The Alternating Knot 10₈₀

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Acknowledgement

KnotPlot

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

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

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

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 3 3 3 / NotAvailable 1

Alexander Polynomial: 3t^-3 - 9t^-2 + 15t^-1 - 17 + 15t - 9t² + 3t³

Conway Polynomial: 1 + 6z² + 9z⁴ + 3z⁶

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

Determinant and Signature: {71, -6}

Jones Polynomial: q^-13 - 3q^-12 + 6q^-11 - 10q^-10 + 11q^-9 - 12q^-8 + 11q^-7 - 8q^-6 + 6q^-5 - 2q^-4 + q^-3

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

A2 (sl(3)) Invariant: q^-40 + q^-38 - q^-36 + q^-34 - 3q^-32 - 2q^-30 - q^-28 - 3q^-26 + 3q^-24 - q^-22 + 3q^-20 + 2q^-18 + 3q^-14 - q^-12 + q^-10

HOMFLY-PT Polynomial: 2a⁶ + 5a⁶z² + 4a⁶z⁴ + a⁶z⁶ + 3a⁸ + 9a⁸z² + 8a⁸z⁴ + 2a⁸z⁶ - 6a¹⁰ - 9a¹⁰z² - 3a¹⁰z⁴ + 2a¹² + a¹²z²

Kauffman Polynomial: - 2a⁶ + 5a⁶z² - 4a⁶z⁴ + a⁶z⁶ + a⁷z + 2a⁷z³ - 5a⁷z⁵ + 2a⁷z⁷ + 3a⁸ - 7a⁸z² + 8a⁸z⁴ - 8a⁸z⁶ + 3a⁸z⁸ - 8a⁹z + 22a⁹z³ - 23a⁹z⁵ + 6a⁹z⁷ + a⁹z⁹ + 6a¹⁰ - 13a¹⁰z² + 13a¹⁰z⁴ - 15a¹⁰z⁶ + 7a¹⁰z⁸ - 12a¹¹z + 29a¹¹z³ - 29a¹¹z⁵ + 10a¹¹z⁷ + a¹¹z⁹ + 2a¹² + 2a¹²z² - 5a¹²z⁴ - a¹²z⁶ + 4a¹²z⁸ - 2a¹³z + 6a¹³z³ - 8a¹³z⁵ + 6a¹³z⁷ + 2a¹⁴z² - 5a¹⁴z⁴ + 5a¹⁴z⁶ + a¹⁵z - 3a¹⁵z³ + 3a¹⁵z⁵ - a¹⁶z² + a¹⁶z⁴

V₂ and V₃, the type 2 and 3 Vassiliev invariants: {6, -12}

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=-6 is the signature of 10₈₀. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

t^rq^j r = -10 r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0

j = -5 1

j = -7 2 1

j = -9 4

j = -11 4 2

j = -13 7 4

j = -15 5 4

j = -17 6 7

j = -19 4 5

j = -21 2 6

j = -23 1 4

j = -25 2

j = -27 1

n Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2 q^-36 - 3q^-35 + 2q^-34 + 6q^-33 - 17q^-32 + 11q^-31 + 23q^-30 - 51q^-29 + 21q^-28 + 58q^-27 - 93q^-26 + 20q^-25 + 95q^-24 - 114q^-23 + 4q^-22 + 112q^-21 - 103q^-20 - 17q^-19 + 103q^-18 - 68q^-17 - 30q^-16 + 71q^-15 - 28q^-14 - 27q^-13 + 33q^-12 - 4q^-11 - 12q^-10 + 8q^-9 + q^-8 - 2q^-7 + q^-6

3 q^-69 - 3q^-68 + 2q^-67 + 2q^-66 - q^-65 - 8q^-64 + 8q^-63 + 15q^-62 - 20q^-61 - 29q^-60 + 40q^-59 + 57q^-58 - 68q^-57 - 107q^-56 + 101q^-55 + 183q^-54 - 135q^-53 - 272q^-52 + 137q^-51 + 395q^-50 - 140q^-49 - 490q^-48 + 94q^-47 + 589q^-46 - 51q^-45 - 635q^-44 - 25q^-43 + 666q^-42 + 89q^-41 - 649q^-40 - 160q^-39 + 608q^-38 + 224q^-37 - 543q^-36 - 272q^-35 + 449q^-34 + 320q^-33 - 361q^-32 - 322q^-31 + 237q^-30 + 327q^-29 - 149q^-28 - 278q^-27 + 47q^-26 + 235q^-25 - 154q^-23 - 48q^-22 + 106q^-21 + 41q^-20 - 46q^-19 - 40q^-18 + 23q^-17 + 21q^-16 - 4q^-15 - 12q^-14 + 3q^-13 + 3q^-12 + q^-11 - 2q^-10 + q^-9

4 q^-112 - 3q^-111 + 2q^-110 + 2q^-109 - 5q^-108 + 8q^-107 - 11q^-106 + 10q^-105 + 5q^-104 - 30q^-103 + 28q^-102 - 18q^-101 + 46q^-100 + 8q^-99 - 130q^-98 + 40q^-97 + 12q^-96 + 203q^-95 + 42q^-94 - 416q^-93 - 93q^-92 + 64q^-91 + 674q^-90 + 309q^-89 - 925q^-88 - 645q^-87 - 101q^-86 + 1516q^-85 + 1130q^-84 - 1345q^-83 - 1654q^-82 - 852q^-81 + 2321q^-80 + 2480q^-79 - 1209q^-78 - 2635q^-77 - 2118q^-76 + 2547q^-75 + 3752q^-74 - 492q^-73 - 3008q^-72 - 3331q^-71 + 2115q^-70 + 4375q^-69 + 388q^-68 - 2697q^-67 - 4028q^-66 + 1340q^-65 + 4274q^-64 + 1124q^-63 - 1953q^-62 - 4181q^-61 + 445q^-60 + 3659q^-59 + 1680q^-58 - 967q^-57 - 3885q^-56 - 487q^-55 + 2639q^-54 + 2001q^-53 + 145q^-52 - 3118q^-51 - 1243q^-50 + 1329q^-49 + 1853q^-48 + 1079q^-47 - 1910q^-46 - 1454q^-45 + 96q^-44 + 1157q^-43 + 1409q^-42 - 663q^-41 - 1014q^-40 - 547q^-39 + 306q^-38 + 1042q^-37 + 72q^-36 - 346q^-35 - 501q^-34 - 162q^-33 + 446q^-32 + 186q^-31 + 30q^-30 - 198q^-29 - 178q^-28 + 95q^-27 + 65q^-26 + 71q^-25 - 29q^-24 - 66q^-23 + 10q^-22 + 2q^-21 + 23q^-20 + 2q^-19 - 13q^-18 + 3q^-17 - 2q^-16 + 3q^-15 + q^-14 - 2q^-13 + q^-12

5 q^-165 - 3q^-164 + 2q^-163 + 2q^-162 - 5q^-161 + 4q^-160 + 5q^-159 - 9q^-158 + 5q^-156 - 14q^-155 + 10q^-154 + 28q^-153 - 8q^-152 - 23q^-151 - 29q^-150 - 34q^-149 + 42q^-148 + 124q^-147 + 79q^-146 - 94q^-145 - 250q^-144 - 230q^-143 + 106q^-142 + 529q^-141 + 571q^-140 - 51q^-139 - 966q^-138 - 1223q^-137 - 229q^-136 + 1508q^-135 + 2362q^-134 + 981q^-133 - 2062q^-132 - 4066q^-131 - 2445q^-130 + 2323q^-129 + 6265q^-128 + 4889q^-127 - 1959q^-126 - 8673q^-125 - 8363q^-124 + 539q^-123 + 10972q^-122 + 12596q^-121 + 2016q^-120 - 12414q^-119 - 17207q^-118 - 5870q^-117 + 12956q^-116 + 21512q^-115 + 10274q^-114 - 11960q^-113 - 24980q^-112 - 15129q^-111 + 10069q^-110 + 27214q^-109 + 19354q^-108 - 7181q^-107 - 28067q^-106 - 22979q^-105 + 4154q^-104 + 27760q^-103 + 25343q^-102 - 1020q^-101 - 26529q^-100 - 26836q^-99 - 1680q^-98 + 24699q^-97 + 27344q^-96 + 4155q^-95 - 22483q^-94 - 27310q^-93 - 6296q^-92 + 19975q^-91 + 26751q^-90 + 8352q^-89 - 17131q^-88 - 25821q^-87 - 10390q^-86 + 13933q^-85 + 24470q^-84 + 12272q^-83 - 10256q^-82 - 22458q^-81 - 14086q^-80 + 6267q^-79 + 19868q^-78 + 15203q^-77 - 2091q^-76 - 16305q^-75 - 15744q^-74 - 1888q^-73 + 12355q^-72 + 14955q^-71 + 5219q^-70 - 7784q^-69 - 13298q^-68 - 7489q^-67 + 3609q^-66 + 10422q^-65 + 8421q^-64 + 170q^-63 - 7264q^-62 - 7995q^-61 - 2604q^-60 + 3830q^-59 + 6550q^-58 + 4003q^-57 - 1224q^-56 - 4525q^-55 - 3951q^-54 - 766q^-53 + 2484q^-52 + 3359q^-51 + 1524q^-50 - 896q^-49 - 2145q^-48 - 1702q^-47 - 131q^-46 + 1214q^-45 + 1274q^-44 + 508q^-43 - 395q^-42 - 823q^-41 - 545q^-40 + 48q^-39 + 376q^-38 + 381q^-37 + 126q^-36 - 153q^-35 - 214q^-34 - 92q^-33 + 9q^-32 + 94q^-31 + 79q^-30 + q^-29 - 39q^-28 - 17q^-27 - 17q^-26 + 5q^-25 + 20q^-24 + 3q^-23 - 7q^-22 + 2q^-21 - 2q^-20 - 2q^-19 + 3q^-18 + q^-17 - 2q^-16 + q^-15

6 q^-228 - 3q^-227 + 2q^-226 + 2q^-225 - 5q^-224 + 4q^-223 + q^-222 + 7q^-221 - 19q^-220 + 21q^-218 - 22q^-217 + 15q^-216 + 14q^-215 + 18q^-214 - 71q^-213 - 30q^-212 + 71q^-211 - 31q^-210 + 71q^-209 + 85q^-208 + 34q^-207 - 270q^-206 - 220q^-205 + 111q^-204 + 26q^-203 + 419q^-202 + 505q^-201 + 183q^-200 - 890q^-199 - 1203q^-198 - 412q^-197 + 20q^-196 + 1748q^-195 + 2573q^-194 + 1578q^-193 - 1945q^-192 - 4536q^-191 - 3949q^-190 - 1926q^-189 + 4402q^-188 + 9270q^-187 + 8545q^-186 - 586q^-185 - 11188q^-184 - 15584q^-183 - 12619q^-182 + 4240q^-181 + 22358q^-180 + 28872q^-179 + 12999q^-178 - 15340q^-177 - 37820q^-176 - 42600q^-175 - 11911q^-174 + 33763q^-173 + 65110q^-172 + 51631q^-171 - 266q^-170 - 59535q^-169 - 93143q^-168 - 58221q^-167 + 23230q^-166 + 101941q^-165 + 113847q^-164 + 48484q^-163 - 57173q^-162 - 143829q^-161 - 129229q^-160 - 22997q^-159 + 112632q^-158 + 174229q^-157 + 121201q^-156 - 18423q^-155 - 165771q^-154 - 195465q^-153 - 91437q^-152 + 86451q^-151 + 203544q^-150 + 186513q^-149 + 40685q^-148 - 150503q^-147 - 229384q^-146 - 151162q^-145 + 40424q^-144 + 196390q^-143 + 220603q^-142 + 91816q^-141 - 115272q^-140 - 228998q^-139 - 183335q^-138 - 1170q^-137 + 169705q^-136 + 224952q^-135 + 121623q^-134 - 80100q^-133 - 210457q^-132 - 192429q^-131 - 30031q^-130 + 139486q^-129 + 214526q^-128 + 136763q^-127 - 49336q^-126 - 186057q^-125 - 191824q^-124 - 53884q^-123 + 107147q^-122 + 198150q^-121 + 148440q^-120 - 15210q^-119 - 154700q^-118 - 186771q^-117 - 81111q^-116 + 65196q^-115 + 172030q^-114 + 157840q^-113 + 27591q^-112 - 108664q^-111 - 170090q^-110 - 108362q^-109 + 10843q^-108 + 127270q^-107 + 153696q^-106 + 71245q^-105 - 46846q^-104 - 130841q^-103 - 119857q^-102 - 44277q^-101 + 63685q^-100 + 122768q^-99 + 95843q^-98 + 15833q^-97 - 69625q^-96 - 100285q^-95 - 77001q^-94 - 907q^-93 + 66914q^-92 + 85123q^-91 + 53714q^-90 - 7204q^-89 - 53396q^-88 - 71461q^-87 - 39229q^-86 + 9834q^-85 + 45719q^-84 + 52431q^-83 + 28402q^-82 - 5435q^-81 - 37853q^-80 - 39416q^-79 - 20827q^-78 + 5859q^-77 + 25578q^-76 + 28626q^-75 + 17736q^-74 - 5285q^-73 - 17888q^-72 - 20292q^-71 - 11545q^-70 + 1384q^-69 + 11758q^-68 + 15251q^-67 + 7345q^-66 - 401q^-65 - 7421q^-64 - 8961q^-63 - 6123q^-62 - 221q^-61 + 5151q^-60 + 5097q^-59 + 3854q^-58 + 354q^-57 - 2259q^-56 - 3576q^-55 - 2344q^-54 + 41q^-53 + 970q^-52 + 1853q^-51 + 1277q^-50 + 387q^-49 - 777q^-48 - 945q^-47 - 464q^-46 - 275q^-45 + 291q^-44 + 418q^-43 + 382q^-42 - 15q^-41 - 145q^-40 - 94q^-39 - 166q^-38 - 27q^-37 + 46q^-36 + 107q^-35 + 11q^-34 - 10q^-33 + 12q^-32 - 33q^-31 - 15q^-30 - 4q^-29 + 22q^-28 - 6q^-26 + 8q^-25 - 3q^-24 - 2q^-23 - 2q^-22 + 3q^-21 + q^-20 - 2q^-19 + q^-18

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, 80]]

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

In[3]:=
GaussCode[Knot[10, 80]]

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

In[4]:=
DTCode[Knot[10, 80]]

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

In[5]:=
br = BR[Knot[10, 80]]

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

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

Out[6]=
{4, 11}

In[7]:=
BraidIndex[Knot[10, 80]]

Out[7]=
4

In[8]:=
Show[DrawMorseLink[Knot[10, 80]]]

Out[8]=
-Graphics-

In[9]:=
#[Knot[10, 80]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}

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

In[10]:=
alex = Alexander[Knot[10, 80]][t]

Out[10]=
3 9 15 2 3 -17 + -- - -- + -- + 15 t - 9 t + 3 t 3 2 t t t

In[11]:=
Conway[Knot[10, 80]][z]

Out[11]=
2 4 6 1 + 6 z + 9 z + 3 z

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

Out[12]=
{Knot[10, 80]}

In[13]:=
{KnotDet[Knot[10, 80]], KnotSignature[Knot[10, 80]]}

Out[13]=
{71, -6}

In[14]:=
Jones[Knot[10, 80]][q]

Out[14]=
-13 3 6 10 11 12 11 8 6 2 -3 q - --- + --- - --- + -- - -- + -- - -- + -- - -- + q 12 11 10 9 8 7 6 5 4 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, 80]}

In[16]:=
A2Invariant[Knot[10, 80]][q]

Out[16]=
-40 -38 -36 -34 3 2 -28 3 3 -22 3 2 q + q - q + q - --- - --- - q - --- + --- - q + --- + --- + 32 30 26 24 20 18 q q q q q q 3 -12 -10 > --- - q + q 14 q

In[17]:=
HOMFLYPT[Knot[10, 80]][a, z]

Out[17]=
6 8 10 12 6 2 8 2 10 2 12 2 6 4 2 a + 3 a - 6 a + 2 a + 5 a z + 9 a z - 9 a z + a z + 4 a z + 8 4 10 4 6 6 8 6 > 8 a z - 3 a z + a z + 2 a z

In[18]:=
Kauffman[Knot[10, 80]][a, z]

Out[18]=
6 8 10 12 7 9 11 13 15 -2 a + 3 a + 6 a + 2 a + a z - 8 a z - 12 a z - 2 a z + a z + 6 2 8 2 10 2 12 2 14 2 16 2 7 3 > 5 a z - 7 a z - 13 a z + 2 a z + 2 a z - a z + 2 a z + 9 3 11 3 13 3 15 3 6 4 8 4 > 22 a z + 29 a z + 6 a z - 3 a z - 4 a z + 8 a z + 10 4 12 4 14 4 16 4 7 5 9 5 11 5 > 13 a z - 5 a z - 5 a z + a z - 5 a z - 23 a z - 29 a z - 13 5 15 5 6 6 8 6 10 6 12 6 14 6 > 8 a z + 3 a z + a z - 8 a z - 15 a z - a z + 5 a z + 7 7 9 7 11 7 13 7 8 8 10 8 12 8 > 2 a z + 6 a z + 10 a z + 6 a z + 3 a z + 7 a z + 4 a z + 9 9 11 9 > a z + a z

In[19]:=
{Vassiliev[2][Knot[10, 80]], Vassiliev[3][Knot[10, 80]]}

Out[19]=
{6, -12}

In[20]:=
Kh[Knot[10, 80]][q, t]

Out[20]=
-7 -5 1 2 1 4 2 6 4 q + q + ------- + ------ + ------ + ------ + ------ + ------ + ------ + 27 10 25 9 23 9 23 8 21 8 21 7 19 7 q t q t q t q t q t q t q t 5 6 7 5 4 7 4 4 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 19 6 17 6 17 5 15 5 15 4 13 4 13 3 11 3 q t q t q t q t q t q t q t q t 2 4 2 > ------ + ----- + ---- 11 2 9 2 7 q t q t q t

In[21]:=
ColouredJones[Knot[10, 80], 2][q]

Out[21]=
-36 3 2 6 17 11 23 51 21 58 93 20 95 q - --- + --- + --- - --- + --- + --- - --- + --- + --- - --- + --- + --- - 35 34 33 32 31 30 29 28 27 26 25 24 q q q q q q q q q q q q 114 4 112 103 17 103 68 30 71 28 27 33 > --- + --- + --- - --- - --- + --- - --- - --- + --- - --- - --- + --- - 23 22 21 20 19 18 17 16 15 14 13 12 q q q q q q q q q q q q 4 12 8 -8 2 -6 > --- - --- + -- + q - -- + q 11 10 9 7 q q q q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 10₈₀

10₇₉
10₈₁

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q^-36 - 3q^-35 + 2q^-34 + 6q^-33 - 17q^-32 + 11q^-31 + 23q^-30 - 51q^-29 + 21q^-28 + 58q^-27 - 93q^-26 + 20q^-25 + 95q^-24 - 114q^-23 + 4q^-22 + 112q^-21 - 103q^-20 - 17q^-19 + 103q^-18 - 68q^-17 - 30q^-16 + 71q^-15 - 28q^-14 - 27q^-13 + 33q^-12 - 4q^-11 - 12q^-10 + 8q^-9 + q^-8 - 2q^-7 + q^-6
3	q^-69 - 3q^-68 + 2q^-67 + 2q^-66 - q^-65 - 8q^-64 + 8q^-63 + 15q^-62 - 20q^-61 - 29q^-60 + 40q^-59 + 57q^-58 - 68q^-57 - 107q^-56 + 101q^-55 + 183q^-54 - 135q^-53 - 272q^-52 + 137q^-51 + 395q^-50 - 140q^-49 - 490q^-48 + 94q^-47 + 589q^-46 - 51q^-45 - 635q^-44 - 25q^-43 + 666q^-42 + 89q^-41 - 649q^-40 - 160q^-39 + 608q^-38 + 224q^-37 - 543q^-36 - 272q^-35 + 449q^-34 + 320q^-33 - 361q^-32 - 322q^-31 + 237q^-30 + 327q^-29 - 149q^-28 - 278q^-27 + 47q^-26 + 235q^-25 - 154q^-23 - 48q^-22 + 106q^-21 + 41q^-20 - 46q^-19 - 40q^-18 + 23q^-17 + 21q^-16 - 4q^-15 - 12q^-14 + 3q^-13 + 3q^-12 + q^-11 - 2q^-10 + q^-9
4	q^-112 - 3q^-111 + 2q^-110 + 2q^-109 - 5q^-108 + 8q^-107 - 11q^-106 + 10q^-105 + 5q^-104 - 30q^-103 + 28q^-102 - 18q^-101 + 46q^-100 + 8q^-99 - 130q^-98 + 40q^-97 + 12q^-96 + 203q^-95 + 42q^-94 - 416q^-93 - 93q^-92 + 64q^-91 + 674q^-90 + 309q^-89 - 925q^-88 - 645q^-87 - 101q^-86 + 1516q^-85 + 1130q^-84 - 1345q^-83 - 1654q^-82 - 852q^-81 + 2321q^-80 + 2480q^-79 - 1209q^-78 - 2635q^-77 - 2118q^-76 + 2547q^-75 + 3752q^-74 - 492q^-73 - 3008q^-72 - 3331q^-71 + 2115q^-70 + 4375q^-69 + 388q^-68 - 2697q^-67 - 4028q^-66 + 1340q^-65 + 4274q^-64 + 1124q^-63 - 1953q^-62 - 4181q^-61 + 445q^-60 + 3659q^-59 + 1680q^-58 - 967q^-57 - 3885q^-56 - 487q^-55 + 2639q^-54 + 2001q^-53 + 145q^-52 - 3118q^-51 - 1243q^-50 + 1329q^-49 + 1853q^-48 + 1079q^-47 - 1910q^-46 - 1454q^-45 + 96q^-44 + 1157q^-43 + 1409q^-42 - 663q^-41 - 1014q^-40 - 547q^-39 + 306q^-38 + 1042q^-37 + 72q^-36 - 346q^-35 - 501q^-34 - 162q^-33 + 446q^-32 + 186q^-31 + 30q^-30 - 198q^-29 - 178q^-28 + 95q^-27 + 65q^-26 + 71q^-25 - 29q^-24 - 66q^-23 + 10q^-22 + 2q^-21 + 23q^-20 + 2q^-19 - 13q^-18 + 3q^-17 - 2q^-16 + 3q^-15 + q^-14 - 2q^-13 + q^-12
5	q^-165 - 3q^-164 + 2q^-163 + 2q^-162 - 5q^-161 + 4q^-160 + 5q^-159 - 9q^-158 + 5q^-156 - 14q^-155 + 10q^-154 + 28q^-153 - 8q^-152 - 23q^-151 - 29q^-150 - 34q^-149 + 42q^-148 + 124q^-147 + 79q^-146 - 94q^-145 - 250q^-144 - 230q^-143 + 106q^-142 + 529q^-141 + 571q^-140 - 51q^-139 - 966q^-138 - 1223q^-137 - 229q^-136 + 1508q^-135 + 2362q^-134 + 981q^-133 - 2062q^-132 - 4066q^-131 - 2445q^-130 + 2323q^-129 + 6265q^-128 + 4889q^-127 - 1959q^-126 - 8673q^-125 - 8363q^-124 + 539q^-123 + 10972q^-122 + 12596q^-121 + 2016q^-120 - 12414q^-119 - 17207q^-118 - 5870q^-117 + 12956q^-116 + 21512q^-115 + 10274q^-114 - 11960q^-113 - 24980q^-112 - 15129q^-111 + 10069q^-110 + 27214q^-109 + 19354q^-108 - 7181q^-107 - 28067q^-106 - 22979q^-105 + 4154q^-104 + 27760q^-103 + 25343q^-102 - 1020q^-101 - 26529q^-100 - 26836q^-99 - 1680q^-98 + 24699q^-97 + 27344q^-96 + 4155q^-95 - 22483q^-94 - 27310q^-93 - 6296q^-92 + 19975q^-91 + 26751q^-90 + 8352q^-89 - 17131q^-88 - 25821q^-87 - 10390q^-86 + 13933q^-85 + 24470q^-84 + 12272q^-83 - 10256q^-82 - 22458q^-81 - 14086q^-80 + 6267q^-79 + 19868q^-78 + 15203q^-77 - 2091q^-76 - 16305q^-75 - 15744q^-74 - 1888q^-73 + 12355q^-72 + 14955q^-71 + 5219q^-70 - 7784q^-69 - 13298q^-68 - 7489q^-67 + 3609q^-66 + 10422q^-65 + 8421q^-64 + 170q^-63 - 7264q^-62 - 7995q^-61 - 2604q^-60 + 3830q^-59 + 6550q^-58 + 4003q^-57 - 1224q^-56 - 4525q^-55 - 3951q^-54 - 766q^-53 + 2484q^-52 + 3359q^-51 + 1524q^-50 - 896q^-49 - 2145q^-48 - 1702q^-47 - 131q^-46 + 1214q^-45 + 1274q^-44 + 508q^-43 - 395q^-42 - 823q^-41 - 545q^-40 + 48q^-39 + 376q^-38 + 381q^-37 + 126q^-36 - 153q^-35 - 214q^-34 - 92q^-33 + 9q^-32 + 94q^-31 + 79q^-30 + q^-29 - 39q^-28 - 17q^-27 - 17q^-26 + 5q^-25 + 20q^-24 + 3q^-23 - 7q^-22 + 2q^-21 - 2q^-20 - 2q^-19 + 3q^-18 + q^-17 - 2q^-16 + q^-15
6	q^-228 - 3q^-227 + 2q^-226 + 2q^-225 - 5q^-224 + 4q^-223 + q^-222 + 7q^-221 - 19q^-220 + 21q^-218 - 22q^-217 + 15q^-216 + 14q^-215 + 18q^-214 - 71q^-213 - 30q^-212 + 71q^-211 - 31q^-210 + 71q^-209 + 85q^-208 + 34q^-207 - 270q^-206 - 220q^-205 + 111q^-204 + 26q^-203 + 419q^-202 + 505q^-201 + 183q^-200 - 890q^-199 - 1203q^-198 - 412q^-197 + 20q^-196 + 1748q^-195 + 2573q^-194 + 1578q^-193 - 1945q^-192 - 4536q^-191 - 3949q^-190 - 1926q^-189 + 4402q^-188 + 9270q^-187 + 8545q^-186 - 586q^-185 - 11188q^-184 - 15584q^-183 - 12619q^-182 + 4240q^-181 + 22358q^-180 + 28872q^-179 + 12999q^-178 - 15340q^-177 - 37820q^-176 - 42600q^-175 - 11911q^-174 + 33763q^-173 + 65110q^-172 + 51631q^-171 - 266q^-170 - 59535q^-169 - 93143q^-168 - 58221q^-167 + 23230q^-166 + 101941q^-165 + 113847q^-164 + 48484q^-163 - 57173q^-162 - 143829q^-161 - 129229q^-160 - 22997q^-159 + 112632q^-158 + 174229q^-157 + 121201q^-156 - 18423q^-155 - 165771q^-154 - 195465q^-153 - 91437q^-152 + 86451q^-151 + 203544q^-150 + 186513q^-149 + 40685q^-148 - 150503q^-147 - 229384q^-146 - 151162q^-145 + 40424q^-144 + 196390q^-143 + 220603q^-142 + 91816q^-141 - 115272q^-140 - 228998q^-139 - 183335q^-138 - 1170q^-137 + 169705q^-136 + 224952q^-135 + 121623q^-134 - 80100q^-133 - 210457q^-132 - 192429q^-131 - 30031q^-130 + 139486q^-129 + 214526q^-128 + 136763q^-127 - 49336q^-126 - 186057q^-125 - 191824q^-124 - 53884q^-123 + 107147q^-122 + 198150q^-121 + 148440q^-120 - 15210q^-119 - 154700q^-118 - 186771q^-117 - 81111q^-116 + 65196q^-115 + 172030q^-114 + 157840q^-113 + 27591q^-112 - 108664q^-111 - 170090q^-110 - 108362q^-109 + 10843q^-108 + 127270q^-107 + 153696q^-106 + 71245q^-105 - 46846q^-104 - 130841q^-103 - 119857q^-102 - 44277q^-101 + 63685q^-100 + 122768q^-99 + 95843q^-98 + 15833q^-97 - 69625q^-96 - 100285q^-95 - 77001q^-94 - 907q^-93 + 66914q^-92 + 85123q^-91 + 53714q^-90 - 7204q^-89 - 53396q^-88 - 71461q^-87 - 39229q^-86 + 9834q^-85 + 45719q^-84 + 52431q^-83 + 28402q^-82 - 5435q^-81 - 37853q^-80 - 39416q^-79 - 20827q^-78 + 5859q^-77 + 25578q^-76 + 28626q^-75 + 17736q^-74 - 5285q^-73 - 17888q^-72 - 20292q^-71 - 11545q^-70 + 1384q^-69 + 11758q^-68 + 15251q^-67 + 7345q^-66 - 401q^-65 - 7421q^-64 - 8961q^-63 - 6123q^-62 - 221q^-61 + 5151q^-60 + 5097q^-59 + 3854q^-58 + 354q^-57 - 2259q^-56 - 3576q^-55 - 2344q^-54 + 41q^-53 + 970q^-52 + 1853q^-51 + 1277q^-50 + 387q^-49 - 777q^-48 - 945q^-47 - 464q^-46 - 275q^-45 + 291q^-44 + 418q^-43 + 382q^-42 - 15q^-41 - 145q^-40 - 94q^-39 - 166q^-38 - 27q^-37 + 46q^-36 + 107q^-35 + 11q^-34 - 10q^-33 + 12q^-32 - 33q^-31 - 15q^-30 - 4q^-29 + 22q^-28 - 6q^-26 + 8q^-25 - 3q^-24 - 2q^-23 - 2q^-22 + 3q^-21 + q^-20 - 2q^-19 + q^-18

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	PD[Knot[10, 80]]
Out[2]=	PD[X[1, 4, 2, 5], X[3, 8, 4, 9], X[5, 12, 6, 13], X[13, 18, 14, 19], > X[9, 16, 10, 17], X[17, 10, 18, 11], X[15, 20, 16, 1], X[19, 14, 20, 15], > X[11, 6, 12, 7], X[7, 2, 8, 3]]
In[3]:=	GaussCode[Knot[10, 80]]
Out[3]=	GaussCode[-1, 10, -2, 1, -3, 9, -10, 2, -5, 6, -9, 3, -4, 8, -7, 5, -6, 4, -8, > 7]
In[4]:=	DTCode[Knot[10, 80]]
Out[4]=	DTCode[4, 8, 12, 2, 16, 6, 18, 20, 10, 14]
In[5]:=	br = BR[Knot[10, 80]]
Out[5]=	BR[4, {-1, -1, -1, 2, -1, -1, -3, -2, -2, -2, -3}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{4, 11}
In[7]:=	BraidIndex[Knot[10, 80]]
Out[7]=	4
In[8]:=	Show[DrawMorseLink[Knot[10, 80]]]

Out[8]=	-Graphics-
In[9]:=	#[Knot[10, 80]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Chiral, 3, 3, 3, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 80]][t]
Out[10]=	3 9 15 2 3 -17 + -- - -- + -- + 15 t - 9 t + 3 t 3 2 t t t
In[11]:=	Conway[Knot[10, 80]][z]
Out[11]=	2 4 6 1 + 6 z + 9 z + 3 z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 80]}
In[13]:=	{KnotDet[Knot[10, 80]], KnotSignature[Knot[10, 80]]}
Out[13]=	{71, -6}
In[14]:=	Jones[Knot[10, 80]][q]
Out[14]=	-13 3 6 10 11 12 11 8 6 2 -3 q - --- + --- - --- + -- - -- + -- - -- + -- - -- + q 12 11 10 9 8 7 6 5 4 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, 80]}
In[16]:=	A2Invariant[Knot[10, 80]][q]
Out[16]=	-40 -38 -36 -34 3 2 -28 3 3 -22 3 2 q + q - q + q - --- - --- - q - --- + --- - q + --- + --- + 32 30 26 24 20 18 q q q q q q 3 -12 -10 > --- - q + q 14 q
In[17]:=	HOMFLYPT[Knot[10, 80]][a, z]
Out[17]=	6 8 10 12 6 2 8 2 10 2 12 2 6 4 2 a + 3 a - 6 a + 2 a + 5 a z + 9 a z - 9 a z + a z + 4 a z + 8 4 10 4 6 6 8 6 > 8 a z - 3 a z + a z + 2 a z
In[18]:=	Kauffman[Knot[10, 80]][a, z]
Out[18]=	6 8 10 12 7 9 11 13 15 -2 a + 3 a + 6 a + 2 a + a z - 8 a z - 12 a z - 2 a z + a z + 6 2 8 2 10 2 12 2 14 2 16 2 7 3 > 5 a z - 7 a z - 13 a z + 2 a z + 2 a z - a z + 2 a z + 9 3 11 3 13 3 15 3 6 4 8 4 > 22 a z + 29 a z + 6 a z - 3 a z - 4 a z + 8 a z + 10 4 12 4 14 4 16 4 7 5 9 5 11 5 > 13 a z - 5 a z - 5 a z + a z - 5 a z - 23 a z - 29 a z - 13 5 15 5 6 6 8 6 10 6 12 6 14 6 > 8 a z + 3 a z + a z - 8 a z - 15 a z - a z + 5 a z + 7 7 9 7 11 7 13 7 8 8 10 8 12 8 > 2 a z + 6 a z + 10 a z + 6 a z + 3 a z + 7 a z + 4 a z + 9 9 11 9 > a z + a z
In[19]:=	{Vassiliev[2][Knot[10, 80]], Vassiliev[3][Knot[10, 80]]}
Out[19]=	{6, -12}
In[20]:=	Kh[Knot[10, 80]][q, t]
Out[20]=	-7 -5 1 2 1 4 2 6 4 q + q + ------- + ------ + ------ + ------ + ------ + ------ + ------ + 27 10 25 9 23 9 23 8 21 8 21 7 19 7 q t q t q t q t q t q t q t 5 6 7 5 4 7 4 4 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 19 6 17 6 17 5 15 5 15 4 13 4 13 3 11 3 q t q t q t q t q t q t q t q t 2 4 2 > ------ + ----- + ---- 11 2 9 2 7 q t q t q t
In[21]:=	ColouredJones[Knot[10, 80], 2][q]
Out[21]=	-36 3 2 6 17 11 23 51 21 58 93 20 95 q - --- + --- + --- - --- + --- + --- - --- + --- + --- - --- + --- + --- - 35 34 33 32 31 30 29 28 27 26 25 24 q q q q q q q q q q q q 114 4 112 103 17 103 68 30 71 28 27 33 > --- + --- + --- - --- - --- + --- - --- - --- + --- - --- - --- + --- - 23 22 21 20 19 18 17 16 15 14 13 12 q q q q q q q q q q q q 4 12 8 -8 2 -6 > --- - --- + -- + q - -- + q 11 10 9 7 q q q q

The Alternating Knot 1080

The Alternating Knot 10₈₀