The Non Alternating Knot 10₁₄₉

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Acknowledgement

KnotPlot

PD Presentation: X₁₄₂₅ X₃₈₄₉ X_12,6,13,5 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_6,12,7,11 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 10, width is 3
Braid index is 3

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: - t^-3 + 5t^-2 - 9t^-1 + 11 - 9t + 5t² - t³

Conway Polynomial: 1 + 2z² - z⁴ - z⁶

Other knots with the same Alexander/Conway Polynomial: {9₂₀, K11n26, ...}

Determinant and Signature: {41, -4}

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

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

A2 (sl(3)) Invariant: q^-30 - q^-28 + q^-26 - 2q^-22 - 3q^-18 + q^-16 + q^-12 + 3q^-10 + 2q^-6

HOMFLY-PT Polynomial: 4a⁴ + 6a⁴z² + 2a⁴z⁴ - 4a⁶ - 6a⁶z² - 4a⁶z⁴ - a⁶z⁶ + a⁸ + 2a⁸z² + a⁸z⁴

Kauffman Polynomial: 4a⁴ - 7a⁴z² + 3a⁴z⁴ - 3a⁵z + 2a⁵z³ - a⁵z⁵ + a⁵z⁷ + 4a⁶ - 9a⁶z² + 5a⁶z⁴ - a⁶z⁶ + a⁶z⁸ - 3a⁷z + 5a⁷z³ - 6a⁷z⁵ + 4a⁷z⁷ + a⁸ - 4a⁸z⁴ + 3a⁸z⁶ + a⁸z⁸ + a⁹z - a⁹z³ - 2a⁹z⁵ + 3a⁹z⁷ + a¹⁰z² - 5a¹⁰z⁴ + 4a¹⁰z⁶ + a¹¹z - 4a¹¹z³ + 3a¹¹z⁵ - a¹²z² + a¹²z⁴

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=-4 is the signature of 10₁₄₉. 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

j = -3 2

j = -5 2 1

j = -7 4 1

j = -9 3 2

j = -11 4 4

j = -13 3 3

j = -15 2 4

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 + 8q^-25 - 14q^-24 + 27q^-22 - 29q^-21 - 10q^-20 + 50q^-19 - 37q^-18 - 22q^-17 + 61q^-16 - 34q^-15 - 29q^-14 + 55q^-13 - 21q^-12 - 28q^-11 + 36q^-10 - 6q^-9 - 18q^-8 + 14q^-7 + 2q^-6 - 6q^-5 + 2q^-4 + q^-3

3 q^-54 - 3q^-53 + q^-52 + 4q^-51 + q^-50 - 11q^-49 - 3q^-48 + 24q^-47 + 7q^-46 - 40q^-45 - 22q^-44 + 62q^-43 + 48q^-42 - 84q^-41 - 86q^-40 + 101q^-39 + 127q^-38 - 103q^-37 - 176q^-36 + 105q^-35 + 210q^-34 - 90q^-33 - 241q^-32 + 78q^-31 + 252q^-30 - 55q^-29 - 257q^-28 + 34q^-27 + 248q^-26 - 11q^-25 - 227q^-24 - 16q^-23 + 201q^-22 + 35q^-21 - 159q^-20 - 60q^-19 + 126q^-18 + 58q^-17 - 75q^-16 - 65q^-15 + 46q^-14 + 47q^-13 - 13q^-12 - 36q^-11 + 4q^-10 + 15q^-9 + 7q^-8 - 9q^-7 - q^-6 + 2q^-4

4 q^-88 - 3q^-87 + q^-86 + 4q^-85 - 3q^-84 + 4q^-83 - 14q^-82 + 5q^-81 + 20q^-80 - 9q^-79 + 7q^-78 - 56q^-77 + 9q^-76 + 79q^-75 + 10q^-74 + 9q^-73 - 182q^-72 - 38q^-71 + 190q^-70 + 140q^-69 + 91q^-68 - 415q^-67 - 251q^-66 + 251q^-65 + 404q^-64 + 370q^-63 - 623q^-62 - 622q^-61 + 123q^-60 + 652q^-59 + 807q^-58 - 658q^-57 - 972q^-56 - 151q^-55 + 743q^-54 + 1193q^-53 - 538q^-52 - 1144q^-51 - 416q^-50 + 682q^-49 + 1396q^-48 - 362q^-47 - 1135q^-46 - 592q^-45 + 532q^-44 + 1416q^-43 - 164q^-42 - 987q^-41 - 699q^-40 + 310q^-39 + 1289q^-38 + 63q^-37 - 718q^-36 - 729q^-35 + 27q^-34 + 1008q^-33 + 271q^-32 - 354q^-31 - 631q^-30 - 226q^-29 + 604q^-28 + 337q^-27 - 17q^-26 - 385q^-25 - 310q^-24 + 211q^-23 + 222q^-22 + 140q^-21 - 119q^-20 - 206q^-19 + q^-18 + 59q^-17 + 104q^-16 + 12q^-15 - 63q^-14 - 26q^-13 - 10q^-12 + 26q^-11 + 17q^-10 - 4q^-9 - 4q^-8 - 6q^-7 + 2q^-5 + q^-4

5 q^-130 - 3q^-129 + q^-128 + 4q^-127 - 3q^-126 + q^-124 - 6q^-123 + q^-122 + 15q^-121 - 2q^-120 - 14q^-119 - 10q^-118 - 9q^-117 + 20q^-116 + 50q^-115 + 24q^-114 - 56q^-113 - 107q^-112 - 61q^-111 + 73q^-110 + 217q^-109 + 188q^-108 - 73q^-107 - 394q^-106 - 418q^-105 - 7q^-104 + 572q^-103 + 809q^-102 + 277q^-101 - 736q^-100 - 1339q^-99 - 764q^-98 + 745q^-97 + 1938q^-96 + 1518q^-95 - 525q^-94 - 2504q^-93 - 2461q^-92 + 4q^-91 + 2936q^-90 + 3485q^-89 + 745q^-88 - 3094q^-87 - 4455q^-86 - 1697q^-85 + 3034q^-84 + 5272q^-83 + 2612q^-82 - 2713q^-81 - 5816q^-80 - 3525q^-79 + 2307q^-78 + 6146q^-77 + 4196q^-76 - 1810q^-75 - 6226q^-74 - 4746q^-73 + 1348q^-72 + 6180q^-71 + 5063q^-70 - 898q^-69 - 5985q^-68 - 5288q^-67 + 490q^-66 + 5713q^-65 + 5381q^-64 - 65q^-63 - 5339q^-62 - 5409q^-61 - 384q^-60 + 4853q^-59 + 5365q^-58 + 870q^-57 - 4234q^-56 - 5204q^-55 - 1410q^-54 + 3466q^-53 + 4953q^-52 + 1894q^-51 - 2586q^-50 - 4448q^-49 - 2368q^-48 + 1611q^-47 + 3864q^-46 + 2585q^-45 - 682q^-44 - 2986q^-43 - 2650q^-42 - 183q^-41 + 2157q^-40 + 2379q^-39 + 743q^-38 - 1191q^-37 - 1958q^-36 - 1084q^-35 + 491q^-34 + 1353q^-33 + 1076q^-32 + 93q^-31 - 803q^-30 - 904q^-29 - 313q^-28 + 303q^-27 + 594q^-26 + 409q^-25 - 41q^-24 - 325q^-23 - 277q^-22 - 111q^-21 + 100q^-20 + 196q^-19 + 100q^-18 - 17q^-17 - 57q^-16 - 73q^-15 - 32q^-14 + 25q^-13 + 31q^-12 + 11q^-11 + 9q^-10 - 9q^-9 - 9q^-8 - 3q^-7 + 2q^-6 + 2q^-4

6 q^-180 - 3q^-179 + q^-178 + 4q^-177 - 3q^-176 - 3q^-174 + 9q^-173 - 10q^-172 - 4q^-171 + 22q^-170 - 12q^-169 - 8q^-168 - 15q^-167 + 30q^-166 - 10q^-165 - q^-164 + 68q^-163 - 35q^-162 - 64q^-161 - 94q^-160 + 59q^-159 + 18q^-158 + 100q^-157 + 276q^-156 - 22q^-155 - 245q^-154 - 481q^-153 - 151q^-152 - 41q^-151 + 454q^-150 + 1100q^-149 + 553q^-148 - 310q^-147 - 1474q^-146 - 1398q^-145 - 1136q^-144 + 620q^-143 + 2969q^-142 + 2980q^-141 + 1251q^-140 - 2288q^-139 - 4210q^-138 - 5099q^-137 - 1590q^-136 + 4589q^-135 + 7780q^-134 + 6715q^-133 - 4q^-132 - 6688q^-131 - 12184q^-130 - 8654q^-129 + 2399q^-128 + 12298q^-127 + 15768q^-126 + 7799q^-125 - 4878q^-124 - 18904q^-123 - 19428q^-122 - 5598q^-121 + 12338q^-120 + 24136q^-119 + 19069q^-118 + 2559q^-117 - 20983q^-116 - 29031q^-115 - 16533q^-114 + 7113q^-113 + 27725q^-112 + 28704q^-111 + 12234q^-110 - 18087q^-109 - 33750q^-108 - 25487q^-107 - 7q^-106 + 26588q^-105 + 33650q^-104 + 19844q^-103 - 13285q^-102 - 33983q^-101 - 30221q^-100 - 5642q^-99 + 23419q^-98 + 34648q^-97 + 24016q^-96 - 9069q^-95 - 32050q^-94 - 31763q^-93 - 9201q^-92 + 19989q^-91 + 33694q^-90 + 26000q^-89 - 5473q^-88 - 29176q^-87 - 31830q^-86 - 12047q^-85 + 15978q^-84 + 31596q^-83 + 27269q^-82 - 1144q^-81 - 24815q^-80 - 30885q^-79 - 15352q^-78 + 10132q^-77 + 27584q^-76 + 27892q^-75 + 4709q^-74 - 17790q^-73 - 27853q^-72 - 18630q^-71 + 2105q^-70 + 20425q^-69 + 26268q^-68 + 10875q^-67 - 8119q^-66 - 21236q^-65 - 19536q^-64 - 6174q^-63 + 10341q^-62 + 20532q^-61 + 14200q^-60 + 1632q^-59 - 11376q^-58 - 15783q^-57 - 10932q^-56 + 383q^-55 + 11251q^-54 + 12173q^-53 + 7270q^-52 - 1777q^-51 - 8228q^-50 - 9796q^-49 - 5114q^-48 + 2410q^-47 + 6177q^-46 + 6832q^-45 + 3236q^-44 - 1208q^-43 - 4841q^-42 - 4813q^-41 - 1890q^-40 + 790q^-39 + 3045q^-38 + 3033q^-37 + 1687q^-36 - 686q^-35 - 1913q^-34 - 1748q^-33 - 1065q^-32 + 217q^-31 + 991q^-30 + 1223q^-29 + 538q^-28 - 78q^-27 - 442q^-26 - 607q^-25 - 367q^-24 - 47q^-23 + 273q^-22 + 246q^-21 + 172q^-20 + 58q^-19 - 75q^-18 - 118q^-17 - 94q^-16 - 9q^-15 + 15q^-14 + 32q^-13 + 32q^-12 + 17q^-11 - 4q^-10 - 12q^-9 - 6q^-8 - 4q^-7 + 2q^-4 + q^-3

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, 149]]

Out[2]=
PD[X[1, 4, 2, 5], X[3, 8, 4, 9], X[12, 6, 13, 5], 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[6, 12, 7, 11], X[7, 2, 8, 3]]

In[3]:=
GaussCode[Knot[10, 149]]

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, 149]]

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

In[5]:=
br = BR[Knot[10, 149]]

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

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

Out[6]=
{3, 10}

In[7]:=
BraidIndex[Knot[10, 149]]

Out[7]=
3

In[8]:=
Show[DrawMorseLink[Knot[10, 149]]]

Out[8]=
-Graphics-

In[9]:=
#[Knot[10, 149]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}

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

In[10]:=
alex = Alexander[Knot[10, 149]][t]

Out[10]=
-3 5 9 2 3 11 - t + -- - - - 9 t + 5 t - t 2 t t

In[11]:=
Conway[Knot[10, 149]][z]

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

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

Out[12]=
{Knot[9, 20], Knot[10, 149], Knot[11, NonAlternating, 26]}

In[13]:=
{KnotDet[Knot[10, 149]], KnotSignature[Knot[10, 149]]}

Out[13]=
{41, -4}

In[14]:=
Jones[Knot[10, 149]][q]

Out[14]=
-10 3 5 7 7 7 6 3 2 q - -- + -- - -- + -- - -- + -- - -- + -- 9 8 7 6 5 4 3 2 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, 149]}

In[16]:=
A2Invariant[Knot[10, 149]][q]

Out[16]=
-30 -28 -26 2 3 -16 -12 3 2 q - q + q - --- - --- + q + q + --- + -- 22 18 10 6 q q q q

In[17]:=
HOMFLYPT[Knot[10, 149]][a, z]

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

In[18]:=
Kauffman[Knot[10, 149]][a, z]

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

In[19]:=
{Vassiliev[2][Knot[10, 149]], Vassiliev[3][Knot[10, 149]]}

Out[19]=
{2, -2}

In[20]:=
Kh[Knot[10, 149]][q, t]

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

In[21]:=
ColouredJones[Knot[10, 149], 2][q]

Out[21]=
-28 3 -26 8 14 27 29 10 50 37 22 61 34 q - --- + q + --- - --- + --- - --- - --- + --- - --- - --- + --- - --- - 27 25 24 22 21 20 19 18 17 16 15 q q q q q q q q q q q 29 55 21 28 36 6 18 14 2 6 2 -3 > --- + --- - --- - --- + --- - -- - -- + -- + -- - -- + -- + q 14 13 12 11 10 9 8 7 6 5 4 q q q q q q q q q q q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 10₁₄₉

10₁₄₈
10₁₅₀

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q^-28 - 3q^-27 + q^-26 + 8q^-25 - 14q^-24 + 27q^-22 - 29q^-21 - 10q^-20 + 50q^-19 - 37q^-18 - 22q^-17 + 61q^-16 - 34q^-15 - 29q^-14 + 55q^-13 - 21q^-12 - 28q^-11 + 36q^-10 - 6q^-9 - 18q^-8 + 14q^-7 + 2q^-6 - 6q^-5 + 2q^-4 + q^-3
3	q^-54 - 3q^-53 + q^-52 + 4q^-51 + q^-50 - 11q^-49 - 3q^-48 + 24q^-47 + 7q^-46 - 40q^-45 - 22q^-44 + 62q^-43 + 48q^-42 - 84q^-41 - 86q^-40 + 101q^-39 + 127q^-38 - 103q^-37 - 176q^-36 + 105q^-35 + 210q^-34 - 90q^-33 - 241q^-32 + 78q^-31 + 252q^-30 - 55q^-29 - 257q^-28 + 34q^-27 + 248q^-26 - 11q^-25 - 227q^-24 - 16q^-23 + 201q^-22 + 35q^-21 - 159q^-20 - 60q^-19 + 126q^-18 + 58q^-17 - 75q^-16 - 65q^-15 + 46q^-14 + 47q^-13 - 13q^-12 - 36q^-11 + 4q^-10 + 15q^-9 + 7q^-8 - 9q^-7 - q^-6 + 2q^-4
4	q^-88 - 3q^-87 + q^-86 + 4q^-85 - 3q^-84 + 4q^-83 - 14q^-82 + 5q^-81 + 20q^-80 - 9q^-79 + 7q^-78 - 56q^-77 + 9q^-76 + 79q^-75 + 10q^-74 + 9q^-73 - 182q^-72 - 38q^-71 + 190q^-70 + 140q^-69 + 91q^-68 - 415q^-67 - 251q^-66 + 251q^-65 + 404q^-64 + 370q^-63 - 623q^-62 - 622q^-61 + 123q^-60 + 652q^-59 + 807q^-58 - 658q^-57 - 972q^-56 - 151q^-55 + 743q^-54 + 1193q^-53 - 538q^-52 - 1144q^-51 - 416q^-50 + 682q^-49 + 1396q^-48 - 362q^-47 - 1135q^-46 - 592q^-45 + 532q^-44 + 1416q^-43 - 164q^-42 - 987q^-41 - 699q^-40 + 310q^-39 + 1289q^-38 + 63q^-37 - 718q^-36 - 729q^-35 + 27q^-34 + 1008q^-33 + 271q^-32 - 354q^-31 - 631q^-30 - 226q^-29 + 604q^-28 + 337q^-27 - 17q^-26 - 385q^-25 - 310q^-24 + 211q^-23 + 222q^-22 + 140q^-21 - 119q^-20 - 206q^-19 + q^-18 + 59q^-17 + 104q^-16 + 12q^-15 - 63q^-14 - 26q^-13 - 10q^-12 + 26q^-11 + 17q^-10 - 4q^-9 - 4q^-8 - 6q^-7 + 2q^-5 + q^-4
5	q^-130 - 3q^-129 + q^-128 + 4q^-127 - 3q^-126 + q^-124 - 6q^-123 + q^-122 + 15q^-121 - 2q^-120 - 14q^-119 - 10q^-118 - 9q^-117 + 20q^-116 + 50q^-115 + 24q^-114 - 56q^-113 - 107q^-112 - 61q^-111 + 73q^-110 + 217q^-109 + 188q^-108 - 73q^-107 - 394q^-106 - 418q^-105 - 7q^-104 + 572q^-103 + 809q^-102 + 277q^-101 - 736q^-100 - 1339q^-99 - 764q^-98 + 745q^-97 + 1938q^-96 + 1518q^-95 - 525q^-94 - 2504q^-93 - 2461q^-92 + 4q^-91 + 2936q^-90 + 3485q^-89 + 745q^-88 - 3094q^-87 - 4455q^-86 - 1697q^-85 + 3034q^-84 + 5272q^-83 + 2612q^-82 - 2713q^-81 - 5816q^-80 - 3525q^-79 + 2307q^-78 + 6146q^-77 + 4196q^-76 - 1810q^-75 - 6226q^-74 - 4746q^-73 + 1348q^-72 + 6180q^-71 + 5063q^-70 - 898q^-69 - 5985q^-68 - 5288q^-67 + 490q^-66 + 5713q^-65 + 5381q^-64 - 65q^-63 - 5339q^-62 - 5409q^-61 - 384q^-60 + 4853q^-59 + 5365q^-58 + 870q^-57 - 4234q^-56 - 5204q^-55 - 1410q^-54 + 3466q^-53 + 4953q^-52 + 1894q^-51 - 2586q^-50 - 4448q^-49 - 2368q^-48 + 1611q^-47 + 3864q^-46 + 2585q^-45 - 682q^-44 - 2986q^-43 - 2650q^-42 - 183q^-41 + 2157q^-40 + 2379q^-39 + 743q^-38 - 1191q^-37 - 1958q^-36 - 1084q^-35 + 491q^-34 + 1353q^-33 + 1076q^-32 + 93q^-31 - 803q^-30 - 904q^-29 - 313q^-28 + 303q^-27 + 594q^-26 + 409q^-25 - 41q^-24 - 325q^-23 - 277q^-22 - 111q^-21 + 100q^-20 + 196q^-19 + 100q^-18 - 17q^-17 - 57q^-16 - 73q^-15 - 32q^-14 + 25q^-13 + 31q^-12 + 11q^-11 + 9q^-10 - 9q^-9 - 9q^-8 - 3q^-7 + 2q^-6 + 2q^-4
6	q^-180 - 3q^-179 + q^-178 + 4q^-177 - 3q^-176 - 3q^-174 + 9q^-173 - 10q^-172 - 4q^-171 + 22q^-170 - 12q^-169 - 8q^-168 - 15q^-167 + 30q^-166 - 10q^-165 - q^-164 + 68q^-163 - 35q^-162 - 64q^-161 - 94q^-160 + 59q^-159 + 18q^-158 + 100q^-157 + 276q^-156 - 22q^-155 - 245q^-154 - 481q^-153 - 151q^-152 - 41q^-151 + 454q^-150 + 1100q^-149 + 553q^-148 - 310q^-147 - 1474q^-146 - 1398q^-145 - 1136q^-144 + 620q^-143 + 2969q^-142 + 2980q^-141 + 1251q^-140 - 2288q^-139 - 4210q^-138 - 5099q^-137 - 1590q^-136 + 4589q^-135 + 7780q^-134 + 6715q^-133 - 4q^-132 - 6688q^-131 - 12184q^-130 - 8654q^-129 + 2399q^-128 + 12298q^-127 + 15768q^-126 + 7799q^-125 - 4878q^-124 - 18904q^-123 - 19428q^-122 - 5598q^-121 + 12338q^-120 + 24136q^-119 + 19069q^-118 + 2559q^-117 - 20983q^-116 - 29031q^-115 - 16533q^-114 + 7113q^-113 + 27725q^-112 + 28704q^-111 + 12234q^-110 - 18087q^-109 - 33750q^-108 - 25487q^-107 - 7q^-106 + 26588q^-105 + 33650q^-104 + 19844q^-103 - 13285q^-102 - 33983q^-101 - 30221q^-100 - 5642q^-99 + 23419q^-98 + 34648q^-97 + 24016q^-96 - 9069q^-95 - 32050q^-94 - 31763q^-93 - 9201q^-92 + 19989q^-91 + 33694q^-90 + 26000q^-89 - 5473q^-88 - 29176q^-87 - 31830q^-86 - 12047q^-85 + 15978q^-84 + 31596q^-83 + 27269q^-82 - 1144q^-81 - 24815q^-80 - 30885q^-79 - 15352q^-78 + 10132q^-77 + 27584q^-76 + 27892q^-75 + 4709q^-74 - 17790q^-73 - 27853q^-72 - 18630q^-71 + 2105q^-70 + 20425q^-69 + 26268q^-68 + 10875q^-67 - 8119q^-66 - 21236q^-65 - 19536q^-64 - 6174q^-63 + 10341q^-62 + 20532q^-61 + 14200q^-60 + 1632q^-59 - 11376q^-58 - 15783q^-57 - 10932q^-56 + 383q^-55 + 11251q^-54 + 12173q^-53 + 7270q^-52 - 1777q^-51 - 8228q^-50 - 9796q^-49 - 5114q^-48 + 2410q^-47 + 6177q^-46 + 6832q^-45 + 3236q^-44 - 1208q^-43 - 4841q^-42 - 4813q^-41 - 1890q^-40 + 790q^-39 + 3045q^-38 + 3033q^-37 + 1687q^-36 - 686q^-35 - 1913q^-34 - 1748q^-33 - 1065q^-32 + 217q^-31 + 991q^-30 + 1223q^-29 + 538q^-28 - 78q^-27 - 442q^-26 - 607q^-25 - 367q^-24 - 47q^-23 + 273q^-22 + 246q^-21 + 172q^-20 + 58q^-19 - 75q^-18 - 118q^-17 - 94q^-16 - 9q^-15 + 15q^-14 + 32q^-13 + 32q^-12 + 17q^-11 - 4q^-10 - 12q^-9 - 6q^-8 - 4q^-7 + 2q^-4 + q^-3

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	PD[Knot[10, 149]]
Out[2]=	PD[X[1, 4, 2, 5], X[3, 8, 4, 9], X[12, 6, 13, 5], 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[6, 12, 7, 11], X[7, 2, 8, 3]]
In[3]:=	GaussCode[Knot[10, 149]]
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, 149]]
Out[4]=	DTCode[4, 8, -12, 2, 16, -6, 18, 20, 10, 14]
In[5]:=	br = BR[Knot[10, 149]]
Out[5]=	BR[3, {-1, -1, -1, -1, -2, 1, -2, 1, -2, -2}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{3, 10}
In[7]:=	BraidIndex[Knot[10, 149]]
Out[7]=	3
In[8]:=	Show[DrawMorseLink[Knot[10, 149]]]

Out[8]=	-Graphics-
In[9]:=	#[Knot[10, 149]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Chiral, 2, 3, 3, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 149]][t]
Out[10]=	-3 5 9 2 3 11 - t + -- - - - 9 t + 5 t - t 2 t t
In[11]:=	Conway[Knot[10, 149]][z]
Out[11]=	2 4 6 1 + 2 z - z - z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[9, 20], Knot[10, 149], Knot[11, NonAlternating, 26]}
In[13]:=	{KnotDet[Knot[10, 149]], KnotSignature[Knot[10, 149]]}
Out[13]=	{41, -4}
In[14]:=	Jones[Knot[10, 149]][q]
Out[14]=	-10 3 5 7 7 7 6 3 2 q - -- + -- - -- + -- - -- + -- - -- + -- 9 8 7 6 5 4 3 2 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, 149]}
In[16]:=	A2Invariant[Knot[10, 149]][q]
Out[16]=	-30 -28 -26 2 3 -16 -12 3 2 q - q + q - --- - --- + q + q + --- + -- 22 18 10 6 q q q q
In[17]:=	HOMFLYPT[Knot[10, 149]][a, z]
Out[17]=	4 6 8 4 2 6 2 8 2 4 4 6 4 8 4 4 a - 4 a + a + 6 a z - 6 a z + 2 a z + 2 a z - 4 a z + a z - 6 6 > a z
In[18]:=	Kauffman[Knot[10, 149]][a, z]
Out[18]=	4 6 8 5 7 9 11 4 2 6 2 4 a + 4 a + a - 3 a z - 3 a z + a z + a z - 7 a z - 9 a z + 10 2 12 2 5 3 7 3 9 3 11 3 4 4 > a z - a z + 2 a z + 5 a z - a z - 4 a z + 3 a z + 6 4 8 4 10 4 12 4 5 5 7 5 9 5 > 5 a z - 4 a z - 5 a z + a z - a z - 6 a z - 2 a z + 11 5 6 6 8 6 10 6 5 7 7 7 9 7 6 8 > 3 a z - a z + 3 a z + 4 a z + a z + 4 a z + 3 a z + a z + 8 8 > a z
In[19]:=	{Vassiliev[2][Knot[10, 149]], Vassiliev[3][Knot[10, 149]]}
Out[19]=	{2, -2}
In[20]:=	Kh[Knot[10, 149]][q, t]
Out[20]=	-5 2 1 2 1 3 2 4 3 q + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 3 21 8 19 7 17 7 17 6 15 6 15 5 13 5 q q t q t q t q t q t q t q t 3 4 4 3 2 4 1 2 > ------ + ------ + ------ + ----- + ----- + ----- + ---- + ---- 13 4 11 4 11 3 9 3 9 2 7 2 7 5 q t q t q t q t q t q t q t q t
In[21]:=	ColouredJones[Knot[10, 149], 2][q]
Out[21]=	-28 3 -26 8 14 27 29 10 50 37 22 61 34 q - --- + q + --- - --- + --- - --- - --- + --- - --- - --- + --- - --- - 27 25 24 22 21 20 19 18 17 16 15 q q q q q q q q q q q 29 55 21 28 36 6 18 14 2 6 2 -3 > --- + --- - --- - --- + --- - -- - -- + -- + -- - -- + -- + q 14 13 12 11 10 9 8 7 6 5 4 q q q q q q q q q q q

The Non Alternating Knot 10149

The Non Alternating Knot 10₁₄₉