The Alternating Knot 10₇₈

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Acknowledgement

KnotPlot

PD Presentation: X₁₄₂₅ X₃₈₄₉ X_5,14,6,15 X_11,17,12,16 X_15,13,16,12 X_17,20,18,1 X_9,18,10,19 X_19,10,20,11 X_13,6,14,7 X₇₂₈₃

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

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

Minimum Braid Representative:

Length is 12, 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 2 3 3 / NotAvailable 1

Alexander Polynomial: - t^-3 + 7t^-2 - 16t^-1 + 21 - 16t + 7t² - t³

Conway Polynomial: 1 + 3z² + z⁴ - z⁶

Other knots with the same Alexander/Conway Polynomial: {K11n98, K11n105, ...}

Determinant and Signature: {69, -4}

Jones Polynomial: q^-10 - 3q^-9 + 5q^-8 - 9q^-7 + 11q^-6 - 11q^-5 + 11q^-4 - 8q^-3 + 6q^-2 - 3q^-1 + 1

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

A2 (sl(3)) Invariant: q^-32 + q^-30 - 2q^-28 - q^-26 - q^-24 - 3q^-22 + 2q^-20 + 2q^-16 + 2q^-14 - q^-12 + 3q^-10 - 2q^-8 + q^-6 + q^-4 - q^-2 + 1

HOMFLY-PT Polynomial: a² + 2a²z² + a²z⁴ - a⁴ - 3a⁴z² - 3a⁴z⁴ - a⁴z⁶ + 4a⁶ + 7a⁶z² + 3a⁶z⁴ - 4a⁸ - 3a⁸z² + a¹⁰

Kauffman Polynomial: - a² + 3a²z² - 3a²z⁴ + a²z⁶ - a³z + 7a³z³ - 9a³z⁵ + 3a³z⁷ - a⁴ + 6a⁴z² - 4a⁴z⁴ - 5a⁴z⁶ + 3a⁴z⁸ - 3a⁵z + 15a⁵z³ - 21a⁵z⁵ + 6a⁵z⁷ + a⁵z⁹ - 4a⁶ + 11a⁶z² - 7a⁶z⁴ - 8a⁶z⁶ + 6a⁶z⁸ + 2a⁷z + 5a⁷z³ - 15a⁷z⁵ + 7a⁷z⁷ + a⁷z⁹ - 4a⁸ + 10a⁸z² - 10a⁸z⁴ + 2a⁸z⁶ + 3a⁸z⁸ + 6a⁹z - 7a⁹z³ + 4a⁹z⁷ - a¹⁰ + a¹⁰z² - 3a¹⁰z⁴ + 4a¹⁰z⁶ + 2a¹¹z - 4a¹¹z³ + 3a¹¹z⁵ - a¹²z² + a¹²z⁴

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=-4 is the signature of 10₇₈. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

t^rq^j r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2

j = 1 1

j = -1 2

j = -3 4 1

j = -5 5 3

j = -7 6 3

j = -9 5 5

j = -11 6 6

j = -13 3 5

j = -15 2 6

j = -17 1 3

j = -19 2

j = -21 1

n Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2 q^-28 - 3q^-27 + q^-26 + 7q^-25 - 14q^-24 + 7q^-23 + 20q^-22 - 41q^-21 + 18q^-20 + 45q^-19 - 79q^-18 + 24q^-17 + 75q^-16 - 104q^-15 + 16q^-14 + 93q^-13 - 100q^-12 - q^-11 + 91q^-10 - 73q^-9 - 16q^-8 + 69q^-7 - 37q^-6 - 21q^-5 + 37q^-4 - 10q^-3 - 13q^-2 + 11q^-1 - 3q + q²

3 q^-54 - 3q^-53 + q^-52 + 3q^-51 + 2q^-50 - 9q^-49 + q^-48 + 14q^-47 - 7q^-46 - 21q^-45 + 15q^-44 + 40q^-43 - 40q^-42 - 60q^-41 + 64q^-40 + 105q^-39 - 107q^-38 - 150q^-37 + 131q^-36 + 232q^-35 - 170q^-34 - 299q^-33 + 174q^-32 + 378q^-31 - 166q^-30 - 441q^-29 + 141q^-28 + 478q^-27 - 92q^-26 - 504q^-25 + 50q^-24 + 486q^-23 + 19q^-22 - 474q^-21 - 62q^-20 + 416q^-19 + 126q^-18 - 371q^-17 - 153q^-16 + 288q^-15 + 191q^-14 - 221q^-13 - 190q^-12 + 139q^-11 + 182q^-10 - 75q^-9 - 149q^-8 + 21q^-7 + 115q^-6 + 6q^-5 - 71q^-4 - 21q^-3 + 39q^-2 + 20q^-1 - 17 - 13q + 6q² + 5q³ - 3q⁵ + q⁶

4 q^-88 - 3q^-87 + q^-86 + 3q^-85 - 2q^-84 + 7q^-83 - 15q^-82 + 5q^-81 + 8q^-80 - 15q^-79 + 29q^-78 - 33q^-77 + 23q^-76 + 9q^-75 - 71q^-74 + 65q^-73 - 30q^-72 + 98q^-71 + 3q^-70 - 243q^-69 + 68q^-68 + 33q^-67 + 335q^-66 + 34q^-65 - 622q^-64 - 88q^-63 + 137q^-62 + 855q^-61 + 237q^-60 - 1186q^-59 - 543q^-58 + 114q^-57 + 1619q^-56 + 768q^-55 - 1692q^-54 - 1249q^-53 - 225q^-52 + 2311q^-51 + 1542q^-50 - 1828q^-49 - 1871q^-48 - 846q^-47 + 2579q^-46 + 2234q^-45 - 1539q^-44 - 2109q^-43 - 1481q^-42 + 2368q^-41 + 2574q^-40 - 1003q^-39 - 1935q^-38 - 1935q^-37 + 1832q^-36 + 2556q^-35 - 384q^-34 - 1487q^-33 - 2165q^-32 + 1114q^-31 + 2250q^-30 + 227q^-29 - 862q^-28 - 2147q^-27 + 337q^-26 + 1685q^-25 + 679q^-24 - 157q^-23 - 1796q^-22 - 296q^-21 + 925q^-20 + 786q^-19 + 425q^-18 - 1150q^-17 - 559q^-16 + 215q^-15 + 527q^-14 + 642q^-13 - 467q^-12 - 426q^-11 - 167q^-10 + 152q^-9 + 484q^-8 - 56q^-7 - 151q^-6 - 186q^-5 - 55q^-4 + 207q^-3 + 42q^-2 + 5q^-1 - 73 - 63q + 48q² + 15q³ + 20q⁴ - 10q⁵ - 20q⁶ + 6q⁷ + 5q⁹ - 3q¹¹ + q¹²

5 q^-130 - 3q^-129 + q^-128 + 3q^-127 - 2q^-126 + 3q^-125 + q^-124 - 11q^-123 - q^-122 + 10q^-121 - 5q^-120 + 12q^-119 + 11q^-118 - 19q^-117 - 17q^-116 - 7q^-115 - 8q^-114 + 34q^-113 + 64q^-112 + 15q^-111 - 70q^-110 - 122q^-109 - 72q^-108 + 97q^-107 + 260q^-106 + 210q^-105 - 150q^-104 - 513q^-103 - 413q^-102 + 166q^-101 + 848q^-100 + 879q^-99 - 129q^-98 - 1416q^-97 - 1543q^-96 - 46q^-95 + 2044q^-94 + 2662q^-93 + 489q^-92 - 2876q^-91 - 4079q^-90 - 1346q^-89 + 3574q^-88 + 6027q^-87 + 2674q^-86 - 4187q^-85 - 8067q^-84 - 4577q^-83 + 4233q^-82 + 10360q^-81 + 6902q^-80 - 3915q^-79 - 12209q^-78 - 9509q^-77 + 2846q^-76 + 13721q^-75 + 12114q^-74 - 1426q^-73 - 14485q^-72 - 14357q^-71 - 404q^-70 + 14575q^-69 + 16123q^-68 + 2285q^-67 - 14083q^-66 - 17210q^-65 - 4006q^-64 + 13016q^-63 + 17702q^-62 + 5584q^-61 - 11764q^-60 - 17634q^-59 - 6755q^-58 + 10152q^-57 + 17149q^-56 + 7858q^-55 - 8590q^-54 - 16326q^-53 - 8586q^-52 + 6696q^-51 + 15252q^-50 + 9370q^-49 - 4919q^-48 - 13878q^-47 - 9783q^-46 + 2792q^-45 + 12248q^-44 + 10197q^-43 - 869q^-42 - 10306q^-41 - 10049q^-40 - 1232q^-39 + 8094q^-38 + 9720q^-37 + 2866q^-36 - 5714q^-35 - 8692q^-34 - 4271q^-33 + 3298q^-32 + 7379q^-31 + 4963q^-30 - 1114q^-29 - 5569q^-28 - 5126q^-27 - 654q^-26 + 3713q^-25 + 4588q^-24 + 1841q^-23 - 1889q^-22 - 3676q^-21 - 2372q^-20 + 481q^-19 + 2460q^-18 + 2330q^-17 + 526q^-16 - 1363q^-15 - 1890q^-14 - 947q^-13 + 453q^-12 + 1246q^-11 + 1021q^-10 + 96q^-9 - 670q^-8 - 792q^-7 - 339q^-6 + 234q^-5 + 498q^-4 + 354q^-3 + 10q^-2 - 247q^-1 - 266 - 84q + 88q² + 136q³ + 95q⁴ - 6q⁵ - 73q⁶ - 56q⁷ - 3q⁸ + 15q⁹ + 24q¹⁰ + 20q¹¹ - 10q¹² - 13q¹³ - q¹⁴ + 5q¹⁷ - 3q¹⁹ + q²⁰

6 q^-180 - 3q^-179 + q^-178 + 3q^-177 - 2q^-176 + 3q^-175 - 3q^-174 + 5q^-173 - 17q^-172 + q^-171 + 20q^-170 - 12q^-169 + 16q^-168 + 16q^-166 - 64q^-165 - 16q^-164 + 56q^-163 - 31q^-162 + 57q^-161 + 45q^-160 + 53q^-159 - 192q^-158 - 105q^-157 + 86q^-156 - 56q^-155 + 206q^-154 + 240q^-153 + 164q^-152 - 507q^-151 - 466q^-150 - 10q^-149 - 75q^-148 + 724q^-147 + 947q^-146 + 513q^-145 - 1236q^-144 - 1700q^-143 - 773q^-142 - 160q^-141 + 2249q^-140 + 3229q^-139 + 1863q^-138 - 2622q^-137 - 5131q^-136 - 3864q^-135 - 1184q^-134 + 5608q^-133 + 9300q^-132 + 6575q^-131 - 3839q^-130 - 12368q^-129 - 12561q^-128 - 6202q^-127 + 10109q^-126 + 21589q^-125 + 19006q^-124 - 1132q^-123 - 22676q^-122 - 29958q^-121 - 20559q^-120 + 11098q^-119 + 38927q^-118 + 42539q^-117 + 11806q^-116 - 30082q^-115 - 53940q^-114 - 47522q^-113 + 1206q^-112 + 53566q^-111 + 73539q^-110 + 38032q^-109 - 25953q^-108 - 74838q^-107 - 81793q^-106 - 22378q^-105 + 55677q^-104 + 100433q^-103 + 71011q^-102 - 7603q^-101 - 82112q^-100 - 110636q^-99 - 52366q^-98 + 42840q^-97 + 112533q^-96 + 98016q^-95 + 17489q^-94 - 73977q^-93 - 124197q^-92 - 76852q^-91 + 22424q^-90 + 108825q^-89 + 110931q^-88 + 38902q^-87 - 57530q^-86 - 122656q^-85 - 89748q^-84 + 3219q^-83 + 95965q^-82 + 111176q^-81 + 52519q^-80 - 39902q^-79 - 112292q^-78 - 93390q^-77 - 12244q^-76 + 79605q^-75 + 104478q^-74 + 61055q^-73 - 22458q^-72 - 97465q^-71 - 92520q^-70 - 26534q^-69 + 60290q^-68 + 93735q^-67 + 67801q^-66 - 3050q^-65 - 77893q^-64 - 88186q^-63 - 41293q^-62 + 36285q^-61 + 77422q^-60 + 71658q^-59 + 18441q^-58 - 51845q^-57 - 77196q^-56 - 53313q^-55 + 8682q^-54 + 53235q^-53 + 67591q^-52 + 37030q^-51 - 21155q^-50 - 56423q^-49 - 55896q^-48 - 16022q^-47 + 23351q^-46 + 51694q^-45 + 44747q^-44 + 6501q^-43 - 28428q^-42 - 44708q^-41 - 28813q^-40 - 3643q^-39 + 27093q^-38 + 37278q^-37 + 21464q^-36 - 2739q^-35 - 23755q^-34 - 25635q^-33 - 17836q^-32 + 4209q^-31 + 19700q^-30 + 20320q^-29 + 10689q^-28 - 4203q^-27 - 12513q^-26 - 16786q^-25 - 7307q^-24 + 3545q^-23 + 9987q^-22 + 10428q^-21 + 5024q^-20 - 771q^-19 - 8074q^-18 - 7082q^-17 - 3451q^-16 + 1127q^-15 + 4209q^-14 + 4632q^-13 + 3406q^-12 - 1234q^-11 - 2571q^-10 - 2939q^-9 - 1674q^-8 - 63q^-7 + 1444q^-6 + 2323q^-5 + 744q^-4 + 100q^-3 - 799q^-2 - 955q^-1 - 844 - 130q + 666q² + 376q³ + 422q⁴ + 67q⁵ - 121q⁶ - 365q⁷ - 230q⁸ + 72q⁹ + 15q¹⁰ + 135q¹¹ + 86q¹² + 59q¹³ - 73q¹⁴ - 67q¹⁵ + 4q¹⁶ - 27q¹⁷ + 15q¹⁸ + 15q¹⁹ + 29q²⁰ - 10q²¹ - 13q²² + 6q²³ - 7q²⁴ + 5q²⁷ - 3q²⁹ + q³⁰

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, 78]]

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

In[3]:=
GaussCode[Knot[10, 78]]

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

In[4]:=
DTCode[Knot[10, 78]]

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

In[5]:=
br = BR[Knot[10, 78]]

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

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

Out[6]=
{5, 12}

In[7]:=
BraidIndex[Knot[10, 78]]

Out[7]=
5

In[8]:=
Show[DrawMorseLink[Knot[10, 78]]]

Out[8]=
-Graphics-

In[9]:=
#[Knot[10, 78]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}

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

In[10]:=
alex = Alexander[Knot[10, 78]][t]

Out[10]=
-3 7 16 2 3 21 - t + -- - -- - 16 t + 7 t - t 2 t t

In[11]:=
Conway[Knot[10, 78]][z]

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

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

Out[12]=
{Knot[10, 78], Knot[11, NonAlternating, 98], Knot[11, NonAlternating, 105]}

In[13]:=
{KnotDet[Knot[10, 78]], KnotSignature[Knot[10, 78]]}

Out[13]=
{69, -4}

In[14]:=
Jones[Knot[10, 78]][q]

Out[14]=
-10 3 5 9 11 11 11 8 6 3 1 + q - -- + -- - -- + -- - -- + -- - -- + -- - - 9 8 7 6 5 4 3 2 q q q q q q q q q

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

Out[15]=
{Knot[10, 78]}

In[16]:=
A2Invariant[Knot[10, 78]][q]

Out[16]=
-32 -30 2 -26 -24 3 2 2 2 -12 3 2 1 + q + q - --- - q - q - --- + --- + --- + --- - q + --- - -- + 28 22 20 16 14 10 8 q q q q q q q -6 -4 -2 > q + q - q

In[17]:=
HOMFLYPT[Knot[10, 78]][a, z]

Out[17]=
2 4 6 8 10 2 2 4 2 6 2 8 2 2 4 a - a + 4 a - 4 a + a + 2 a z - 3 a z + 7 a z - 3 a z + a z - 4 4 6 4 4 6 > 3 a z + 3 a z - a z

In[18]:=
Kauffman[Knot[10, 78]][a, z]

Out[18]=
2 4 6 8 10 3 5 7 9 11 -a - a - 4 a - 4 a - a - a z - 3 a z + 2 a z + 6 a z + 2 a z + 2 2 4 2 6 2 8 2 10 2 12 2 3 3 > 3 a z + 6 a z + 11 a z + 10 a z + a z - a z + 7 a z + 5 3 7 3 9 3 11 3 2 4 4 4 6 4 > 15 a z + 5 a z - 7 a z - 4 a z - 3 a z - 4 a z - 7 a z - 8 4 10 4 12 4 3 5 5 5 7 5 11 5 > 10 a z - 3 a z + a z - 9 a z - 21 a z - 15 a z + 3 a z + 2 6 4 6 6 6 8 6 10 6 3 7 5 7 > a z - 5 a z - 8 a z + 2 a z + 4 a z + 3 a z + 6 a z + 7 7 9 7 4 8 6 8 8 8 5 9 7 9 > 7 a z + 4 a z + 3 a z + 6 a z + 3 a z + a z + a z

In[19]:=
{Vassiliev[2][Knot[10, 78]], Vassiliev[3][Knot[10, 78]]}

Out[19]=
{3, -5}

In[20]:=
Kh[Knot[10, 78]][q, t]

Out[20]=
3 4 1 2 1 3 2 6 3 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 5 3 21 8 19 7 17 7 17 6 15 6 15 5 13 5 q q q t q t q t q t q t q t q t 5 6 6 5 5 6 3 5 t 2 t > ------ + ------ + ------ + ----- + ----- + ----- + ---- + ---- + -- + --- + 13 4 11 4 11 3 9 3 9 2 7 2 7 5 3 q q t q t q t q t q t q t q t q t q 2 > q t

In[21]:=
ColouredJones[Knot[10, 78], 2][q]

Out[21]=
-28 3 -26 7 14 7 20 41 18 45 79 24 75 q - --- + q + --- - --- + --- + --- - --- + --- + --- - --- + --- + --- - 27 25 24 23 22 21 20 19 18 17 16 q q q q q q q q q q q 104 16 93 100 -11 91 73 16 69 37 21 37 10 > --- + --- + --- - --- - q + --- - -- - -- + -- - -- - -- + -- - -- - 15 14 13 12 10 9 8 7 6 5 4 3 q q q q q q q q q q q q 13 11 2 > -- + -- - 3 q + q 2 q q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 10₇₈

10₇₇
10₇₉

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q^-28 - 3q^-27 + q^-26 + 7q^-25 - 14q^-24 + 7q^-23 + 20q^-22 - 41q^-21 + 18q^-20 + 45q^-19 - 79q^-18 + 24q^-17 + 75q^-16 - 104q^-15 + 16q^-14 + 93q^-13 - 100q^-12 - q^-11 + 91q^-10 - 73q^-9 - 16q^-8 + 69q^-7 - 37q^-6 - 21q^-5 + 37q^-4 - 10q^-3 - 13q^-2 + 11q^-1 - 3q + q²
3	q^-54 - 3q^-53 + q^-52 + 3q^-51 + 2q^-50 - 9q^-49 + q^-48 + 14q^-47 - 7q^-46 - 21q^-45 + 15q^-44 + 40q^-43 - 40q^-42 - 60q^-41 + 64q^-40 + 105q^-39 - 107q^-38 - 150q^-37 + 131q^-36 + 232q^-35 - 170q^-34 - 299q^-33 + 174q^-32 + 378q^-31 - 166q^-30 - 441q^-29 + 141q^-28 + 478q^-27 - 92q^-26 - 504q^-25 + 50q^-24 + 486q^-23 + 19q^-22 - 474q^-21 - 62q^-20 + 416q^-19 + 126q^-18 - 371q^-17 - 153q^-16 + 288q^-15 + 191q^-14 - 221q^-13 - 190q^-12 + 139q^-11 + 182q^-10 - 75q^-9 - 149q^-8 + 21q^-7 + 115q^-6 + 6q^-5 - 71q^-4 - 21q^-3 + 39q^-2 + 20q^-1 - 17 - 13q + 6q² + 5q³ - 3q⁵ + q⁶
4	q^-88 - 3q^-87 + q^-86 + 3q^-85 - 2q^-84 + 7q^-83 - 15q^-82 + 5q^-81 + 8q^-80 - 15q^-79 + 29q^-78 - 33q^-77 + 23q^-76 + 9q^-75 - 71q^-74 + 65q^-73 - 30q^-72 + 98q^-71 + 3q^-70 - 243q^-69 + 68q^-68 + 33q^-67 + 335q^-66 + 34q^-65 - 622q^-64 - 88q^-63 + 137q^-62 + 855q^-61 + 237q^-60 - 1186q^-59 - 543q^-58 + 114q^-57 + 1619q^-56 + 768q^-55 - 1692q^-54 - 1249q^-53 - 225q^-52 + 2311q^-51 + 1542q^-50 - 1828q^-49 - 1871q^-48 - 846q^-47 + 2579q^-46 + 2234q^-45 - 1539q^-44 - 2109q^-43 - 1481q^-42 + 2368q^-41 + 2574q^-40 - 1003q^-39 - 1935q^-38 - 1935q^-37 + 1832q^-36 + 2556q^-35 - 384q^-34 - 1487q^-33 - 2165q^-32 + 1114q^-31 + 2250q^-30 + 227q^-29 - 862q^-28 - 2147q^-27 + 337q^-26 + 1685q^-25 + 679q^-24 - 157q^-23 - 1796q^-22 - 296q^-21 + 925q^-20 + 786q^-19 + 425q^-18 - 1150q^-17 - 559q^-16 + 215q^-15 + 527q^-14 + 642q^-13 - 467q^-12 - 426q^-11 - 167q^-10 + 152q^-9 + 484q^-8 - 56q^-7 - 151q^-6 - 186q^-5 - 55q^-4 + 207q^-3 + 42q^-2 + 5q^-1 - 73 - 63q + 48q² + 15q³ + 20q⁴ - 10q⁵ - 20q⁶ + 6q⁷ + 5q⁹ - 3q¹¹ + q¹²
5	q^-130 - 3q^-129 + q^-128 + 3q^-127 - 2q^-126 + 3q^-125 + q^-124 - 11q^-123 - q^-122 + 10q^-121 - 5q^-120 + 12q^-119 + 11q^-118 - 19q^-117 - 17q^-116 - 7q^-115 - 8q^-114 + 34q^-113 + 64q^-112 + 15q^-111 - 70q^-110 - 122q^-109 - 72q^-108 + 97q^-107 + 260q^-106 + 210q^-105 - 150q^-104 - 513q^-103 - 413q^-102 + 166q^-101 + 848q^-100 + 879q^-99 - 129q^-98 - 1416q^-97 - 1543q^-96 - 46q^-95 + 2044q^-94 + 2662q^-93 + 489q^-92 - 2876q^-91 - 4079q^-90 - 1346q^-89 + 3574q^-88 + 6027q^-87 + 2674q^-86 - 4187q^-85 - 8067q^-84 - 4577q^-83 + 4233q^-82 + 10360q^-81 + 6902q^-80 - 3915q^-79 - 12209q^-78 - 9509q^-77 + 2846q^-76 + 13721q^-75 + 12114q^-74 - 1426q^-73 - 14485q^-72 - 14357q^-71 - 404q^-70 + 14575q^-69 + 16123q^-68 + 2285q^-67 - 14083q^-66 - 17210q^-65 - 4006q^-64 + 13016q^-63 + 17702q^-62 + 5584q^-61 - 11764q^-60 - 17634q^-59 - 6755q^-58 + 10152q^-57 + 17149q^-56 + 7858q^-55 - 8590q^-54 - 16326q^-53 - 8586q^-52 + 6696q^-51 + 15252q^-50 + 9370q^-49 - 4919q^-48 - 13878q^-47 - 9783q^-46 + 2792q^-45 + 12248q^-44 + 10197q^-43 - 869q^-42 - 10306q^-41 - 10049q^-40 - 1232q^-39 + 8094q^-38 + 9720q^-37 + 2866q^-36 - 5714q^-35 - 8692q^-34 - 4271q^-33 + 3298q^-32 + 7379q^-31 + 4963q^-30 - 1114q^-29 - 5569q^-28 - 5126q^-27 - 654q^-26 + 3713q^-25 + 4588q^-24 + 1841q^-23 - 1889q^-22 - 3676q^-21 - 2372q^-20 + 481q^-19 + 2460q^-18 + 2330q^-17 + 526q^-16 - 1363q^-15 - 1890q^-14 - 947q^-13 + 453q^-12 + 1246q^-11 + 1021q^-10 + 96q^-9 - 670q^-8 - 792q^-7 - 339q^-6 + 234q^-5 + 498q^-4 + 354q^-3 + 10q^-2 - 247q^-1 - 266 - 84q + 88q² + 136q³ + 95q⁴ - 6q⁵ - 73q⁶ - 56q⁷ - 3q⁸ + 15q⁹ + 24q¹⁰ + 20q¹¹ - 10q¹² - 13q¹³ - q¹⁴ + 5q¹⁷ - 3q¹⁹ + q²⁰
6	q^-180 - 3q^-179 + q^-178 + 3q^-177 - 2q^-176 + 3q^-175 - 3q^-174 + 5q^-173 - 17q^-172 + q^-171 + 20q^-170 - 12q^-169 + 16q^-168 + 16q^-166 - 64q^-165 - 16q^-164 + 56q^-163 - 31q^-162 + 57q^-161 + 45q^-160 + 53q^-159 - 192q^-158 - 105q^-157 + 86q^-156 - 56q^-155 + 206q^-154 + 240q^-153 + 164q^-152 - 507q^-151 - 466q^-150 - 10q^-149 - 75q^-148 + 724q^-147 + 947q^-146 + 513q^-145 - 1236q^-144 - 1700q^-143 - 773q^-142 - 160q^-141 + 2249q^-140 + 3229q^-139 + 1863q^-138 - 2622q^-137 - 5131q^-136 - 3864q^-135 - 1184q^-134 + 5608q^-133 + 9300q^-132 + 6575q^-131 - 3839q^-130 - 12368q^-129 - 12561q^-128 - 6202q^-127 + 10109q^-126 + 21589q^-125 + 19006q^-124 - 1132q^-123 - 22676q^-122 - 29958q^-121 - 20559q^-120 + 11098q^-119 + 38927q^-118 + 42539q^-117 + 11806q^-116 - 30082q^-115 - 53940q^-114 - 47522q^-113 + 1206q^-112 + 53566q^-111 + 73539q^-110 + 38032q^-109 - 25953q^-108 - 74838q^-107 - 81793q^-106 - 22378q^-105 + 55677q^-104 + 100433q^-103 + 71011q^-102 - 7603q^-101 - 82112q^-100 - 110636q^-99 - 52366q^-98 + 42840q^-97 + 112533q^-96 + 98016q^-95 + 17489q^-94 - 73977q^-93 - 124197q^-92 - 76852q^-91 + 22424q^-90 + 108825q^-89 + 110931q^-88 + 38902q^-87 - 57530q^-86 - 122656q^-85 - 89748q^-84 + 3219q^-83 + 95965q^-82 + 111176q^-81 + 52519q^-80 - 39902q^-79 - 112292q^-78 - 93390q^-77 - 12244q^-76 + 79605q^-75 + 104478q^-74 + 61055q^-73 - 22458q^-72 - 97465q^-71 - 92520q^-70 - 26534q^-69 + 60290q^-68 + 93735q^-67 + 67801q^-66 - 3050q^-65 - 77893q^-64 - 88186q^-63 - 41293q^-62 + 36285q^-61 + 77422q^-60 + 71658q^-59 + 18441q^-58 - 51845q^-57 - 77196q^-56 - 53313q^-55 + 8682q^-54 + 53235q^-53 + 67591q^-52 + 37030q^-51 - 21155q^-50 - 56423q^-49 - 55896q^-48 - 16022q^-47 + 23351q^-46 + 51694q^-45 + 44747q^-44 + 6501q^-43 - 28428q^-42 - 44708q^-41 - 28813q^-40 - 3643q^-39 + 27093q^-38 + 37278q^-37 + 21464q^-36 - 2739q^-35 - 23755q^-34 - 25635q^-33 - 17836q^-32 + 4209q^-31 + 19700q^-30 + 20320q^-29 + 10689q^-28 - 4203q^-27 - 12513q^-26 - 16786q^-25 - 7307q^-24 + 3545q^-23 + 9987q^-22 + 10428q^-21 + 5024q^-20 - 771q^-19 - 8074q^-18 - 7082q^-17 - 3451q^-16 + 1127q^-15 + 4209q^-14 + 4632q^-13 + 3406q^-12 - 1234q^-11 - 2571q^-10 - 2939q^-9 - 1674q^-8 - 63q^-7 + 1444q^-6 + 2323q^-5 + 744q^-4 + 100q^-3 - 799q^-2 - 955q^-1 - 844 - 130q + 666q² + 376q³ + 422q⁴ + 67q⁵ - 121q⁶ - 365q⁷ - 230q⁸ + 72q⁹ + 15q¹⁰ + 135q¹¹ + 86q¹² + 59q¹³ - 73q¹⁴ - 67q¹⁵ + 4q¹⁶ - 27q¹⁷ + 15q¹⁸ + 15q¹⁹ + 29q²⁰ - 10q²¹ - 13q²² + 6q²³ - 7q²⁴ + 5q²⁷ - 3q²⁹ + q³⁰

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	PD[Knot[10, 78]]
Out[2]=	PD[X[1, 4, 2, 5], X[3, 8, 4, 9], X[5, 14, 6, 15], X[11, 17, 12, 16], > X[15, 13, 16, 12], X[17, 20, 18, 1], X[9, 18, 10, 19], X[19, 10, 20, 11], > X[13, 6, 14, 7], X[7, 2, 8, 3]]
In[3]:=	GaussCode[Knot[10, 78]]
Out[3]=	GaussCode[-1, 10, -2, 1, -3, 9, -10, 2, -7, 8, -4, 5, -9, 3, -5, 4, -6, 7, -8, > 6]
In[4]:=	DTCode[Knot[10, 78]]
Out[4]=	DTCode[4, 8, 14, 2, 18, 16, 6, 12, 20, 10]
In[5]:=	br = BR[Knot[10, 78]]
Out[5]=	BR[5, {-1, -1, -2, 1, -2, -1, 3, -2, -4, 3, -4, -4}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{5, 12}
In[7]:=	BraidIndex[Knot[10, 78]]
Out[7]=	5
In[8]:=	Show[DrawMorseLink[Knot[10, 78]]]

Out[8]=	-Graphics-
In[9]:=	#[Knot[10, 78]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 2, 3, 3, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 78]][t]
Out[10]=	-3 7 16 2 3 21 - t + -- - -- - 16 t + 7 t - t 2 t t
In[11]:=	Conway[Knot[10, 78]][z]
Out[11]=	2 4 6 1 + 3 z + z - z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 78], Knot[11, NonAlternating, 98], Knot[11, NonAlternating, 105]}
In[13]:=	{KnotDet[Knot[10, 78]], KnotSignature[Knot[10, 78]]}
Out[13]=	{69, -4}
In[14]:=	Jones[Knot[10, 78]][q]
Out[14]=	-10 3 5 9 11 11 11 8 6 3 1 + q - -- + -- - -- + -- - -- + -- - -- + -- - - 9 8 7 6 5 4 3 2 q q q q q q q q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] \|\| (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[10, 78]}
In[16]:=	A2Invariant[Knot[10, 78]][q]
Out[16]=	-32 -30 2 -26 -24 3 2 2 2 -12 3 2 1 + q + q - --- - q - q - --- + --- + --- + --- - q + --- - -- + 28 22 20 16 14 10 8 q q q q q q q -6 -4 -2 > q + q - q
In[17]:=	HOMFLYPT[Knot[10, 78]][a, z]
Out[17]=	2 4 6 8 10 2 2 4 2 6 2 8 2 2 4 a - a + 4 a - 4 a + a + 2 a z - 3 a z + 7 a z - 3 a z + a z - 4 4 6 4 4 6 > 3 a z + 3 a z - a z
In[18]:=	Kauffman[Knot[10, 78]][a, z]
Out[18]=	2 4 6 8 10 3 5 7 9 11 -a - a - 4 a - 4 a - a - a z - 3 a z + 2 a z + 6 a z + 2 a z + 2 2 4 2 6 2 8 2 10 2 12 2 3 3 > 3 a z + 6 a z + 11 a z + 10 a z + a z - a z + 7 a z + 5 3 7 3 9 3 11 3 2 4 4 4 6 4 > 15 a z + 5 a z - 7 a z - 4 a z - 3 a z - 4 a z - 7 a z - 8 4 10 4 12 4 3 5 5 5 7 5 11 5 > 10 a z - 3 a z + a z - 9 a z - 21 a z - 15 a z + 3 a z + 2 6 4 6 6 6 8 6 10 6 3 7 5 7 > a z - 5 a z - 8 a z + 2 a z + 4 a z + 3 a z + 6 a z + 7 7 9 7 4 8 6 8 8 8 5 9 7 9 > 7 a z + 4 a z + 3 a z + 6 a z + 3 a z + a z + a z
In[19]:=	{Vassiliev[2][Knot[10, 78]], Vassiliev[3][Knot[10, 78]]}
Out[19]=	{3, -5}
In[20]:=	Kh[Knot[10, 78]][q, t]
Out[20]=	3 4 1 2 1 3 2 6 3 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 5 3 21 8 19 7 17 7 17 6 15 6 15 5 13 5 q q q t q t q t q t q t q t q t 5 6 6 5 5 6 3 5 t 2 t > ------ + ------ + ------ + ----- + ----- + ----- + ---- + ---- + -- + --- + 13 4 11 4 11 3 9 3 9 2 7 2 7 5 3 q q t q t q t q t q t q t q t q t q 2 > q t
In[21]:=	ColouredJones[Knot[10, 78], 2][q]
Out[21]=	-28 3 -26 7 14 7 20 41 18 45 79 24 75 q - --- + q + --- - --- + --- + --- - --- + --- + --- - --- + --- + --- - 27 25 24 23 22 21 20 19 18 17 16 q q q q q q q q q q q 104 16 93 100 -11 91 73 16 69 37 21 37 10 > --- + --- + --- - --- - q + --- - -- - -- + -- - -- - -- + -- - -- - 15 14 13 12 10 9 8 7 6 5 4 3 q q q q q q q q q q q q 13 11 2 > -- + -- - 3 q + q 2 q q

The Alternating Knot 1078

The Alternating Knot 10₇₈