The Non Alternating Knot 10₁₄₈

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Acknowledgement

KnotPlot

PD Presentation: X₄₂₅₁ X₈₄₉₃ 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 X₂₈₃₇

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 - 3t^-2 + 7t^-1 - 9 + 7t - 3t² + t³

Conway Polynomial: 1 + 4z² + 3z⁴ + z⁶

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

Determinant and Signature: {31, -2}

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

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

A2 (sl(3)) Invariant: - q^-24 - 2q^-20 - q^-18 + q^-16 + 3q^-12 + q^-10 + 2q^-8 + q^-6 - q^-4 + q^-2 - 1

HOMFLY-PT Polynomial: - a² - 2a²z² - a²z⁴ + 5a⁴ + 9a⁴z² + 5a⁴z⁴ + a⁴z⁶ - 3a⁶ - 3a⁶z² - a⁶z⁴

Kauffman Polynomial: - az + az³ + a² - 3a²z² + 3a²z⁴ - 3a³z + 6a³z³ - 2a³z⁵ + a³z⁷ + 5a⁴ - 11a⁴z² + 10a⁴z⁴ - 3a⁴z⁶ + a⁴z⁸ - 5a⁵z + 9a⁵z³ - 7a⁵z⁵ + 3a⁵z⁷ + 3a⁶ - 6a⁶z² + 2a⁶z⁴ - a⁶z⁶ + a⁶z⁸ - a⁷z + a⁷z³ - 4a⁷z⁵ + 2a⁷z⁷ + 2a⁸z² - 5a⁸z⁴ + 2a⁸z⁶ + 2a⁹z - 3a⁹z³ + a⁹z⁵

V₂ and V₃, the type 2 and 3 Vassiliev invariants: {4, -7}

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 10₁₄₈. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

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

j = 1 1

j = -1 2

j = -3 3 2

j = -5 3 1

j = -7 2 3

j = -9 3 3

j = -11 1 2

j = -13 1 3

j = -15 1

j = -17 1

n Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2 q^-23 - 2q^-22 + 6q^-20 - 7q^-19 - 5q^-18 + 17q^-17 - 9q^-16 - 16q^-15 + 26q^-14 - 5q^-13 - 26q^-12 + 29q^-11 + q^-10 - 30q^-9 + 26q^-8 + 5q^-7 - 24q^-6 + 15q^-5 + 6q^-4 - 12q^-3 + 4q^-2 + 3q^-1 - 2

3 - q^-45 + 2q^-44 - 2q^-42 - 3q^-41 + 6q^-40 + 6q^-39 - 8q^-38 - 15q^-37 + 10q^-36 + 25q^-35 - 2q^-34 - 41q^-33 - 6q^-32 + 47q^-31 + 26q^-30 - 53q^-29 - 45q^-28 + 51q^-27 + 64q^-26 - 44q^-25 - 82q^-24 + 36q^-23 + 93q^-22 - 23q^-21 - 107q^-20 + 19q^-19 + 105q^-18 - 4q^-17 - 111q^-16 + q^-15 + 97q^-14 + 16q^-13 - 90q^-12 - 18q^-11 + 67q^-10 + 28q^-9 - 48q^-8 - 26q^-7 + 25q^-6 + 25q^-5 - 12q^-4 - 14q^-3 + 8q^-1 + 3 - 3q - q² - q³ + q⁴

4 q^-74 - 2q^-73 + 2q^-71 - q^-70 + 4q^-69 - 8q^-68 - 2q^-67 + 8q^-66 + q^-65 + 16q^-64 - 24q^-63 - 20q^-62 + 9q^-61 + 12q^-60 + 62q^-59 - 27q^-58 - 57q^-57 - 36q^-56 - 11q^-55 + 141q^-54 + 34q^-53 - 50q^-52 - 113q^-51 - 123q^-50 + 167q^-49 + 134q^-48 + 61q^-47 - 135q^-46 - 285q^-45 + 89q^-44 + 189q^-43 + 224q^-42 - 68q^-41 - 406q^-40 - 43q^-39 + 173q^-38 + 363q^-37 + 31q^-36 - 463q^-35 - 156q^-34 + 130q^-33 + 445q^-32 + 111q^-31 - 473q^-30 - 232q^-29 + 85q^-28 + 476q^-27 + 172q^-26 - 437q^-25 - 277q^-24 + 22q^-23 + 445q^-22 + 227q^-21 - 329q^-20 - 283q^-19 - 67q^-18 + 333q^-17 + 249q^-16 - 161q^-15 - 214q^-14 - 137q^-13 + 159q^-12 + 197q^-11 - 12q^-10 - 92q^-9 - 125q^-8 + 19q^-7 + 90q^-6 + 36q^-5 - q^-4 - 53q^-3 - 23q^-2 + 14q^-1 + 16 + 14q - 6q² - 9q³ - 2q⁴ + 3q⁶ + q⁷ - q⁸

5 - q^-110 + 2q^-109 - 2q^-107 + q^-106 - 2q^-104 + 4q^-103 + 3q^-102 - 7q^-101 - 4q^-100 + q^-99 + 14q^-97 + 15q^-96 - 8q^-95 - 29q^-94 - 26q^-93 - 9q^-92 + 36q^-91 + 70q^-90 + 40q^-89 - 31q^-88 - 103q^-87 - 112q^-86 - 18q^-85 + 124q^-84 + 191q^-83 + 126q^-82 - 73q^-81 - 277q^-80 - 270q^-79 - 42q^-78 + 263q^-77 + 436q^-76 + 266q^-75 - 182q^-74 - 551q^-73 - 509q^-72 - 44q^-71 + 566q^-70 + 784q^-69 + 338q^-68 - 476q^-67 - 979q^-66 - 685q^-65 + 264q^-64 + 1103q^-63 + 1031q^-62 + q^-61 - 1127q^-60 - 1325q^-59 - 302q^-58 + 1081q^-57 + 1564q^-56 + 586q^-55 - 1008q^-54 - 1718q^-53 - 833q^-52 + 899q^-51 + 1852q^-50 + 1023q^-49 - 825q^-48 - 1904q^-47 - 1179q^-46 + 721q^-45 + 1982q^-44 + 1293q^-43 - 673q^-42 - 1971q^-41 - 1396q^-40 + 548q^-39 + 2002q^-38 + 1485q^-37 - 467q^-36 - 1915q^-35 - 1562q^-34 + 267q^-33 + 1836q^-32 + 1622q^-31 - 100q^-30 - 1615q^-29 - 1622q^-28 - 164q^-27 + 1363q^-26 + 1555q^-25 + 373q^-24 - 1000q^-23 - 1394q^-22 - 568q^-21 + 637q^-20 + 1145q^-19 + 645q^-18 - 265q^-17 - 840q^-16 - 649q^-15 + 15q^-14 + 518q^-13 + 526q^-12 + 162q^-11 - 245q^-10 - 381q^-9 - 199q^-8 + 65q^-7 + 205q^-6 + 174q^-5 + 32q^-4 - 84q^-3 - 114q^-2 - 51q^-1 + 20 + 47q + 39q² + 11q³ - 18q⁴ - 21q⁵ - 4q⁶ + 5q⁸ + 5q⁹ - 2q¹¹

6 q^-153 - 2q^-152 + 2q^-150 - q^-149 - 2q^-147 + 6q^-146 - 5q^-145 - 4q^-144 + 9q^-143 - 11q^-140 + 10q^-139 - 13q^-138 - 13q^-137 + 25q^-136 + 19q^-135 + 19q^-134 - 23q^-133 + 9q^-132 - 60q^-131 - 70q^-130 + 19q^-129 + 59q^-128 + 113q^-127 + 45q^-126 + 89q^-125 - 113q^-124 - 239q^-123 - 172q^-122 - 59q^-121 + 164q^-120 + 238q^-119 + 479q^-118 + 150q^-117 - 233q^-116 - 503q^-115 - 591q^-114 - 330q^-113 + 32q^-112 + 930q^-111 + 947q^-110 + 596q^-109 - 163q^-108 - 981q^-107 - 1432q^-106 - 1286q^-105 + 334q^-104 + 1371q^-103 + 2042q^-102 + 1521q^-101 + 69q^-100 - 1883q^-99 - 3153q^-98 - 1815q^-97 + 60q^-96 + 2560q^-95 + 3644q^-94 + 2775q^-93 - 403q^-92 - 3881q^-91 - 4317q^-90 - 2865q^-89 + 1092q^-88 + 4541q^-87 + 5710q^-86 + 2560q^-85 - 2706q^-84 - 5631q^-83 - 5888q^-82 - 1639q^-81 + 3776q^-80 + 7506q^-79 + 5480q^-78 - 520q^-77 - 5545q^-76 - 7852q^-75 - 4206q^-74 + 2280q^-73 + 8086q^-72 + 7406q^-71 + 1435q^-70 - 4884q^-69 - 8752q^-68 - 5874q^-67 + 1020q^-66 + 8112q^-65 + 8400q^-64 + 2667q^-63 - 4331q^-62 - 9112q^-61 - 6790q^-60 + 232q^-59 + 8032q^-58 + 8936q^-57 + 3449q^-56 - 3926q^-55 - 9266q^-54 - 7437q^-53 - 457q^-52 + 7768q^-51 + 9296q^-50 + 4294q^-49 - 3214q^-48 - 9068q^-47 - 8066q^-46 - 1573q^-45 + 6793q^-44 + 9250q^-43 + 5398q^-42 - 1660q^-41 - 7915q^-40 - 8316q^-39 - 3208q^-38 + 4586q^-37 + 8079q^-36 + 6208q^-35 + 674q^-34 - 5327q^-33 - 7328q^-32 - 4559q^-31 + 1472q^-30 + 5367q^-29 + 5678q^-28 + 2652q^-27 - 1916q^-26 - 4761q^-25 - 4410q^-24 - 1060q^-23 + 2003q^-22 + 3550q^-21 + 2936q^-20 + 607q^-19 - 1725q^-18 - 2671q^-17 - 1699q^-16 - 241q^-15 + 1119q^-14 + 1681q^-13 + 1168q^-12 + 87q^-11 - 781q^-10 - 909q^-9 - 660q^-8 - 128q^-7 + 391q^-6 + 563q^-5 + 356q^-4 + 46q^-3 - 148q^-2 - 251q^-1 - 212 - 54q + 81q² + 107q³ + 76q⁴ + 37q⁵ - 11q⁶ - 49q⁷ - 32q⁸ - 8q⁹ + 3q¹⁰ + 6q¹¹ + 8q¹² + 9q¹³ - 3q¹⁴ - 2q¹⁵ - q¹⁸ - q¹⁹ + q²⁰

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, 148]]

Out[2]=
PD[X[4, 2, 5, 1], X[8, 4, 9, 3], 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[2, 8, 3, 7]]

In[3]:=
GaussCode[Knot[10, 148]]

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, 148]]

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

In[5]:=
br = BR[Knot[10, 148]]

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

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

Out[6]=
{3, 10}

In[7]:=
BraidIndex[Knot[10, 148]]

Out[7]=
3

In[8]:=
Show[DrawMorseLink[Knot[10, 148]]]

Out[8]=
-Graphics-

In[9]:=
#[Knot[10, 148]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}

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

In[10]:=
alex = Alexander[Knot[10, 148]][t]

Out[10]=
-3 3 7 2 3 -9 + t - -- + - + 7 t - 3 t + t 2 t t

In[11]:=
Conway[Knot[10, 148]][z]

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

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

Out[12]=
{Knot[10, 148]}

In[13]:=
{KnotDet[Knot[10, 148]], KnotSignature[Knot[10, 148]]}

Out[13]=
{31, -2}

In[14]:=
Jones[Knot[10, 148]][q]

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

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

Out[15]=
{Knot[10, 148]}

In[16]:=
A2Invariant[Knot[10, 148]][q]

Out[16]=
-24 2 -18 -16 3 -10 2 -6 -4 -2 -1 - q - --- - q + q + --- + q + -- + q - q + q 20 12 8 q q q

In[17]:=
HOMFLYPT[Knot[10, 148]][a, z]

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

In[18]:=
Kauffman[Knot[10, 148]][a, z]

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

In[19]:=
{Vassiliev[2][Knot[10, 148]], Vassiliev[3][Knot[10, 148]]}

Out[19]=
{4, -7}

In[20]:=
Kh[Knot[10, 148]][q, t]

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

In[21]:=
ColouredJones[Knot[10, 148], 2][q]

Out[21]=
-23 2 6 7 5 17 9 16 26 5 26 29 -2 + q - --- + --- - --- - --- + --- - --- - --- + --- - --- - --- + --- + 22 20 19 18 17 16 15 14 13 12 11 q q q q q q q q q q q -10 30 26 5 24 15 6 12 4 3 > q - -- + -- + -- - -- + -- + -- - -- + -- + - 9 8 7 6 5 4 3 2 q q q q q q q q q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 10₁₄₈

10₁₄₇
10₁₄₉

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q^-23 - 2q^-22 + 6q^-20 - 7q^-19 - 5q^-18 + 17q^-17 - 9q^-16 - 16q^-15 + 26q^-14 - 5q^-13 - 26q^-12 + 29q^-11 + q^-10 - 30q^-9 + 26q^-8 + 5q^-7 - 24q^-6 + 15q^-5 + 6q^-4 - 12q^-3 + 4q^-2 + 3q^-1 - 2
3	- q^-45 + 2q^-44 - 2q^-42 - 3q^-41 + 6q^-40 + 6q^-39 - 8q^-38 - 15q^-37 + 10q^-36 + 25q^-35 - 2q^-34 - 41q^-33 - 6q^-32 + 47q^-31 + 26q^-30 - 53q^-29 - 45q^-28 + 51q^-27 + 64q^-26 - 44q^-25 - 82q^-24 + 36q^-23 + 93q^-22 - 23q^-21 - 107q^-20 + 19q^-19 + 105q^-18 - 4q^-17 - 111q^-16 + q^-15 + 97q^-14 + 16q^-13 - 90q^-12 - 18q^-11 + 67q^-10 + 28q^-9 - 48q^-8 - 26q^-7 + 25q^-6 + 25q^-5 - 12q^-4 - 14q^-3 + 8q^-1 + 3 - 3q - q² - q³ + q⁴
4	q^-74 - 2q^-73 + 2q^-71 - q^-70 + 4q^-69 - 8q^-68 - 2q^-67 + 8q^-66 + q^-65 + 16q^-64 - 24q^-63 - 20q^-62 + 9q^-61 + 12q^-60 + 62q^-59 - 27q^-58 - 57q^-57 - 36q^-56 - 11q^-55 + 141q^-54 + 34q^-53 - 50q^-52 - 113q^-51 - 123q^-50 + 167q^-49 + 134q^-48 + 61q^-47 - 135q^-46 - 285q^-45 + 89q^-44 + 189q^-43 + 224q^-42 - 68q^-41 - 406q^-40 - 43q^-39 + 173q^-38 + 363q^-37 + 31q^-36 - 463q^-35 - 156q^-34 + 130q^-33 + 445q^-32 + 111q^-31 - 473q^-30 - 232q^-29 + 85q^-28 + 476q^-27 + 172q^-26 - 437q^-25 - 277q^-24 + 22q^-23 + 445q^-22 + 227q^-21 - 329q^-20 - 283q^-19 - 67q^-18 + 333q^-17 + 249q^-16 - 161q^-15 - 214q^-14 - 137q^-13 + 159q^-12 + 197q^-11 - 12q^-10 - 92q^-9 - 125q^-8 + 19q^-7 + 90q^-6 + 36q^-5 - q^-4 - 53q^-3 - 23q^-2 + 14q^-1 + 16 + 14q - 6q² - 9q³ - 2q⁴ + 3q⁶ + q⁷ - q⁸
5	- q^-110 + 2q^-109 - 2q^-107 + q^-106 - 2q^-104 + 4q^-103 + 3q^-102 - 7q^-101 - 4q^-100 + q^-99 + 14q^-97 + 15q^-96 - 8q^-95 - 29q^-94 - 26q^-93 - 9q^-92 + 36q^-91 + 70q^-90 + 40q^-89 - 31q^-88 - 103q^-87 - 112q^-86 - 18q^-85 + 124q^-84 + 191q^-83 + 126q^-82 - 73q^-81 - 277q^-80 - 270q^-79 - 42q^-78 + 263q^-77 + 436q^-76 + 266q^-75 - 182q^-74 - 551q^-73 - 509q^-72 - 44q^-71 + 566q^-70 + 784q^-69 + 338q^-68 - 476q^-67 - 979q^-66 - 685q^-65 + 264q^-64 + 1103q^-63 + 1031q^-62 + q^-61 - 1127q^-60 - 1325q^-59 - 302q^-58 + 1081q^-57 + 1564q^-56 + 586q^-55 - 1008q^-54 - 1718q^-53 - 833q^-52 + 899q^-51 + 1852q^-50 + 1023q^-49 - 825q^-48 - 1904q^-47 - 1179q^-46 + 721q^-45 + 1982q^-44 + 1293q^-43 - 673q^-42 - 1971q^-41 - 1396q^-40 + 548q^-39 + 2002q^-38 + 1485q^-37 - 467q^-36 - 1915q^-35 - 1562q^-34 + 267q^-33 + 1836q^-32 + 1622q^-31 - 100q^-30 - 1615q^-29 - 1622q^-28 - 164q^-27 + 1363q^-26 + 1555q^-25 + 373q^-24 - 1000q^-23 - 1394q^-22 - 568q^-21 + 637q^-20 + 1145q^-19 + 645q^-18 - 265q^-17 - 840q^-16 - 649q^-15 + 15q^-14 + 518q^-13 + 526q^-12 + 162q^-11 - 245q^-10 - 381q^-9 - 199q^-8 + 65q^-7 + 205q^-6 + 174q^-5 + 32q^-4 - 84q^-3 - 114q^-2 - 51q^-1 + 20 + 47q + 39q² + 11q³ - 18q⁴ - 21q⁵ - 4q⁶ + 5q⁸ + 5q⁹ - 2q¹¹
6	q^-153 - 2q^-152 + 2q^-150 - q^-149 - 2q^-147 + 6q^-146 - 5q^-145 - 4q^-144 + 9q^-143 - 11q^-140 + 10q^-139 - 13q^-138 - 13q^-137 + 25q^-136 + 19q^-135 + 19q^-134 - 23q^-133 + 9q^-132 - 60q^-131 - 70q^-130 + 19q^-129 + 59q^-128 + 113q^-127 + 45q^-126 + 89q^-125 - 113q^-124 - 239q^-123 - 172q^-122 - 59q^-121 + 164q^-120 + 238q^-119 + 479q^-118 + 150q^-117 - 233q^-116 - 503q^-115 - 591q^-114 - 330q^-113 + 32q^-112 + 930q^-111 + 947q^-110 + 596q^-109 - 163q^-108 - 981q^-107 - 1432q^-106 - 1286q^-105 + 334q^-104 + 1371q^-103 + 2042q^-102 + 1521q^-101 + 69q^-100 - 1883q^-99 - 3153q^-98 - 1815q^-97 + 60q^-96 + 2560q^-95 + 3644q^-94 + 2775q^-93 - 403q^-92 - 3881q^-91 - 4317q^-90 - 2865q^-89 + 1092q^-88 + 4541q^-87 + 5710q^-86 + 2560q^-85 - 2706q^-84 - 5631q^-83 - 5888q^-82 - 1639q^-81 + 3776q^-80 + 7506q^-79 + 5480q^-78 - 520q^-77 - 5545q^-76 - 7852q^-75 - 4206q^-74 + 2280q^-73 + 8086q^-72 + 7406q^-71 + 1435q^-70 - 4884q^-69 - 8752q^-68 - 5874q^-67 + 1020q^-66 + 8112q^-65 + 8400q^-64 + 2667q^-63 - 4331q^-62 - 9112q^-61 - 6790q^-60 + 232q^-59 + 8032q^-58 + 8936q^-57 + 3449q^-56 - 3926q^-55 - 9266q^-54 - 7437q^-53 - 457q^-52 + 7768q^-51 + 9296q^-50 + 4294q^-49 - 3214q^-48 - 9068q^-47 - 8066q^-46 - 1573q^-45 + 6793q^-44 + 9250q^-43 + 5398q^-42 - 1660q^-41 - 7915q^-40 - 8316q^-39 - 3208q^-38 + 4586q^-37 + 8079q^-36 + 6208q^-35 + 674q^-34 - 5327q^-33 - 7328q^-32 - 4559q^-31 + 1472q^-30 + 5367q^-29 + 5678q^-28 + 2652q^-27 - 1916q^-26 - 4761q^-25 - 4410q^-24 - 1060q^-23 + 2003q^-22 + 3550q^-21 + 2936q^-20 + 607q^-19 - 1725q^-18 - 2671q^-17 - 1699q^-16 - 241q^-15 + 1119q^-14 + 1681q^-13 + 1168q^-12 + 87q^-11 - 781q^-10 - 909q^-9 - 660q^-8 - 128q^-7 + 391q^-6 + 563q^-5 + 356q^-4 + 46q^-3 - 148q^-2 - 251q^-1 - 212 - 54q + 81q² + 107q³ + 76q⁴ + 37q⁵ - 11q⁶ - 49q⁷ - 32q⁸ - 8q⁹ + 3q¹⁰ + 6q¹¹ + 8q¹² + 9q¹³ - 3q¹⁴ - 2q¹⁵ - q¹⁸ - q¹⁹ + q²⁰

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	PD[Knot[10, 148]]
Out[2]=	PD[X[4, 2, 5, 1], X[8, 4, 9, 3], 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[2, 8, 3, 7]]
In[3]:=	GaussCode[Knot[10, 148]]
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, 148]]
Out[4]=	DTCode[4, 8, -12, 2, -16, -6, -18, -20, -10, -14]
In[5]:=	br = BR[Knot[10, 148]]
Out[5]=	BR[3, {-1, -1, -1, -1, -2, 1, 1, -2, 1, -2}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{3, 10}
In[7]:=	BraidIndex[Knot[10, 148]]
Out[7]=	3
In[8]:=	Show[DrawMorseLink[Knot[10, 148]]]

Out[8]=	-Graphics-
In[9]:=	#[Knot[10, 148]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Chiral, 2, 3, 3, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 148]][t]
Out[10]=	-3 3 7 2 3 -9 + t - -- + - + 7 t - 3 t + t 2 t t
In[11]:=	Conway[Knot[10, 148]][z]
Out[11]=	2 4 6 1 + 4 z + 3 z + z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 148]}
In[13]:=	{KnotDet[Knot[10, 148]], KnotSignature[Knot[10, 148]]}
Out[13]=	{31, -2}
In[14]:=	Jones[Knot[10, 148]][q]
Out[14]=	-8 2 4 5 5 6 4 3 -1 - q + -- - -- + -- - -- + -- - -- + - 7 6 5 4 3 2 q q q q q q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] \|\| (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[10, 148]}
In[16]:=	A2Invariant[Knot[10, 148]][q]
Out[16]=	-24 2 -18 -16 3 -10 2 -6 -4 -2 -1 - q - --- - q + q + --- + q + -- + q - q + q 20 12 8 q q q
In[17]:=	HOMFLYPT[Knot[10, 148]][a, z]
Out[17]=	2 4 6 2 2 4 2 6 2 2 4 4 4 6 4 -a + 5 a - 3 a - 2 a z + 9 a z - 3 a z - a z + 5 a z - a z + 4 6 > a z
In[18]:=	Kauffman[Knot[10, 148]][a, z]
Out[18]=	2 4 6 3 5 7 9 2 2 4 2 a + 5 a + 3 a - a z - 3 a z - 5 a z - a z + 2 a z - 3 a z - 11 a z - 6 2 8 2 3 3 3 5 3 7 3 9 3 2 4 > 6 a z + 2 a z + a z + 6 a z + 9 a z + a z - 3 a z + 3 a z + 4 4 6 4 8 4 3 5 5 5 7 5 9 5 > 10 a z + 2 a z - 5 a z - 2 a z - 7 a z - 4 a z + a z - 4 6 6 6 8 6 3 7 5 7 7 7 4 8 6 8 > 3 a z - a z + 2 a z + a z + 3 a z + 2 a z + a z + a z
In[19]:=	{Vassiliev[2][Knot[10, 148]], Vassiliev[3][Knot[10, 148]]}
Out[19]=	{4, -7}
In[20]:=	Kh[Knot[10, 148]][q, t]
Out[20]=	2 2 1 1 1 3 1 2 3 3 -- + - + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 3 q 17 7 15 6 13 6 13 5 11 5 11 4 9 4 9 3 q q t q t q t q t q t q t q t q t 2 3 3 1 3 > ----- + ----- + ----- + ---- + ---- + q t 7 3 7 2 5 2 5 3 q t q t q t q t q t
In[21]:=	ColouredJones[Knot[10, 148], 2][q]
Out[21]=	-23 2 6 7 5 17 9 16 26 5 26 29 -2 + q - --- + --- - --- - --- + --- - --- - --- + --- - --- - --- + --- + 22 20 19 18 17 16 15 14 13 12 11 q q q q q q q q q q q -10 30 26 5 24 15 6 12 4 3 > q - -- + -- + -- - -- + -- + -- - -- + -- + - 9 8 7 6 5 4 3 2 q q q q q q q q q

The Non Alternating Knot 10148

The Non Alternating Knot 10₁₄₈