The Alternating Knot 9₃₁

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Acknowledgement

KnotPlot

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

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

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

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 2 / 4--6 1

Alexander Polynomial: t^-3 - 5t^-2 + 13t^-1 - 17 + 13t - 5t² + t³

Conway Polynomial: 1 + 2z² + z⁴ + z⁶

Other knots with the same Alexander/Conway Polynomial: {K11n11, K11n22, K11n112, K11n127, ...}

Determinant and Signature: {55, -2}

Jones Polynomial: q^-7 - 4q^-6 + 6q^-5 - 8q^-4 + 10q^-3 - 9q^-2 + 8q^-1 - 5 + 3q - q²

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

A2 (sl(3)) Invariant: q^-22 - q^-20 - 2q^-18 + q^-16 - 2q^-14 + q^-12 + q^-10 + 3q^-6 - q^-4 + 3q^-2 - q² + q⁴ - q⁶

HOMFLY-PT Polynomial: - 1 - 2z² - z⁴ + 4a² + 7a²z² + 4a²z⁴ + a²z⁶ - 2a⁴ - 4a⁴z² - 2a⁴z⁴ + a⁶z²

Kauffman Polynomial: a^-1z - 2a^-1z³ + a^-1z⁵ - 1 + 5z² - 7z⁴ + 3z⁶ + 3az - 3az³ - 3az⁵ + 3az⁷ - 4a² + 15a²z² - 21a²z⁴ + 8a²z⁶ + a²z⁸ + 5a³z - 5a³z³ - 7a³z⁵ + 7a³z⁷ - 2a⁴ + 13a⁴z² - 23a⁴z⁴ + 11a⁴z⁶ + a⁴z⁸ + 3a⁵z - 8a⁵z³ + a⁵z⁵ + 4a⁵z⁷ + 3a⁶z² - 8a⁶z⁴ + 6a⁶z⁶ - 4a⁷z³ + 4a⁷z⁵ + a⁸z⁴

V₂ and V₃, the type 2 and 3 Vassiliev invariants: {2, -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 9₃₁. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

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

j = 5 1

j = 3 2

j = 1 3 1

j = -1 5 2

j = -3 5 4

j = -5 5 4

j = -7 3 5

j = -9 3 5

j = -11 1 3

j = -13 3

j = -15 1

n Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2 q^-20 - 4q^-19 + 2q^-18 + 12q^-17 - 21q^-16 + q^-15 + 38q^-14 - 46q^-13 - 7q^-12 + 71q^-11 - 65q^-10 - 21q^-9 + 93q^-8 - 66q^-7 - 32q^-6 + 91q^-5 - 48q^-4 - 35q^-3 + 66q^-2 - 24q^-1 - 27 + 33q - 6q² - 13q³ + 10q⁴ - 3q⁶ + q⁷

3 q^-39 - 4q^-38 + 2q^-37 + 8q^-36 - q^-35 - 20q^-34 - 5q^-33 + 46q^-32 + 8q^-31 - 71q^-30 - 32q^-29 + 116q^-28 + 58q^-27 - 155q^-26 - 108q^-25 + 204q^-24 + 160q^-23 - 238q^-22 - 222q^-21 + 262q^-20 + 285q^-19 - 277q^-18 - 333q^-17 + 265q^-16 + 380q^-15 - 256q^-14 - 393q^-13 + 213q^-12 + 409q^-11 - 179q^-10 - 389q^-9 + 123q^-8 + 369q^-7 - 80q^-6 - 318q^-5 + 26q^-4 + 273q^-3 + 4q^-2 - 207q^-1 - 34 + 153q + 41q² - 98q³ - 42q⁴ + 58q⁵ + 33q⁶ - 29q⁷ - 23q⁸ + 13q⁹ + 13q¹⁰ - 5q¹¹ - 5q¹² + 3q¹⁴ - q¹⁵

4 q^-64 - 4q^-63 + 2q^-62 + 8q^-61 - 5q^-60 - 26q^-58 + 13q^-57 + 47q^-56 - 11q^-55 - 4q^-54 - 117q^-53 + 22q^-52 + 168q^-51 + 27q^-50 - 11q^-49 - 354q^-48 - 27q^-47 + 396q^-46 + 208q^-45 + 45q^-44 - 786q^-43 - 259q^-42 + 665q^-41 + 608q^-40 + 279q^-39 - 1327q^-38 - 729q^-37 + 811q^-36 + 1135q^-35 + 741q^-34 - 1774q^-33 - 1317q^-32 + 731q^-31 + 1574q^-30 + 1308q^-29 - 1966q^-28 - 1803q^-27 + 463q^-26 + 1772q^-25 + 1790q^-24 - 1879q^-23 - 2044q^-22 + 114q^-21 + 1699q^-20 + 2067q^-19 - 1550q^-18 - 2012q^-17 - 257q^-16 + 1393q^-15 + 2120q^-14 - 1046q^-13 - 1736q^-12 - 581q^-11 + 916q^-10 + 1926q^-9 - 479q^-8 - 1258q^-7 - 757q^-6 + 381q^-5 + 1496q^-4 - 24q^-3 - 694q^-2 - 698q^-1 - 31 + 932q + 179q² - 230q³ - 455q⁴ - 197q⁵ + 434q⁶ + 156q⁷ + 5q⁸ - 197q⁹ - 163q¹⁰ + 141q¹¹ + 65q¹² + 48q¹³ - 52q¹⁴ - 74q¹⁵ + 32q¹⁶ + 12q¹⁷ + 23q¹⁸ - 6q¹⁹ - 20q²⁰ + 5q²¹ + 5q²³ - 3q²⁵ + q²⁶

5 q^-95 - 4q^-94 + 2q^-93 + 8q^-92 - 5q^-91 - 4q^-90 - 6q^-89 - 8q^-88 + 14q^-87 + 38q^-86 + 3q^-85 - 44q^-84 - 58q^-83 - 27q^-82 + 72q^-81 + 140q^-80 + 82q^-79 - 135q^-78 - 292q^-77 - 157q^-76 + 191q^-75 + 488q^-74 + 401q^-73 - 233q^-72 - 874q^-71 - 729q^-70 + 232q^-69 + 1256q^-68 + 1357q^-67 - 36q^-66 - 1849q^-65 - 2164q^-64 - 337q^-63 + 2319q^-62 + 3297q^-61 + 1065q^-60 - 2802q^-59 - 4572q^-58 - 2081q^-57 + 2988q^-56 + 5982q^-55 + 3432q^-54 - 2962q^-53 - 7300q^-52 - 4978q^-51 + 2583q^-50 + 8454q^-49 + 6608q^-48 - 1930q^-47 - 9332q^-46 - 8162q^-45 + 1094q^-44 + 9834q^-43 + 9526q^-42 - 78q^-41 - 10078q^-40 - 10622q^-39 - 850q^-38 + 9913q^-37 + 11379q^-36 + 1898q^-35 - 9620q^-34 - 11871q^-33 - 2685q^-32 + 8966q^-31 + 12014q^-30 + 3587q^-29 - 8242q^-28 - 11966q^-27 - 4226q^-26 + 7224q^-25 + 11585q^-24 + 4979q^-23 - 6142q^-22 - 11023q^-21 - 5471q^-20 + 4798q^-19 + 10161q^-18 + 5986q^-17 - 3457q^-16 - 9091q^-15 - 6158q^-14 + 1973q^-13 + 7754q^-12 + 6241q^-11 - 682q^-10 - 6289q^-9 - 5876q^-8 - 527q^-7 + 4704q^-6 + 5366q^-5 + 1357q^-4 - 3230q^-3 - 4480q^-2 - 1899q^-1 + 1879 + 3554q + 2036q² - 850q³ - 2531q⁴ - 1908q⁵ + 125q⁶ + 1656q⁷ + 1555q⁸ + 274q⁹ - 933q¹⁰ - 1148q¹¹ - 423q¹² + 453q¹³ + 743q¹⁴ + 400q¹⁵ - 148q¹⁶ - 435q¹⁷ - 311q¹⁸ + 11q¹⁹ + 229q²⁰ + 196q²¹ + 33q²² - 92q²³ - 116q²⁴ - 45q²⁵ + 44q²⁶ + 59q²⁷ + 18q²⁸ - 6q²⁹ - 21q³⁰ - 23q³¹ + 6q³² + 13q³³ + 2q³⁴ - 5q³⁷ + 3q³⁹ - q⁴⁰

6 q^-132 - 4q^-131 + 2q^-130 + 8q^-129 - 5q^-128 - 4q^-127 - 10q^-126 + 12q^-125 - 7q^-124 + 5q^-123 + 52q^-122 - 27q^-121 - 38q^-120 - 62q^-119 + 30q^-118 + 9q^-117 + 62q^-116 + 209q^-115 - 65q^-114 - 188q^-113 - 307q^-112 + 7q^-111 + 46q^-110 + 344q^-109 + 770q^-108 + 4q^-107 - 578q^-106 - 1170q^-105 - 461q^-104 - 53q^-103 + 1162q^-102 + 2462q^-101 + 860q^-100 - 1094q^-99 - 3313q^-98 - 2500q^-97 - 1271q^-96 + 2500q^-95 + 6368q^-94 + 4208q^-93 - 463q^-92 - 6870q^-91 - 7710q^-90 - 5902q^-89 + 2804q^-88 + 12711q^-87 + 12156q^-86 + 4182q^-85 - 9916q^-84 - 16395q^-83 - 16397q^-82 - 1246q^-81 + 19092q^-80 + 24803q^-79 + 15450q^-78 - 8609q^-77 - 25661q^-76 - 32402q^-75 - 12206q^-74 + 21191q^-73 + 38503q^-72 + 32408q^-71 - 383q^-70 - 30788q^-69 - 49484q^-68 - 28624q^-67 + 16453q^-66 + 48063q^-65 + 50088q^-64 + 13089q^-63 - 29244q^-62 - 62142q^-61 - 45404q^-60 + 6528q^-59 + 50837q^-58 + 63153q^-57 + 27105q^-56 - 22494q^-55 - 67781q^-54 - 57815q^-53 - 4580q^-52 + 47919q^-51 + 69463q^-50 + 37884q^-49 - 13791q^-48 - 67262q^-47 - 64351q^-46 - 14105q^-45 + 41739q^-44 + 69948q^-43 + 44639q^-42 - 5048q^-41 - 62441q^-40 - 66031q^-39 - 21818q^-38 + 33477q^-37 + 66103q^-36 + 48411q^-35 + 3859q^-34 - 54014q^-33 - 63964q^-32 - 28580q^-31 + 22824q^-30 + 58226q^-29 + 49748q^-28 + 13471q^-27 - 41627q^-26 - 57914q^-25 - 34155q^-24 + 9710q^-23 + 45798q^-22 + 47556q^-21 + 22695q^-20 - 25659q^-19 - 46928q^-18 - 36402q^-17 - 3836q^-16 + 29361q^-15 + 40118q^-14 + 28404q^-13 - 8919q^-12 - 31464q^-11 - 32809q^-10 - 13740q^-9 + 12152q^-8 + 27611q^-7 + 27561q^-6 + 3854q^-5 - 14998q^-4 - 23412q^-3 - 16582q^-2 - 834q^-1 + 13631 + 20363q + 9173q² - 2596q³ - 12013q⁴ - 12771q⁵ - 6415q⁶ + 3157q⁷ + 10912q⁸ + 7804q⁹ + 2952q¹⁰ - 3466q¹¹ - 6496q¹² - 5813q¹³ - 1464q¹⁴ + 3835q¹⁵ + 3922q¹⁶ + 3153q¹⁷ + 288q¹⁸ - 1903q¹⁹ - 3035q²⁰ - 1834q²¹ + 642q²² + 1094q²³ + 1581q²⁴ + 798q²⁵ - 59q²⁶ - 1017q²⁷ - 914q²⁸ - 75q²⁹ + 50q³⁰ + 471q³¹ + 380q³² + 216q³³ - 223q³⁴ - 283q³⁵ - 53q³⁶ - 84q³⁷ + 80q³⁸ + 97q³⁹ + 109q⁴⁰ - 36q⁴¹ - 62q⁴² - 3q⁴³ - 35q⁴⁴ + 6q⁴⁵ + 12q⁴⁶ + 32q⁴⁷ - 6q⁴⁸ - 13q⁴⁹ + 5q⁵⁰ - 7q⁵¹ + 5q⁵⁴ - 3q⁵⁶ + q⁵⁷

7 q^-175 - 4q^-174 + 2q^-173 + 8q^-172 - 5q^-171 - 4q^-170 - 10q^-169 + 8q^-168 + 13q^-167 - 16q^-166 + 19q^-165 + 22q^-164 - 21q^-163 - 32q^-162 - 58q^-161 + 12q^-160 + 84q^-159 + 13q^-158 + 91q^-157 + 54q^-156 - 111q^-155 - 153q^-154 - 291q^-153 - 45q^-152 + 306q^-151 + 312q^-150 + 498q^-149 + 219q^-148 - 417q^-147 - 724q^-146 - 1183q^-145 - 598q^-144 + 738q^-143 + 1479q^-142 + 2273q^-141 + 1377q^-140 - 821q^-139 - 2533q^-138 - 4364q^-137 - 3319q^-136 + 506q^-135 + 4294q^-134 + 7817q^-133 + 6644q^-132 + 769q^-131 - 5882q^-130 - 12795q^-129 - 12926q^-128 - 4537q^-127 + 7342q^-126 + 20015q^-125 + 22388q^-124 + 11484q^-123 - 6640q^-122 - 28010q^-121 - 36511q^-120 - 24388q^-119 + 2513q^-118 + 36778q^-117 + 54726q^-116 + 43408q^-115 + 7899q^-114 - 42871q^-113 - 76524q^-112 - 70626q^-111 - 26370q^-110 + 44780q^-109 + 99484q^-108 + 104380q^-107 + 54577q^-106 - 38596q^-105 - 120792q^-104 - 143625q^-103 - 92456q^-102 + 22975q^-101 + 136842q^-100 + 184498q^-99 + 138565q^-98 + 3495q^-97 - 144791q^-96 - 223562q^-95 - 189620q^-94 - 39569q^-93 + 142628q^-92 + 256697q^-91 + 241727q^-90 + 83046q^-89 - 130280q^-88 - 281264q^-87 - 290581q^-86 - 130107q^-85 + 109056q^-84 + 295683q^-83 + 332631q^-82 + 176913q^-81 - 81393q^-80 - 300125q^-79 - 365865q^-78 - 220045q^-77 + 50789q^-76 + 296150q^-75 + 388976q^-74 + 256729q^-73 - 19511q^-72 - 285424q^-71 - 403208q^-70 - 286451q^-69 - 9332q^-68 + 270954q^-67 + 408982q^-66 + 308177q^-65 + 35629q^-64 - 253431q^-63 - 409006q^-62 - 324132q^-61 - 58083q^-60 + 235404q^-59 + 403707q^-58 + 334089q^-57 + 78338q^-56 - 215553q^-55 - 395109q^-54 - 341020q^-53 - 96510q^-52 + 195129q^-51 + 382831q^-50 + 344198q^-49 + 114478q^-48 - 171729q^-47 - 367246q^-46 - 345612q^-45 - 132536q^-44 + 145965q^-43 + 347125q^-42 + 343582q^-41 + 151346q^-40 - 115679q^-39 - 322041q^-38 - 338536q^-37 - 169976q^-36 + 82021q^-35 + 290651q^-34 + 327952q^-33 + 187731q^-32 - 44536q^-31 - 253245q^-30 - 311536q^-29 - 201908q^-28 + 5971q^-27 + 209431q^-26 + 287116q^-25 + 211005q^-24 + 32323q^-23 - 161687q^-22 - 255401q^-21 - 211910q^-20 - 66028q^-19 + 111601q^-18 + 215933q^-17 + 204020q^-16 + 93241q^-15 - 63419q^-14 - 172048q^-13 - 186238q^-12 - 110076q^-11 + 20176q^-10 + 125608q^-9 + 160268q^-8 + 116494q^-7 + 14371q^-6 - 81621q^-5 - 128455q^-4 - 111581q^-3 - 38228q^-2 + 42739q^-1 + 94484 + 98221q + 50763q² - 12618q³ - 62159q⁴ - 78875q⁵ - 53130q⁶ - 7965q⁷ + 34751q⁸ + 57676q⁹ + 47751q¹⁰ + 18959q¹¹ - 14077q¹² - 37695q¹³ - 38050q¹⁴ - 22298q¹⁵ + 863q¹⁶ + 21458q¹⁷ + 26846q¹⁸ + 20328q¹⁹ + 6061q²⁰ - 9805q²¹ - 16819q²² - 15843q²³ - 8147q²⁴ + 2824q²⁵ + 9072q²⁶ + 10657q²⁷ + 7458q²⁸ + 722q²⁹ - 3948q³⁰ - 6380q³¹ - 5643q³² - 1803q³³ + 1186q³⁴ + 3273q³⁵ + 3579q³⁶ + 1744q³⁷ + 161q³⁸ - 1402q³⁹ - 2126q⁴⁰ - 1271q⁴¹ - 427q⁴² + 499q⁴³ + 1028q⁴⁴ + 689q⁴⁵ + 477q⁴⁶ - 29q⁴⁷ - 517q⁴⁸ - 409q⁴⁹ - 287q⁵⁰ - 6q⁵¹ + 200q⁵² + 120q⁵³ + 172q⁵⁴ + 96q⁵⁵ - 83q⁵⁶ - 87q⁵⁷ - 90q⁵⁸ - 17q⁵⁹ + 43q⁶⁰ - 5q⁶¹ + 31q⁶² + 35q⁶³ - 6q⁶⁴ - 12q⁶⁵ - 23q⁶⁶ - 3q⁶⁷ + 13q⁶⁸ - 5q⁶⁹ + 7q⁷¹ - 5q⁷⁴ + 3q⁷⁶ - q⁷⁷

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, 31]]

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

In[3]:=
GaussCode[Knot[9, 31]]

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

In[4]:=
DTCode[Knot[9, 31]]

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

In[5]:=
br = BR[Knot[9, 31]]

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

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

Out[6]=
{4, 9}

In[7]:=
BraidIndex[Knot[9, 31]]

Out[7]=
4

In[8]:=
Show[DrawMorseLink[Knot[9, 31]]]

Out[8]=
-Graphics-

In[9]:=
#[Knot[9, 31]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}

Out[9]=
{Reversible, 2, 3, 2, {4, 6}, 1}

In[10]:=
alex = Alexander[Knot[9, 31]][t]

Out[10]=
-3 5 13 2 3 -17 + t - -- + -- + 13 t - 5 t + t 2 t t

In[11]:=
Conway[Knot[9, 31]][z]

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

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

Out[12]=
{Knot[9, 31], Knot[11, NonAlternating, 11], Knot[11, NonAlternating, 22], > Knot[11, NonAlternating, 112], Knot[11, NonAlternating, 127]}

In[13]:=
{KnotDet[Knot[9, 31]], KnotSignature[Knot[9, 31]]}

Out[13]=
{55, -2}

In[14]:=
Jones[Knot[9, 31]][q]

Out[14]=
-7 4 6 8 10 9 8 2 -5 + q - -- + -- - -- + -- - -- + - + 3 q - q 6 5 4 3 2 q q q q q q

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

Out[15]=
{Knot[9, 31]}

In[16]:=
A2Invariant[Knot[9, 31]][q]

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

In[17]:=
HOMFLYPT[Knot[9, 31]][a, z]

Out[17]=
2 4 2 2 2 4 2 6 2 4 2 4 4 4 -1 + 4 a - 2 a - 2 z + 7 a z - 4 a z + a z - z + 4 a z - 2 a z + 2 6 > a z

In[18]:=
Kauffman[Knot[9, 31]][a, z]

Out[18]=
2 4 z 3 5 2 2 2 4 2 -1 - 4 a - 2 a + - + 3 a z + 5 a z + 3 a z + 5 z + 15 a z + 13 a z + a 3 6 2 2 z 3 3 3 5 3 7 3 4 2 4 > 3 a z - ---- - 3 a z - 5 a z - 8 a z - 4 a z - 7 z - 21 a z - a 5 4 4 6 4 8 4 z 5 3 5 5 5 7 5 > 23 a z - 8 a z + a z + -- - 3 a z - 7 a z + a z + 4 a z + a 6 2 6 4 6 6 6 7 3 7 5 7 2 8 > 3 z + 8 a z + 11 a z + 6 a z + 3 a z + 7 a z + 4 a z + a z + 4 8 > a z

In[19]:=
{Vassiliev[2][Knot[9, 31]], Vassiliev[3][Knot[9, 31]]}

Out[19]=
{2, -2}

In[20]:=
Kh[Knot[9, 31]][q, t]

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

In[21]:=
ColouredJones[Knot[9, 31], 2][q]

Out[21]=
-20 4 2 12 21 -15 38 46 7 71 65 21 -27 + q - --- + --- + --- - --- + q + --- - --- - --- + --- - --- - -- + 19 18 17 16 14 13 12 11 10 9 q q q q q q q q q q 93 66 32 91 48 35 66 24 2 3 4 > -- - -- - -- + -- - -- - -- + -- - -- + 33 q - 6 q - 13 q + 10 q - 8 7 6 5 4 3 2 q q q q q q q q 6 7 > 3 q + q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 9₃₁

9₃₀
9₃₂

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q^-20 - 4q^-19 + 2q^-18 + 12q^-17 - 21q^-16 + q^-15 + 38q^-14 - 46q^-13 - 7q^-12 + 71q^-11 - 65q^-10 - 21q^-9 + 93q^-8 - 66q^-7 - 32q^-6 + 91q^-5 - 48q^-4 - 35q^-3 + 66q^-2 - 24q^-1 - 27 + 33q - 6q² - 13q³ + 10q⁴ - 3q⁶ + q⁷
3	q^-39 - 4q^-38 + 2q^-37 + 8q^-36 - q^-35 - 20q^-34 - 5q^-33 + 46q^-32 + 8q^-31 - 71q^-30 - 32q^-29 + 116q^-28 + 58q^-27 - 155q^-26 - 108q^-25 + 204q^-24 + 160q^-23 - 238q^-22 - 222q^-21 + 262q^-20 + 285q^-19 - 277q^-18 - 333q^-17 + 265q^-16 + 380q^-15 - 256q^-14 - 393q^-13 + 213q^-12 + 409q^-11 - 179q^-10 - 389q^-9 + 123q^-8 + 369q^-7 - 80q^-6 - 318q^-5 + 26q^-4 + 273q^-3 + 4q^-2 - 207q^-1 - 34 + 153q + 41q² - 98q³ - 42q⁴ + 58q⁵ + 33q⁶ - 29q⁷ - 23q⁸ + 13q⁹ + 13q¹⁰ - 5q¹¹ - 5q¹² + 3q¹⁴ - q¹⁵
4	q^-64 - 4q^-63 + 2q^-62 + 8q^-61 - 5q^-60 - 26q^-58 + 13q^-57 + 47q^-56 - 11q^-55 - 4q^-54 - 117q^-53 + 22q^-52 + 168q^-51 + 27q^-50 - 11q^-49 - 354q^-48 - 27q^-47 + 396q^-46 + 208q^-45 + 45q^-44 - 786q^-43 - 259q^-42 + 665q^-41 + 608q^-40 + 279q^-39 - 1327q^-38 - 729q^-37 + 811q^-36 + 1135q^-35 + 741q^-34 - 1774q^-33 - 1317q^-32 + 731q^-31 + 1574q^-30 + 1308q^-29 - 1966q^-28 - 1803q^-27 + 463q^-26 + 1772q^-25 + 1790q^-24 - 1879q^-23 - 2044q^-22 + 114q^-21 + 1699q^-20 + 2067q^-19 - 1550q^-18 - 2012q^-17 - 257q^-16 + 1393q^-15 + 2120q^-14 - 1046q^-13 - 1736q^-12 - 581q^-11 + 916q^-10 + 1926q^-9 - 479q^-8 - 1258q^-7 - 757q^-6 + 381q^-5 + 1496q^-4 - 24q^-3 - 694q^-2 - 698q^-1 - 31 + 932q + 179q² - 230q³ - 455q⁴ - 197q⁵ + 434q⁶ + 156q⁷ + 5q⁸ - 197q⁹ - 163q¹⁰ + 141q¹¹ + 65q¹² + 48q¹³ - 52q¹⁴ - 74q¹⁵ + 32q¹⁶ + 12q¹⁷ + 23q¹⁸ - 6q¹⁹ - 20q²⁰ + 5q²¹ + 5q²³ - 3q²⁵ + q²⁶
5	q^-95 - 4q^-94 + 2q^-93 + 8q^-92 - 5q^-91 - 4q^-90 - 6q^-89 - 8q^-88 + 14q^-87 + 38q^-86 + 3q^-85 - 44q^-84 - 58q^-83 - 27q^-82 + 72q^-81 + 140q^-80 + 82q^-79 - 135q^-78 - 292q^-77 - 157q^-76 + 191q^-75 + 488q^-74 + 401q^-73 - 233q^-72 - 874q^-71 - 729q^-70 + 232q^-69 + 1256q^-68 + 1357q^-67 - 36q^-66 - 1849q^-65 - 2164q^-64 - 337q^-63 + 2319q^-62 + 3297q^-61 + 1065q^-60 - 2802q^-59 - 4572q^-58 - 2081q^-57 + 2988q^-56 + 5982q^-55 + 3432q^-54 - 2962q^-53 - 7300q^-52 - 4978q^-51 + 2583q^-50 + 8454q^-49 + 6608q^-48 - 1930q^-47 - 9332q^-46 - 8162q^-45 + 1094q^-44 + 9834q^-43 + 9526q^-42 - 78q^-41 - 10078q^-40 - 10622q^-39 - 850q^-38 + 9913q^-37 + 11379q^-36 + 1898q^-35 - 9620q^-34 - 11871q^-33 - 2685q^-32 + 8966q^-31 + 12014q^-30 + 3587q^-29 - 8242q^-28 - 11966q^-27 - 4226q^-26 + 7224q^-25 + 11585q^-24 + 4979q^-23 - 6142q^-22 - 11023q^-21 - 5471q^-20 + 4798q^-19 + 10161q^-18 + 5986q^-17 - 3457q^-16 - 9091q^-15 - 6158q^-14 + 1973q^-13 + 7754q^-12 + 6241q^-11 - 682q^-10 - 6289q^-9 - 5876q^-8 - 527q^-7 + 4704q^-6 + 5366q^-5 + 1357q^-4 - 3230q^-3 - 4480q^-2 - 1899q^-1 + 1879 + 3554q + 2036q² - 850q³ - 2531q⁴ - 1908q⁵ + 125q⁶ + 1656q⁷ + 1555q⁸ + 274q⁹ - 933q¹⁰ - 1148q¹¹ - 423q¹² + 453q¹³ + 743q¹⁴ + 400q¹⁵ - 148q¹⁶ - 435q¹⁷ - 311q¹⁸ + 11q¹⁹ + 229q²⁰ + 196q²¹ + 33q²² - 92q²³ - 116q²⁴ - 45q²⁵ + 44q²⁶ + 59q²⁷ + 18q²⁸ - 6q²⁹ - 21q³⁰ - 23q³¹ + 6q³² + 13q³³ + 2q³⁴ - 5q³⁷ + 3q³⁹ - q⁴⁰
6	q^-132 - 4q^-131 + 2q^-130 + 8q^-129 - 5q^-128 - 4q^-127 - 10q^-126 + 12q^-125 - 7q^-124 + 5q^-123 + 52q^-122 - 27q^-121 - 38q^-120 - 62q^-119 + 30q^-118 + 9q^-117 + 62q^-116 + 209q^-115 - 65q^-114 - 188q^-113 - 307q^-112 + 7q^-111 + 46q^-110 + 344q^-109 + 770q^-108 + 4q^-107 - 578q^-106 - 1170q^-105 - 461q^-104 - 53q^-103 + 1162q^-102 + 2462q^-101 + 860q^-100 - 1094q^-99 - 3313q^-98 - 2500q^-97 - 1271q^-96 + 2500q^-95 + 6368q^-94 + 4208q^-93 - 463q^-92 - 6870q^-91 - 7710q^-90 - 5902q^-89 + 2804q^-88 + 12711q^-87 + 12156q^-86 + 4182q^-85 - 9916q^-84 - 16395q^-83 - 16397q^-82 - 1246q^-81 + 19092q^-80 + 24803q^-79 + 15450q^-78 - 8609q^-77 - 25661q^-76 - 32402q^-75 - 12206q^-74 + 21191q^-73 + 38503q^-72 + 32408q^-71 - 383q^-70 - 30788q^-69 - 49484q^-68 - 28624q^-67 + 16453q^-66 + 48063q^-65 + 50088q^-64 + 13089q^-63 - 29244q^-62 - 62142q^-61 - 45404q^-60 + 6528q^-59 + 50837q^-58 + 63153q^-57 + 27105q^-56 - 22494q^-55 - 67781q^-54 - 57815q^-53 - 4580q^-52 + 47919q^-51 + 69463q^-50 + 37884q^-49 - 13791q^-48 - 67262q^-47 - 64351q^-46 - 14105q^-45 + 41739q^-44 + 69948q^-43 + 44639q^-42 - 5048q^-41 - 62441q^-40 - 66031q^-39 - 21818q^-38 + 33477q^-37 + 66103q^-36 + 48411q^-35 + 3859q^-34 - 54014q^-33 - 63964q^-32 - 28580q^-31 + 22824q^-30 + 58226q^-29 + 49748q^-28 + 13471q^-27 - 41627q^-26 - 57914q^-25 - 34155q^-24 + 9710q^-23 + 45798q^-22 + 47556q^-21 + 22695q^-20 - 25659q^-19 - 46928q^-18 - 36402q^-17 - 3836q^-16 + 29361q^-15 + 40118q^-14 + 28404q^-13 - 8919q^-12 - 31464q^-11 - 32809q^-10 - 13740q^-9 + 12152q^-8 + 27611q^-7 + 27561q^-6 + 3854q^-5 - 14998q^-4 - 23412q^-3 - 16582q^-2 - 834q^-1 + 13631 + 20363q + 9173q² - 2596q³ - 12013q⁴ - 12771q⁵ - 6415q⁶ + 3157q⁷ + 10912q⁸ + 7804q⁹ + 2952q¹⁰ - 3466q¹¹ - 6496q¹² - 5813q¹³ - 1464q¹⁴ + 3835q¹⁵ + 3922q¹⁶ + 3153q¹⁷ + 288q¹⁸ - 1903q¹⁹ - 3035q²⁰ - 1834q²¹ + 642q²² + 1094q²³ + 1581q²⁴ + 798q²⁵ - 59q²⁶ - 1017q²⁷ - 914q²⁸ - 75q²⁹ + 50q³⁰ + 471q³¹ + 380q³² + 216q³³ - 223q³⁴ - 283q³⁵ - 53q³⁶ - 84q³⁷ + 80q³⁸ + 97q³⁹ + 109q⁴⁰ - 36q⁴¹ - 62q⁴² - 3q⁴³ - 35q⁴⁴ + 6q⁴⁵ + 12q⁴⁶ + 32q⁴⁷ - 6q⁴⁸ - 13q⁴⁹ + 5q⁵⁰ - 7q⁵¹ + 5q⁵⁴ - 3q⁵⁶ + q⁵⁷
7	q^-175 - 4q^-174 + 2q^-173 + 8q^-172 - 5q^-171 - 4q^-170 - 10q^-169 + 8q^-168 + 13q^-167 - 16q^-166 + 19q^-165 + 22q^-164 - 21q^-163 - 32q^-162 - 58q^-161 + 12q^-160 + 84q^-159 + 13q^-158 + 91q^-157 + 54q^-156 - 111q^-155 - 153q^-154 - 291q^-153 - 45q^-152 + 306q^-151 + 312q^-150 + 498q^-149 + 219q^-148 - 417q^-147 - 724q^-146 - 1183q^-145 - 598q^-144 + 738q^-143 + 1479q^-142 + 2273q^-141 + 1377q^-140 - 821q^-139 - 2533q^-138 - 4364q^-137 - 3319q^-136 + 506q^-135 + 4294q^-134 + 7817q^-133 + 6644q^-132 + 769q^-131 - 5882q^-130 - 12795q^-129 - 12926q^-128 - 4537q^-127 + 7342q^-126 + 20015q^-125 + 22388q^-124 + 11484q^-123 - 6640q^-122 - 28010q^-121 - 36511q^-120 - 24388q^-119 + 2513q^-118 + 36778q^-117 + 54726q^-116 + 43408q^-115 + 7899q^-114 - 42871q^-113 - 76524q^-112 - 70626q^-111 - 26370q^-110 + 44780q^-109 + 99484q^-108 + 104380q^-107 + 54577q^-106 - 38596q^-105 - 120792q^-104 - 143625q^-103 - 92456q^-102 + 22975q^-101 + 136842q^-100 + 184498q^-99 + 138565q^-98 + 3495q^-97 - 144791q^-96 - 223562q^-95 - 189620q^-94 - 39569q^-93 + 142628q^-92 + 256697q^-91 + 241727q^-90 + 83046q^-89 - 130280q^-88 - 281264q^-87 - 290581q^-86 - 130107q^-85 + 109056q^-84 + 295683q^-83 + 332631q^-82 + 176913q^-81 - 81393q^-80 - 300125q^-79 - 365865q^-78 - 220045q^-77 + 50789q^-76 + 296150q^-75 + 388976q^-74 + 256729q^-73 - 19511q^-72 - 285424q^-71 - 403208q^-70 - 286451q^-69 - 9332q^-68 + 270954q^-67 + 408982q^-66 + 308177q^-65 + 35629q^-64 - 253431q^-63 - 409006q^-62 - 324132q^-61 - 58083q^-60 + 235404q^-59 + 403707q^-58 + 334089q^-57 + 78338q^-56 - 215553q^-55 - 395109q^-54 - 341020q^-53 - 96510q^-52 + 195129q^-51 + 382831q^-50 + 344198q^-49 + 114478q^-48 - 171729q^-47 - 367246q^-46 - 345612q^-45 - 132536q^-44 + 145965q^-43 + 347125q^-42 + 343582q^-41 + 151346q^-40 - 115679q^-39 - 322041q^-38 - 338536q^-37 - 169976q^-36 + 82021q^-35 + 290651q^-34 + 327952q^-33 + 187731q^-32 - 44536q^-31 - 253245q^-30 - 311536q^-29 - 201908q^-28 + 5971q^-27 + 209431q^-26 + 287116q^-25 + 211005q^-24 + 32323q^-23 - 161687q^-22 - 255401q^-21 - 211910q^-20 - 66028q^-19 + 111601q^-18 + 215933q^-17 + 204020q^-16 + 93241q^-15 - 63419q^-14 - 172048q^-13 - 186238q^-12 - 110076q^-11 + 20176q^-10 + 125608q^-9 + 160268q^-8 + 116494q^-7 + 14371q^-6 - 81621q^-5 - 128455q^-4 - 111581q^-3 - 38228q^-2 + 42739q^-1 + 94484 + 98221q + 50763q² - 12618q³ - 62159q⁴ - 78875q⁵ - 53130q⁶ - 7965q⁷ + 34751q⁸ + 57676q⁹ + 47751q¹⁰ + 18959q¹¹ - 14077q¹² - 37695q¹³ - 38050q¹⁴ - 22298q¹⁵ + 863q¹⁶ + 21458q¹⁷ + 26846q¹⁸ + 20328q¹⁹ + 6061q²⁰ - 9805q²¹ - 16819q²² - 15843q²³ - 8147q²⁴ + 2824q²⁵ + 9072q²⁶ + 10657q²⁷ + 7458q²⁸ + 722q²⁹ - 3948q³⁰ - 6380q³¹ - 5643q³² - 1803q³³ + 1186q³⁴ + 3273q³⁵ + 3579q³⁶ + 1744q³⁷ + 161q³⁸ - 1402q³⁹ - 2126q⁴⁰ - 1271q⁴¹ - 427q⁴² + 499q⁴³ + 1028q⁴⁴ + 689q⁴⁵ + 477q⁴⁶ - 29q⁴⁷ - 517q⁴⁸ - 409q⁴⁹ - 287q⁵⁰ - 6q⁵¹ + 200q⁵² + 120q⁵³ + 172q⁵⁴ + 96q⁵⁵ - 83q⁵⁶ - 87q⁵⁷ - 90q⁵⁸ - 17q⁵⁹ + 43q⁶⁰ - 5q⁶¹ + 31q⁶² + 35q⁶³ - 6q⁶⁴ - 12q⁶⁵ - 23q⁶⁶ - 3q⁶⁷ + 13q⁶⁸ - 5q⁶⁹ + 7q⁷¹ - 5q⁷⁴ + 3q⁷⁶ - q⁷⁷

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	PD[Knot[9, 31]]
Out[2]=	PD[X[1, 4, 2, 5], X[3, 10, 4, 11], X[11, 1, 12, 18], X[5, 13, 6, 12], > X[17, 7, 18, 6], X[7, 14, 8, 15], X[13, 16, 14, 17], X[15, 8, 16, 9], > X[9, 2, 10, 3]]
In[3]:=	GaussCode[Knot[9, 31]]
Out[3]=	GaussCode[-1, 9, -2, 1, -4, 5, -6, 8, -9, 2, -3, 4, -7, 6, -8, 7, -5, 3]
In[4]:=	DTCode[Knot[9, 31]]
Out[4]=	DTCode[4, 10, 12, 14, 2, 18, 16, 8, 6]
In[5]:=	br = BR[Knot[9, 31]]
Out[5]=	BR[4, {-1, -1, 2, -1, 2, -3, 2, -3, -3}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{4, 9}
In[7]:=	BraidIndex[Knot[9, 31]]
Out[7]=	4
In[8]:=	Show[DrawMorseLink[Knot[9, 31]]]

Out[8]=	-Graphics-
In[9]:=	#[Knot[9, 31]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 2, 3, 2, {4, 6}, 1}
In[10]:=	alex = Alexander[Knot[9, 31]][t]
Out[10]=	-3 5 13 2 3 -17 + t - -- + -- + 13 t - 5 t + t 2 t t
In[11]:=	Conway[Knot[9, 31]][z]
Out[11]=	2 4 6 1 + 2 z + z + z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[9, 31], Knot[11, NonAlternating, 11], Knot[11, NonAlternating, 22], > Knot[11, NonAlternating, 112], Knot[11, NonAlternating, 127]}
In[13]:=	{KnotDet[Knot[9, 31]], KnotSignature[Knot[9, 31]]}
Out[13]=	{55, -2}
In[14]:=	Jones[Knot[9, 31]][q]
Out[14]=	-7 4 6 8 10 9 8 2 -5 + q - -- + -- - -- + -- - -- + - + 3 q - q 6 5 4 3 2 q q q q q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] \|\| (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[9, 31]}
In[16]:=	A2Invariant[Knot[9, 31]][q]
Out[16]=	-22 -20 2 -16 2 -12 -10 3 -4 3 2 4 6 q - q - --- + q - --- + q + q + -- - q + -- - q + q - q 18 14 6 2 q q q q
In[17]:=	HOMFLYPT[Knot[9, 31]][a, z]
Out[17]=	2 4 2 2 2 4 2 6 2 4 2 4 4 4 -1 + 4 a - 2 a - 2 z + 7 a z - 4 a z + a z - z + 4 a z - 2 a z + 2 6 > a z
In[18]:=	Kauffman[Knot[9, 31]][a, z]
Out[18]=	2 4 z 3 5 2 2 2 4 2 -1 - 4 a - 2 a + - + 3 a z + 5 a z + 3 a z + 5 z + 15 a z + 13 a z + a 3 6 2 2 z 3 3 3 5 3 7 3 4 2 4 > 3 a z - ---- - 3 a z - 5 a z - 8 a z - 4 a z - 7 z - 21 a z - a 5 4 4 6 4 8 4 z 5 3 5 5 5 7 5 > 23 a z - 8 a z + a z + -- - 3 a z - 7 a z + a z + 4 a z + a 6 2 6 4 6 6 6 7 3 7 5 7 2 8 > 3 z + 8 a z + 11 a z + 6 a z + 3 a z + 7 a z + 4 a z + a z + 4 8 > a z
In[19]:=	{Vassiliev[2][Knot[9, 31]], Vassiliev[3][Knot[9, 31]]}
Out[19]=	{2, -2}
In[20]:=	Kh[Knot[9, 31]][q, t]
Out[20]=	4 5 1 3 1 3 3 5 3 5 -- + - + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + 3 q 15 6 13 5 11 5 11 4 9 4 9 3 7 3 7 2 q q t q t q t q t q t q t q t q t 5 4 5 2 t 2 3 2 5 3 > ----- + ---- + ---- + --- + 3 q t + q t + 2 q t + q t 5 2 5 3 q q t q t q t
In[21]:=	ColouredJones[Knot[9, 31], 2][q]
Out[21]=	-20 4 2 12 21 -15 38 46 7 71 65 21 -27 + q - --- + --- + --- - --- + q + --- - --- - --- + --- - --- - -- + 19 18 17 16 14 13 12 11 10 9 q q q q q q q q q q 93 66 32 91 48 35 66 24 2 3 4 > -- - -- - -- + -- - -- - -- + -- - -- + 33 q - 6 q - 13 q + 10 q - 8 7 6 5 4 3 2 q q q q q q q q 6 7 > 3 q + q

The Alternating Knot 931

The Alternating Knot 9₃₁