷¹ + 78q⁷² - 345q⁷³ - 671q⁷⁴ - 558q⁷⁵ - 285q⁷⁶ + 107q⁷⁷ + 399q⁷⁸ + 231q⁷⁹ + 173q⁸⁰ - 13q⁸¹ - 103q⁸² - 163q⁸³ - 87q⁸⁴ + 25q⁸⁵ + 17q⁸⁶ + 54q⁸⁷ + 32q⁸⁸ + 18q⁸⁹ - 22q⁹⁰ - 20q⁹¹ + q⁹² - 7q⁹³ + 3q⁹⁴ + 3q⁹⁵ + 5q⁹⁶ - q⁹⁷ - 3q⁹⁸ + q⁹⁹

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, 93]]

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

In[3]:=
GaussCode[Knot[10, 93]]

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

In[4]:=
DTCode[Knot[10, 93]]

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

In[5]:=
br = BR[Knot[10, 93]]

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

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

Out[6]=
{4, 11}

In[7]:=
BraidIndex[Knot[10, 93]]

Out[7]=
4

In[8]:=
Show[DrawMorseLink[Knot[10, 93]]]

Out[8]=
-Graphics-

In[9]:=
#[Knot[10, 93]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}

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

In[10]:=
alex = Alexander[Knot[10, 93]][t]

Out[10]=
2 8 15 2 3 -17 + -- - -- + -- + 15 t - 8 t + 2 t 3 2 t t t

In[11]:=
Conway[Knot[10, 93]][z]

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

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

Out[12]=
{Knot[10, 93]}

In[13]:=
{KnotDet[Knot[10, 93]], KnotSignature[Knot[10, 93]]}

Out[13]=
{67, -2}

In[14]:=
Jones[Knot[10, 93]][q]

Out[14]=
-6 3 6 9 10 11 2 3 4 -10 - q + -- - -- + -- - -- + -- + 8 q - 5 q + 3 q - q 5 4 3 2 q q q q q

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

Out[15]=
{Knot[10, 93]}

In[16]:=
A2Invariant[Knot[10, 93]][q]

Out[16]=
-18 -16 -14 -12 2 -8 3 2 4 10 12 1 - q + q - q - q + --- - q + -- - 2 q + 2 q + q - q 10 6 q q

In[17]:=
HOMFLYPT[Knot[10, 93]][a, z]

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

In[18]:=
Kauffman[Knot[10, 93]][a, z]

Out[18]=
2 2 4 2 z 6 z 3 5 2 6 z 2 2 -2 a - a - --- - --- - 6 a z - a z + a z - 6 z - ---- + 7 a z + 3 a 2 a a 3 3 4 2 5 z 18 z 3 3 3 5 3 7 3 4 > 7 a z + ---- + ----- + 25 a z + 7 a z - 4 a z + a z + 28 z + 3 a a 4 5 5 17 z 2 4 4 4 6 4 4 z 10 z 5 3 5 > ----- - 6 a z - 14 a z + 3 a z - ---- - ----- - 29 a z - 17 a z + 2 3 a a a 6 7 7 5 5 6 13 z 2 6 4 6 z 3 z 7 > 6 a z - 31 z - ----- - 9 a z + 9 a z + -- - ---- + 5 a z + 2 3 a a a 8 9 3 7 8 3 z 2 8 2 z 9 > 9 a z + 9 z + ---- + 6 a z + ---- + 2 a z 2 a a

In[19]:=
{Vassiliev[2][Knot[10, 93]], Vassiliev[3][Knot[10, 93]]}

Out[19]=
{1, -1}

In[20]:=
Kh[Knot[10, 93]][q, t]

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

In[21]:=
ColouredJones[Knot[10, 93], 2][q]

Out[21]=
-17 3 3 3 14 16 4 35 39 7 63 57 21 35 + q - --- + --- + --- - --- + --- + --- - --- + -- + -- - -- + -- + -- - 16 15 14 13 12 11 10 9 8 7 6 5 q q q q q q q q q q q q 85 54 41 88 2 3 4 5 6 7 > -- + -- + -- - -- + 52 q - 71 q + 11 q + 48 q - 42 q - 6 q + 31 q - 4 3 2 q q q q 8 9 10 11 12 13 > 14 q - 9 q + 11 q - q - 3 q + q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 10₉₃

10₉₂
10₉₄

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q^-17 - 3q^-16 + 3q^-15 + 3q^-14 - 14q^-13 + 16q^-12 + 4q^-11 - 35q^-10 + 39q^-9 + 7q^-8 - 63q^-7 + 57q^-6 + 21q^-5 - 85q^-4 + 54q^-3 + 41q^-2 - 88q^-1 + 35 + 52q - 71q² + 11q³ + 48q⁴ - 42q⁵ - 6q⁶ + 31q⁷ - 14q⁸ - 9q⁹ + 11q¹⁰ - q¹¹ - 3q¹² + q¹³
3	- q^-33 + 3q^-32 - 3q^-31 + 2q^-29 + 2q^-28 - 7q^-27 + q^-26 + 13q^-25 - 10q^-24 - 21q^-23 + 29q^-22 + 32q^-21 - 58q^-20 - 55q^-19 + 97q^-18 + 93q^-17 - 139q^-16 - 140q^-15 + 161q^-14 + 205q^-13 - 172q^-12 - 253q^-11 + 142q^-10 + 303q^-9 - 106q^-8 - 318q^-7 + 44q^-6 + 326q^-5 + 9q^-4 - 304q^-3 - 72q^-2 + 282q^-1 + 117 - 240q - 162q² + 196q³ + 190q⁴ - 137q⁵ - 209q⁶ + 80q⁷ + 202q⁸ - 15q⁹ - 183q¹⁰ - 31q¹¹ + 138q¹² + 66q¹³ - 91q¹⁴ - 74q¹⁵ + 44q¹⁶ + 64q¹⁷ - 11q¹⁸ - 43q¹⁹ - 6q²⁰ + 23q²¹ + 9q²² - 9q²³ - 5q²⁴ + q²⁵ + 3q²⁶ - q²⁷
4	q^-54 - 3q^-53 + 3q^-52 - 5q^-50 + 10q^-49 - 11q^-48 + 8q^-47 - 3q^-46 - 12q^-45 + 39q^-44 - 40q^-43 + q^-42 - q^-41 + 8q^-40 + 115q^-39 - 127q^-38 - 95q^-37 - 10q^-36 + 156q^-35 + 353q^-34 - 293q^-33 - 439q^-32 - 164q^-31 + 469q^-30 + 940q^-29 - 348q^-28 - 1042q^-27 - 697q^-26 + 673q^-25 + 1824q^-24 + 10q^-23 - 1501q^-22 - 1515q^-21 + 389q^-20 + 2489q^-19 + 688q^-18 - 1383q^-17 - 2092q^-16 - 277q^-15 + 2511q^-14 + 1195q^-13 - 796q^-12 - 2102q^-11 - 885q^-10 + 2042q^-9 + 1310q^-8 - 149q^-7 - 1742q^-6 - 1241q^-5 + 1432q^-4 + 1197q^-3 + 403q^-2 - 1264q^-1 - 1447 + 776q + 990q² + 889q³ - 682q⁴ - 1488q⁵ + 70q⁶ + 584q⁷ + 1191q⁸ + 26q⁹ - 1177q¹⁰ - 489q¹¹ - 49q¹² + 1055q¹³ + 601q¹⁴ - 505q¹⁵ - 576q¹⁶ - 593q¹⁷ + 489q¹⁸ + 681q¹⁹ + 124q²⁰ - 203q²¹ - 655q²² - 59q²³ + 316q²⁴ + 297q²⁵ + 164q²⁶ - 323q²⁷ - 199q²⁸ - 20q²⁹ + 121q³⁰ + 200q³¹ - 40q³² - 75q³³ - 77q³⁴ - 15q³⁵ + 74q³⁶ + 18q³⁷ + 4q³⁸ - 21q³⁹ - 19q⁴⁰ + 9q⁴¹ + 3q⁴² + 5q⁴³ - q⁴⁴ - 3q⁴⁵ + q⁴⁶
5	- q^-80 + 3q^-79 - 3q^-78 + 5q^-76 - 7q^-75 - q^-74 + 10q^-73 - 6q^-72 - 4q^-71 + 7q^-70 - 8q^-69 + 5q^-68 + 23q^-67 - 19q^-66 - 46q^-65 - 6q^-64 + 54q^-63 + 101q^-62 + 38q^-61 - 152q^-60 - 278q^-59 - 76q^-58 + 363q^-57 + 596q^-56 + 197q^-55 - 687q^-54 - 1202q^-53 - 505q^-52 + 1149q^-51 + 2213q^-50 + 1134q^-49 - 1678q^-48 - 3654q^-47 - 2296q^-46 + 2039q^-45 + 5542q^-44 + 4140q^-43 - 2009q^-42 - 7605q^-41 - 6671q^-40 + 1260q^-39 + 9464q^-38 + 9698q^-37 + 387q^-36 - 10704q^-35 - 12830q^-34 - 2759q^-33 + 10948q^-32 + 15431q^-31 + 5669q^-30 - 10111q^-29 - 17228q^-28 - 8438q^-27 + 8442q^-26 + 17753q^-25 + 10751q^-24 - 6227q^-23 - 17348q^-22 - 12189q^-21 + 4053q^-20 + 16041q^-19 + 12845q^-18 - 2098q^-17 - 14452q^-16 - 12805q^-15 + 580q^-14 + 12692q^-13 + 12452q^-12 + 683q^-11 - 11120q^-10 - 11949q^-9 - 1745q^-8 + 9512q^-7 + 11522q^-6 + 2905q^-5 - 7972q^-4 - 11067q^-3 - 4104q^-2 + 6142q^-1 + 10507 + 5447q - 4124q² - 9612q³ - 6628q⁴ + 1777q⁵ + 8244q⁶ + 7503q⁷ + 628q⁸ - 6314q⁹ - 7739q¹⁰ - 2894q¹¹ + 3939q¹² + 7185q¹³ + 4603q¹⁴ - 1309q¹⁵ - 5808q¹⁶ - 5542q¹⁷ - 1078q¹⁸ + 3786q¹⁹ + 5416q²⁰ + 2941q²¹ - 1490q²² - 4437q²³ - 3846q²⁴ - 556q²⁵ + 2756q²⁶ + 3782q²⁷ + 2015q²⁸ - 978q²⁹ - 2908q³⁰ - 2568q³¹ - 514q³² + 1614q³³ + 2340q³⁴ + 1365q³⁵ - 376q³⁶ - 1586q³⁷ - 1538q³⁸ - 465q³⁹ + 708q⁴⁰ + 1197q⁴¹ + 812q⁴² - 23q⁴³ - 677q⁴⁴ - 740q⁴⁵ - 307q⁴⁶ + 202q⁴⁷ + 463q⁴⁸ + 371q⁴⁹ + 62q⁵⁰ - 207q⁵¹ - 253q⁵² - 129q⁵³ + 22q⁵⁴ + 123q⁵⁵ + 115q⁵⁶ + 27q⁵⁷ - 41q⁵⁸ - 48q⁵⁹ - 31q⁶⁰ - 6q⁶¹ + 24q⁶² + 17q⁶³ + q⁶⁴ - 3q⁶⁵ - 3q⁶⁶ - 5q⁶⁷ + q⁶⁸ + 3q⁶⁹ - q⁷⁰
6	q^-111 - 3q^-110 + 3q^-109 - 5q^-107 + 7q^-106 - 2q^-105 + 2q^-104 - 12q^-103 + 13q^-102 + 9q^-101 - 32q^-100 + 19q^-99 + q^-98 + 2q^-97 - 8q^-96 + 47q^-95 + 8q^-94 - 140q^-93 - 4q^-92 + 41q^-91 + 98q^-90 + 128q^-89 + 145q^-88 - 161q^-87 - 638q^-86 - 251q^-85 + 256q^-84 + 782q^-83 + 964q^-82 + 450q^-81 - 1076q^-80 - 2623q^-79 - 1556q^-78 + 881q^-77 + 3493q^-76 + 4399q^-75 + 1900q^-74 - 3789q^-73 - 8844q^-72 - 6742q^-71 + 1149q^-70 + 10516q^-69 + 14801q^-68 + 8304q^-67 - 7771q^-66 - 22953q^-65 - 21967q^-64 - 4029q^-63 + 21177q^-62 + 36934q^-61 + 27722q^-60 - 5939q^-59 - 43133q^-58 - 52471q^-57 - 24938q^-56 + 25304q^-55 + 66556q^-54 + 65121q^-53 + 14490q^-52 - 55019q^-51 - 90391q^-50 - 65639q^-49 + 7736q^-48 + 84731q^-47 + 108329q^-46 + 55848q^-45 - 41784q^-44 - 112822q^-43 - 109773q^-42 - 31220q^-41 + 74597q^-40 + 132025q^-39 + 98093q^-38 - 7096q^-37 - 104791q^-36 - 131746q^-35 - 68995q^-34 + 43660q^-33 + 124960q^-32 + 117454q^-31 + 25482q^-30 - 77564q^-29 - 125207q^-28 - 85458q^-27 + 15029q^-26 + 101514q^-25 + 112681q^-24 + 40469q^-23 - 52531q^-22 - 106456q^-21 - 83866q^-20 - 484q^-19 + 80423q^-18 + 100459q^-17 + 44212q^-16 - 36351q^-15 - 90771q^-14 - 79734q^-13 - 10744q^-12 + 64708q^-11 + 92073q^-10 + 49855q^-9 - 20665q^-8 - 77915q^-7 - 80175q^-6 - 26053q^-5 + 45773q^-4 + 84289q^-3 + 60590q^-2 + 2410q^-1 - 58840 - 79226q - 46396q² + 17310q³ + 67282q⁴ + 68012q⁵ + 30543q⁶ - 28022q⁷ - 66026q⁸ - 61270q⁹ - 16628q¹⁰ + 35865q¹¹ + 60034q¹² + 51329q¹³ + 8980q¹⁴ - 35657q¹⁵ - 57393q¹⁶ - 41619q¹⁷ - 2674q¹⁸ + 32157q¹⁹ + 50409q²⁰ + 35598q²¹ + 2436q²² - 31230q²³ - 42448q²⁴ - 29444q²⁵ - 3605q²⁶ + 25837q²⁷ + 36305q²⁸ + 27183q²⁹ + 2206q³⁰ - 19470q³¹ - 29339q³² - 24547q³³ - 4302q³⁴ + 14471q³⁵ + 24979q³⁶ + 19415q³⁷ + 6499q³⁸ - 9175q³⁹ - 19757q⁴⁰ - 17038q⁴¹ - 7167q⁴² + 6295q⁴³ + 13180q⁴⁴ + 14542q⁴⁵ + 7650q⁴⁶ - 2996q⁴⁷ - 9559q⁴⁸ - 11243q⁴⁹ - 6301q⁵⁰ - 429q⁵¹ + 6419q⁵² + 8418q⁵³ + 5626q⁵⁴ + 1116q⁵⁵ - 3550q⁵⁶ - 5186q⁵⁷ - 5153q⁵⁸ - 1471q⁵⁹ + 1731q⁶⁰ + 3403q⁶¹ + 3332q⁶² + 1619q⁶³ - 210q⁶⁴ - 2284q⁶⁵ - 2108q⁶⁶ - 1313q⁶⁷ - 75q⁶⁸ + 906q⁶⁹ + 1287q⁷⁰ + 1123q⁷¹ + 78q⁷² - 345q⁷³ - 671q⁷⁴ - 558q⁷⁵ - 285q⁷⁶ + 107q⁷⁷ + 399q⁷⁸ + 231q⁷⁹ + 173q⁸⁰ - 13q⁸¹ - 103q⁸² - 163q⁸³ - 87q⁸⁴ + 25q⁸⁵ + 17q⁸⁶ + 54q⁸⁷ + 32q⁸⁸ + 18q⁸⁹ - 22q⁹⁰ - 20q⁹¹ + q⁹² - 7q⁹³ + 3q⁹⁴ + 3q⁹⁵ + 5q⁹⁶ - q⁹⁷ - 3q⁹⁸ + q⁹⁹

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	PD[Knot[10, 93]]
Out[2]=	PD[X[6, 2, 7, 1], X[16, 6, 17, 5], X[20, 8, 1, 7], X[18, 13, 19, 14], > X[14, 9, 15, 10], X[10, 3, 11, 4], X[4, 11, 5, 12], X[12, 17, 13, 18], > X[8, 20, 9, 19], X[2, 16, 3, 15]]
In[3]:=	GaussCode[Knot[10, 93]]
Out[3]=	GaussCode[1, -10, 6, -7, 2, -1, 3, -9, 5, -6, 7, -8, 4, -5, 10, -2, 8, -4, 9, > -3]
In[4]:=	DTCode[Knot[10, 93]]
Out[4]=	DTCode[6, 10, 16, 20, 14, 4, 18, 2, 12, 8]
In[5]:=	br = BR[Knot[10, 93]]
Out[5]=	BR[4, {-1, -1, 2, -1, -1, 2, -1, 2, 3, -2, 3}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{4, 11}
In[7]:=	BraidIndex[Knot[10, 93]]
Out[7]=	4
In[8]:=	Show[DrawMorseLink[Knot[10, 93]]]

Out[8]=	-Graphics-
In[9]:=	#[Knot[10, 93]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Chiral, 2, 3, 3, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 93]][t]
Out[10]=	2 8 15 2 3 -17 + -- - -- + -- + 15 t - 8 t + 2 t 3 2 t t t
In[11]:=	Conway[Knot[10, 93]][z]
Out[11]=	2 4 6 1 + z + 4 z + 2 z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 93]}
In[13]:=	{KnotDet[Knot[10, 93]], KnotSignature[Knot[10, 93]]}
Out[13]=	{67, -2}
In[14]:=	Jones[Knot[10, 93]][q]
Out[14]=	-6 3 6 9 10 11 2 3 4 -10 - q + -- - -- + -- - -- + -- + 8 q - 5 q + 3 q - q 5 4 3 2 q q q q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] \|\| (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[10, 93]}
In[16]:=	A2Invariant[Knot[10, 93]][q]
Out[16]=	-18 -16 -14 -12 2 -8 3 2 4 10 12 1 - q + q - q - q + --- - q + -- - 2 q + 2 q + q - q 10 6 q q
In[17]:=	HOMFLYPT[Knot[10, 93]][a, z]
Out[17]=	2 4 2 4 2 2 z 2 2 4 2 4 z 2 4 4 4 2 a - a + 2 z - ---- + 3 a z - 2 a z + 3 z - -- + 3 a z - a z + 2 2 a a 6 2 6 > z + a z
In[18]:=	Kauffman[Knot[10, 93]][a, z]
Out[18]=	2 2 4 2 z 6 z 3 5 2 6 z 2 2 -2 a - a - --- - --- - 6 a z - a z + a z - 6 z - ---- + 7 a z + 3 a 2 a a 3 3 4 2 5 z 18 z 3 3 3 5 3 7 3 4 > 7 a z + ---- + ----- + 25 a z + 7 a z - 4 a z + a z + 28 z + 3 a a 4 5 5 17 z 2 4 4 4 6 4 4 z 10 z 5 3 5 > ----- - 6 a z - 14 a z + 3 a z - ---- - ----- - 29 a z - 17 a z + 2 3 a a a 6 7 7 5 5 6 13 z 2 6 4 6 z 3 z 7 > 6 a z - 31 z - ----- - 9 a z + 9 a z + -- - ---- + 5 a z + 2 3 a a a 8 9 3 7 8 3 z 2 8 2 z 9 > 9 a z + 9 z + ---- + 6 a z + ---- + 2 a z 2 a a
In[19]:=	{Vassiliev[2][Knot[10, 93]], Vassiliev[3][Knot[10, 93]]}
Out[19]=	{1, -1}
In[20]:=	Kh[Knot[10, 93]][q, t]
Out[20]=	6 6 1 2 1 4 2 5 4 5 -- + - + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ---- + 3 q 13 5 11 4 9 4 9 3 7 3 7 2 5 2 5 q q t q t q t q t q t q t q t q t 5 5 t 2 3 2 3 3 5 3 5 4 > ---- + --- + 5 q t + 3 q t + 5 q t + 2 q t + 3 q t + q t + 3 q q t 7 4 9 5 > 2 q t + q t
In[21]:=	ColouredJones[Knot[10, 93], 2][q]
Out[21]=	-17 3 3 3 14 16 4 35 39 7 63 57 21 35 + q - --- + --- + --- - --- + --- + --- - --- + -- + -- - -- + -- + -- - 16 15 14 13 12 11 10 9 8 7 6 5 q q q q q q q q q q q q 85 54 41 88 2 3 4 5 6 7 > -- + -- + -- - -- + 52 q - 71 q + 11 q + 48 q - 42 q - 6 q + 31 q - 4 3 2 q q q q 8 9 10 11 12 13 > 14 q - 9 q + 11 q - q - 3 q + q

The Alternating Knot 1093

The Alternating Knot 10₉₃