The Alternating Knot 7₆

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Acknowledgement

KnotPlot

PD Presentation: X₁₄₂₅ X₃₈₄₉ X_5,12,6,13 X_9,1,10,14 X_13,11,14,10 X_11,6,12,7 X₇₂₈₃

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

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

Minimum Braid Representative:

Length is 7, 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 1 2 2 / 4 1

Alexander Polynomial: - t^-2 + 5t^-1 - 7 + 5t - t²

Conway Polynomial: 1 + z² - z⁴

Other knots with the same Alexander/Conway Polynomial: {10₁₃₃, ...}

Determinant and Signature: {19, -2}

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

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

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

HOMFLY-PT Polynomial: 1 + z² - a² - 2a²z² - a²z⁴ + 2a⁴ + 2a⁴z² - a⁶

Kauffman Polynomial: 1 - 2z² + z⁴ + az - 4az³ + 2az⁵ + a² - 4a²z² + a²z⁴ + a²z⁶ + 2a³z - 6a³z³ + 4a³z⁵ + 2a⁴ - 4a⁴z² + 2a⁴z⁴ + a⁴z⁶ - a⁵z³ + 2a⁵z⁵ + a⁶ - 2a⁶z² + 2a⁶z⁴ - a⁷z + a⁷z³

V₂ and V₃, the type 2 and 3 Vassiliev invariants: {1, -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 7₆. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

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

j = 3 1

j = 1 1

j = -1 2 1

j = -3 2 2

j = -5 2 1

j = -7 1 2

j = -9 1 2

j = -11 1

j = -13 1

n Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2 q^-17 - 2q^-16 + 5q^-14 - 7q^-13 - q^-12 + 12q^-11 - 12q^-10 - 3q^-9 + 17q^-8 - 13q^-7 - 4q^-6 + 16q^-5 - 10q^-4 - 5q^-3 + 12q^-2 - 5q^-1 - 4 + 6q - q² - 2q³ + q⁴

3 - q^-33 + 2q^-32 - 2q^-30 - 2q^-29 + 6q^-28 + 3q^-27 - 10q^-26 - 6q^-25 + 15q^-24 + 10q^-23 - 20q^-22 - 16q^-21 + 25q^-20 + 22q^-19 - 29q^-18 - 25q^-17 + 28q^-16 + 31q^-15 - 31q^-14 - 30q^-13 + 26q^-12 + 33q^-11 - 26q^-10 - 29q^-9 + 19q^-8 + 30q^-7 - 15q^-6 - 26q^-5 + 9q^-4 + 24q^-3 - 5q^-2 - 18q^-1 + 14q + 2q² - 9q³ - 3q⁴ + 5q⁵ + 2q⁶ - q⁷ - 2q⁸ + q⁹

4 q^-54 - 2q^-53 + 2q^-51 - q^-50 + 3q^-49 - 8q^-48 + q^-47 + 9q^-46 - q^-45 + 6q^-44 - 25q^-43 - q^-42 + 25q^-41 + 7q^-40 + 12q^-39 - 56q^-38 - 13q^-37 + 44q^-36 + 28q^-35 + 29q^-34 - 93q^-33 - 36q^-32 + 57q^-31 + 51q^-30 + 52q^-29 - 119q^-28 - 57q^-27 + 57q^-26 + 64q^-25 + 72q^-24 - 127q^-23 - 69q^-22 + 50q^-21 + 66q^-20 + 80q^-19 - 118q^-18 - 69q^-17 + 36q^-16 + 58q^-15 + 83q^-14 - 96q^-13 - 62q^-12 + 16q^-11 + 43q^-10 + 81q^-9 - 64q^-8 - 50q^-7 - 6q^-6 + 23q^-5 + 71q^-4 - 31q^-3 - 32q^-2 - 18q^-1 + 3 + 49q - 6q² - 12q³ - 17q⁴ - 8q⁵ + 24q⁶ + 2q⁷ - 7q⁹ - 7q¹⁰ + 6q¹¹ + q¹² + 2q¹³ - q¹⁴ - 2q¹⁵ + q¹⁶

5 - q^-80 + 2q^-79 - 2q^-77 + q^-76 - q^-74 + 4q^-73 - 8q^-71 - q^-70 + 4q^-69 + 6q^-68 + 10q^-67 - 4q^-66 - 21q^-65 - 16q^-64 + 8q^-63 + 32q^-62 + 32q^-61 - 4q^-60 - 53q^-59 - 56q^-58 - 3q^-57 + 73q^-56 + 90q^-55 + 17q^-54 - 91q^-53 - 129q^-52 - 41q^-51 + 105q^-50 + 170q^-49 + 71q^-48 - 115q^-47 - 206q^-46 - 103q^-45 + 118q^-44 + 234q^-43 + 132q^-42 - 107q^-41 - 262q^-40 - 157q^-39 + 108q^-38 + 265q^-37 + 174q^-36 - 85q^-35 - 280q^-34 - 190q^-33 + 89q^-32 + 266q^-31 + 192q^-30 - 62q^-29 - 269q^-28 - 198q^-27 + 64q^-26 + 244q^-25 + 192q^-24 - 35q^-23 - 233q^-22 - 193q^-21 + 29q^-20 + 201q^-19 + 183q^-18 + 3q^-17 - 179q^-16 - 176q^-15 - 17q^-14 + 139q^-13 + 163q^-12 + 44q^-11 - 107q^-10 - 147q^-9 - 56q^-8 + 68q^-7 + 121q^-6 + 74q^-5 - 37q^-4 - 97q^-3 - 71q^-2 + 7q^-1 + 66 + 68q + 13q² - 41q³ - 53q⁴ - 25q⁵ + 19q⁶ + 39q⁷ + 23q⁸ - 2q⁹ - 22q¹⁰ - 22q¹¹ - 4q¹² + 13q¹³ + 11q¹⁴ + 5q¹⁵ - 9q¹⁷ - 5q¹⁸ + 2q¹⁹ + 2q²⁰ + q²¹ + 2q²² - q²³ - 2q²⁴ + q²⁵

6 q^-111 - 2q^-110 + 2q^-108 - q^-107 - 2q^-105 + 5q^-104 - 5q^-103 - q^-102 + 10q^-101 - 3q^-100 - 4q^-99 - 10q^-98 + 9q^-97 - 9q^-96 + 4q^-95 + 32q^-94 - q^-93 - 16q^-92 - 38q^-91 + 4q^-90 - 23q^-89 + 23q^-88 + 92q^-87 + 25q^-86 - 26q^-85 - 104q^-84 - 45q^-83 - 75q^-82 + 47q^-81 + 210q^-80 + 119q^-79 + 3q^-78 - 192q^-77 - 161q^-76 - 214q^-75 + 35q^-74 + 359q^-73 + 294q^-72 + 116q^-71 - 251q^-70 - 304q^-69 - 434q^-68 - 53q^-67 + 476q^-66 + 493q^-65 + 295q^-64 - 240q^-63 - 409q^-62 - 660q^-61 - 190q^-60 + 517q^-59 + 641q^-58 + 464q^-57 - 177q^-56 - 439q^-55 - 820q^-54 - 314q^-53 + 496q^-52 + 707q^-51 + 572q^-50 - 109q^-49 - 416q^-48 - 889q^-47 - 390q^-46 + 448q^-45 + 709q^-44 + 614q^-43 - 56q^-42 - 371q^-41 - 889q^-40 - 420q^-39 + 387q^-38 + 669q^-37 + 609q^-36 - 9q^-35 - 308q^-34 - 840q^-33 - 428q^-32 + 301q^-31 + 592q^-30 + 574q^-29 + 55q^-28 - 211q^-27 - 750q^-26 - 429q^-25 + 176q^-24 + 470q^-23 + 516q^-22 + 136q^-21 - 77q^-20 - 614q^-19 - 416q^-18 + 28q^-17 + 304q^-16 + 419q^-15 + 207q^-14 + 71q^-13 - 429q^-12 - 361q^-11 - 100q^-10 + 124q^-9 + 275q^-8 + 220q^-7 + 183q^-6 - 225q^-5 - 247q^-4 - 155q^-3 - 19q^-2 + 113q^-1 + 158 + 209q - 60q² - 109q³ - 122q⁴ - 75q⁵ - 7q⁶ + 63q⁷ + 150q⁸ + 16q⁹ - 9q¹⁰ - 52q¹¹ - 53q¹² - 46q¹³ - 3q¹⁴ + 69q¹⁵ + 18q¹⁶ + 20q¹⁷ - 5q¹⁸ - 14q¹⁹ - 29q²⁰ - 16q²¹ + 18q²² + 3q²³ + 11q²⁴ + 4q²⁵ + 3q²⁶ - 9q²⁷ - 7q²⁸ + 4q²⁹ - 2q³⁰ + 2q³¹ + q³² + 2q³³ - q³⁴ - 2q³⁵ + q³⁶

7 - q^-147 + 2q^-146 - 2q^-144 + q^-143 + 2q^-141 - 2q^-140 - 4q^-139 + 6q^-138 - q^-137 - 6q^-136 + 3q^-135 + 2q^-134 + 10q^-133 - q^-132 - 14q^-131 + 7q^-130 - 9q^-129 - 15q^-128 + 8q^-127 + 10q^-126 + 37q^-125 + 14q^-124 - 26q^-123 - 11q^-122 - 48q^-121 - 49q^-120 + 12q^-119 + 34q^-118 + 110q^-117 + 89q^-116 - 13q^-115 - 48q^-114 - 167q^-113 - 170q^-112 - 41q^-111 + 61q^-110 + 265q^-109 + 296q^-108 + 120q^-107 - 50q^-106 - 366q^-105 - 464q^-104 - 259q^-103 - 9q^-102 + 465q^-101 + 671q^-100 + 465q^-99 + 117q^-98 - 545q^-97 - 892q^-96 - 715q^-95 - 285q^-94 + 571q^-93 + 1110q^-92 + 994q^-91 + 514q^-90 - 551q^-89 - 1301q^-88 - 1277q^-87 - 768q^-86 + 486q^-85 + 1432q^-84 + 1536q^-83 + 1031q^-82 - 374q^-81 - 1517q^-80 - 1748q^-79 - 1278q^-78 + 239q^-77 + 1561q^-76 + 1907q^-75 + 1477q^-74 - 114q^-73 - 1541q^-72 - 1998q^-71 - 1657q^-70 - 21q^-69 + 1529q^-68 + 2069q^-67 + 1748q^-66 + 103q^-65 - 1459q^-64 - 2062q^-63 - 1843q^-62 - 210q^-61 + 1433q^-60 + 2076q^-59 + 1855q^-58 + 242q^-57 - 1348q^-56 - 2017q^-55 - 1889q^-54 - 320q^-53 + 1310q^-52 + 1997q^-51 + 1857q^-50 + 335q^-49 - 1215q^-48 - 1903q^-47 - 1852q^-46 - 411q^-45 + 1142q^-44 + 1843q^-43 + 1802q^-42 + 442q^-41 - 1014q^-40 - 1707q^-39 - 1773q^-38 - 533q^-37 + 882q^-36 + 1594q^-35 + 1702q^-34 + 593q^-33 - 698q^-32 - 1409q^-31 - 1640q^-30 - 695q^-29 + 515q^-28 + 1227q^-27 + 1525q^-26 + 764q^-25 - 293q^-24 - 986q^-23 - 1410q^-22 - 830q^-21 + 88q^-20 + 752q^-19 + 1231q^-18 + 846q^-17 + 115q^-16 - 485q^-15 - 1033q^-14 - 838q^-13 - 270q^-12 + 251q^-11 + 804q^-10 + 753q^-9 + 374q^-8 - 19q^-7 - 568q^-6 - 643q^-5 - 423q^-4 - 133q^-3 + 344q^-2 + 483q^-1 + 402 + 244q - 151q² - 328q³ - 334q⁴ - 281q⁵ + 14q⁶ + 177q⁷ + 243q⁸ + 264q⁹ + 61q¹⁰ - 61q¹¹ - 137q¹² - 210q¹³ - 103q¹⁴ - 10q¹⁵ + 66q¹⁶ + 146q¹⁷ + 80q¹⁸ + 41q¹⁹ + 3q²⁰ - 82q²¹ - 67q²² - 51q²³ - 15q²⁴ + 44q²⁵ + 29q²⁶ + 29q²⁷ + 30q²⁸ - 7q²⁹ - 17q³⁰ - 25q³¹ - 20q³² + 9q³³ + q³⁴ + 4q³⁵ + 13q³⁶ + 4q³⁷ + 2q³⁸ - 6q³⁹ - 7q⁴⁰ + 2q⁴¹ - 2q⁴³ + 2q⁴⁴ + q⁴⁵ + 2q⁴⁶ - q⁴⁷ - 2q⁴⁸ + q⁴⁹

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[7, 6]]

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

In[3]:=
GaussCode[Knot[7, 6]]

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

In[4]:=
DTCode[Knot[7, 6]]

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

In[5]:=
br = BR[Knot[7, 6]]

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

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

Out[6]=
{4, 7}

In[7]:=
BraidIndex[Knot[7, 6]]

Out[7]=
4

In[8]:=
Show[DrawMorseLink[Knot[7, 6]]]

Out[8]=
-Graphics-

In[9]:=
#[Knot[7, 6]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}

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

In[10]:=
alex = Alexander[Knot[7, 6]][t]

Out[10]=
-2 5 2 -7 - t + - + 5 t - t t

In[11]:=
Conway[Knot[7, 6]][z]

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

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

Out[12]=
{Knot[7, 6], Knot[10, 133]}

In[13]:=
{KnotDet[Knot[7, 6]], KnotSignature[Knot[7, 6]]}

Out[13]=
{19, -2}

In[14]:=
Jones[Knot[7, 6]][q]

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

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

Out[15]=
{Knot[7, 6]}

In[16]:=
A2Invariant[Knot[7, 6]][q]

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

In[17]:=
HOMFLYPT[Knot[7, 6]][a, z]

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

In[18]:=
Kauffman[Knot[7, 6]][a, z]

Out[18]=
2 4 6 3 7 2 2 2 4 2 6 2 1 + a + 2 a + a + a z + 2 a z - a z - 2 z - 4 a z - 4 a z - 2 a z - 3 3 3 5 3 7 3 4 2 4 4 4 6 4 > 4 a z - 6 a z - a z + a z + z + a z + 2 a z + 2 a z + 5 3 5 5 5 2 6 4 6 > 2 a z + 4 a z + 2 a z + a z + a z

In[19]:=
{Vassiliev[2][Knot[7, 6]], Vassiliev[3][Knot[7, 6]]}

Out[19]=
{1, -2}

In[20]:=
Kh[Knot[7, 6]][q, t]

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

In[21]:=
ColouredJones[Knot[7, 6], 2][q]

Out[21]=
-17 2 5 7 -12 12 12 3 17 13 4 16 10 -4 + q - --- + --- - --- - q + --- - --- - -- + -- - -- - -- + -- - -- - 16 14 13 11 10 9 8 7 6 5 4 q q q q q q q q q q q 5 12 5 2 3 4 > -- + -- - - + 6 q - q - 2 q + q 3 2 q q q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 7₆

7₅
7₇

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q^-17 - 2q^-16 + 5q^-14 - 7q^-13 - q^-12 + 12q^-11 - 12q^-10 - 3q^-9 + 17q^-8 - 13q^-7 - 4q^-6 + 16q^-5 - 10q^-4 - 5q^-3 + 12q^-2 - 5q^-1 - 4 + 6q - q² - 2q³ + q⁴
3	- q^-33 + 2q^-32 - 2q^-30 - 2q^-29 + 6q^-28 + 3q^-27 - 10q^-26 - 6q^-25 + 15q^-24 + 10q^-23 - 20q^-22 - 16q^-21 + 25q^-20 + 22q^-19 - 29q^-18 - 25q^-17 + 28q^-16 + 31q^-15 - 31q^-14 - 30q^-13 + 26q^-12 + 33q^-11 - 26q^-10 - 29q^-9 + 19q^-8 + 30q^-7 - 15q^-6 - 26q^-5 + 9q^-4 + 24q^-3 - 5q^-2 - 18q^-1 + 14q + 2q² - 9q³ - 3q⁴ + 5q⁵ + 2q⁶ - q⁷ - 2q⁸ + q⁹
4	q^-54 - 2q^-53 + 2q^-51 - q^-50 + 3q^-49 - 8q^-48 + q^-47 + 9q^-46 - q^-45 + 6q^-44 - 25q^-43 - q^-42 + 25q^-41 + 7q^-40 + 12q^-39 - 56q^-38 - 13q^-37 + 44q^-36 + 28q^-35 + 29q^-34 - 93q^-33 - 36q^-32 + 57q^-31 + 51q^-30 + 52q^-29 - 119q^-28 - 57q^-27 + 57q^-26 + 64q^-25 + 72q^-24 - 127q^-23 - 69q^-22 + 50q^-21 + 66q^-20 + 80q^-19 - 118q^-18 - 69q^-17 + 36q^-16 + 58q^-15 + 83q^-14 - 96q^-13 - 62q^-12 + 16q^-11 + 43q^-10 + 81q^-9 - 64q^-8 - 50q^-7 - 6q^-6 + 23q^-5 + 71q^-4 - 31q^-3 - 32q^-2 - 18q^-1 + 3 + 49q - 6q² - 12q³ - 17q⁴ - 8q⁵ + 24q⁶ + 2q⁷ - 7q⁹ - 7q¹⁰ + 6q¹¹ + q¹² + 2q¹³ - q¹⁴ - 2q¹⁵ + q¹⁶
5	- q^-80 + 2q^-79 - 2q^-77 + q^-76 - q^-74 + 4q^-73 - 8q^-71 - q^-70 + 4q^-69 + 6q^-68 + 10q^-67 - 4q^-66 - 21q^-65 - 16q^-64 + 8q^-63 + 32q^-62 + 32q^-61 - 4q^-60 - 53q^-59 - 56q^-58 - 3q^-57 + 73q^-56 + 90q^-55 + 17q^-54 - 91q^-53 - 129q^-52 - 41q^-51 + 105q^-50 + 170q^-49 + 71q^-48 - 115q^-47 - 206q^-46 - 103q^-45 + 118q^-44 + 234q^-43 + 132q^-42 - 107q^-41 - 262q^-40 - 157q^-39 + 108q^-38 + 265q^-37 + 174q^-36 - 85q^-35 - 280q^-34 - 190q^-33 + 89q^-32 + 266q^-31 + 192q^-30 - 62q^-29 - 269q^-28 - 198q^-27 + 64q^-26 + 244q^-25 + 192q^-24 - 35q^-23 - 233q^-22 - 193q^-21 + 29q^-20 + 201q^-19 + 183q^-18 + 3q^-17 - 179q^-16 - 176q^-15 - 17q^-14 + 139q^-13 + 163q^-12 + 44q^-11 - 107q^-10 - 147q^-9 - 56q^-8 + 68q^-7 + 121q^-6 + 74q^-5 - 37q^-4 - 97q^-3 - 71q^-2 + 7q^-1 + 66 + 68q + 13q² - 41q³ - 53q⁴ - 25q⁵ + 19q⁶ + 39q⁷ + 23q⁸ - 2q⁹ - 22q¹⁰ - 22q¹¹ - 4q¹² + 13q¹³ + 11q¹⁴ + 5q¹⁵ - 9q¹⁷ - 5q¹⁸ + 2q¹⁹ + 2q²⁰ + q²¹ + 2q²² - q²³ - 2q²⁴ + q²⁵
6	q^-111 - 2q^-110 + 2q^-108 - q^-107 - 2q^-105 + 5q^-104 - 5q^-103 - q^-102 + 10q^-101 - 3q^-100 - 4q^-99 - 10q^-98 + 9q^-97 - 9q^-96 + 4q^-95 + 32q^-94 - q^-93 - 16q^-92 - 38q^-91 + 4q^-90 - 23q^-89 + 23q^-88 + 92q^-87 + 25q^-86 - 26q^-85 - 104q^-84 - 45q^-83 - 75q^-82 + 47q^-81 + 210q^-80 + 119q^-79 + 3q^-78 - 192q^-77 - 161q^-76 - 214q^-75 + 35q^-74 + 359q^-73 + 294q^-72 + 116q^-71 - 251q^-70 - 304q^-69 - 434q^-68 - 53q^-67 + 476q^-66 + 493q^-65 + 295q^-64 - 240q^-63 - 409q^-62 - 660q^-61 - 190q^-60 + 517q^-59 + 641q^-58 + 464q^-57 - 177q^-56 - 439q^-55 - 820q^-54 - 314q^-53 + 496q^-52 + 707q^-51 + 572q^-50 - 109q^-49 - 416q^-48 - 889q^-47 - 390q^-46 + 448q^-45 + 709q^-44 + 614q^-43 - 56q^-42 - 371q^-41 - 889q^-40 - 420q^-39 + 387q^-38 + 669q^-37 + 609q^-36 - 9q^-35 - 308q^-34 - 840q^-33 - 428q^-32 + 301q^-31 + 592q^-30 + 574q^-29 + 55q^-28 - 211q^-27 - 750q^-26 - 429q^-25 + 176q^-24 + 470q^-23 + 516q^-22 + 136q^-21 - 77q^-20 - 614q^-19 - 416q^-18 + 28q^-17 + 304q^-16 + 419q^-15 + 207q^-14 + 71q^-13 - 429q^-12 - 361q^-11 - 100q^-10 + 124q^-9 + 275q^-8 + 220q^-7 + 183q^-6 - 225q^-5 - 247q^-4 - 155q^-3 - 19q^-2 + 113q^-1 + 158 + 209q - 60q² - 109q³ - 122q⁴ - 75q⁵ - 7q⁶ + 63q⁷ + 150q⁸ + 16q⁹ - 9q¹⁰ - 52q¹¹ - 53q¹² - 46q¹³ - 3q¹⁴ + 69q¹⁵ + 18q¹⁶ + 20q¹⁷ - 5q¹⁸ - 14q¹⁹ - 29q²⁰ - 16q²¹ + 18q²² + 3q²³ + 11q²⁴ + 4q²⁵ + 3q²⁶ - 9q²⁷ - 7q²⁸ + 4q²⁹ - 2q³⁰ + 2q³¹ + q³² + 2q³³ - q³⁴ - 2q³⁵ + q³⁶
7	- q^-147 + 2q^-146 - 2q^-144 + q^-143 + 2q^-141 - 2q^-140 - 4q^-139 + 6q^-138 - q^-137 - 6q^-136 + 3q^-135 + 2q^-134 + 10q^-133 - q^-132 - 14q^-131 + 7q^-130 - 9q^-129 - 15q^-128 + 8q^-127 + 10q^-126 + 37q^-125 + 14q^-124 - 26q^-123 - 11q^-122 - 48q^-121 - 49q^-120 + 12q^-119 + 34q^-118 + 110q^-117 + 89q^-116 - 13q^-115 - 48q^-114 - 167q^-113 - 170q^-112 - 41q^-111 + 61q^-110 + 265q^-109 + 296q^-108 + 120q^-107 - 50q^-106 - 366q^-105 - 464q^-104 - 259q^-103 - 9q^-102 + 465q^-101 + 671q^-100 + 465q^-99 + 117q^-98 - 545q^-97 - 892q^-96 - 715q^-95 - 285q^-94 + 571q^-93 + 1110q^-92 + 994q^-91 + 514q^-90 - 551q^-89 - 1301q^-88 - 1277q^-87 - 768q^-86 + 486q^-85 + 1432q^-84 + 1536q^-83 + 1031q^-82 - 374q^-81 - 1517q^-80 - 1748q^-79 - 1278q^-78 + 239q^-77 + 1561q^-76 + 1907q^-75 + 1477q^-74 - 114q^-73 - 1541q^-72 - 1998q^-71 - 1657q^-70 - 21q^-69 + 1529q^-68 + 2069q^-67 + 1748q^-66 + 103q^-65 - 1459q^-64 - 2062q^-63 - 1843q^-62 - 210q^-61 + 1433q^-60 + 2076q^-59 + 1855q^-58 + 242q^-57 - 1348q^-56 - 2017q^-55 - 1889q^-54 - 320q^-53 + 1310q^-52 + 1997q^-51 + 1857q^-50 + 335q^-49 - 1215q^-48 - 1903q^-47 - 1852q^-46 - 411q^-45 + 1142q^-44 + 1843q^-43 + 1802q^-42 + 442q^-41 - 1014q^-40 - 1707q^-39 - 1773q^-38 - 533q^-37 + 882q^-36 + 1594q^-35 + 1702q^-34 + 593q^-33 - 698q^-32 - 1409q^-31 - 1640q^-30 - 695q^-29 + 515q^-28 + 1227q^-27 + 1525q^-26 + 764q^-25 - 293q^-24 - 986q^-23 - 1410q^-22 - 830q^-21 + 88q^-20 + 752q^-19 + 1231q^-18 + 846q^-17 + 115q^-16 - 485q^-15 - 1033q^-14 - 838q^-13 - 270q^-12 + 251q^-11 + 804q^-10 + 753q^-9 + 374q^-8 - 19q^-7 - 568q^-6 - 643q^-5 - 423q^-4 - 133q^-3 + 344q^-2 + 483q^-1 + 402 + 244q - 151q² - 328q³ - 334q⁴ - 281q⁵ + 14q⁶ + 177q⁷ + 243q⁸ + 264q⁹ + 61q¹⁰ - 61q¹¹ - 137q¹² - 210q¹³ - 103q¹⁴ - 10q¹⁵ + 66q¹⁶ + 146q¹⁷ + 80q¹⁸ + 41q¹⁹ + 3q²⁰ - 82q²¹ - 67q²² - 51q²³ - 15q²⁴ + 44q²⁵ + 29q²⁶ + 29q²⁷ + 30q²⁸ - 7q²⁹ - 17q³⁰ - 25q³¹ - 20q³² + 9q³³ + q³⁴ + 4q³⁵ + 13q³⁶ + 4q³⁷ + 2q³⁸ - 6q³⁹ - 7q⁴⁰ + 2q⁴¹ - 2q⁴³ + 2q⁴⁴ + q⁴⁵ + 2q⁴⁶ - q⁴⁷ - 2q⁴⁸ + q⁴⁹

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

Out[8]=	-Graphics-
In[9]:=	#[Knot[7, 6]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 1, 2, 2, 4, 1}
In[10]:=	alex = Alexander[Knot[7, 6]][t]
Out[10]=	-2 5 2 -7 - t + - + 5 t - t t
In[11]:=	Conway[Knot[7, 6]][z]
Out[11]=	2 4 1 + z - z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[7, 6], Knot[10, 133]}
In[13]:=	{KnotDet[Knot[7, 6]], KnotSignature[Knot[7, 6]]}
Out[13]=	{19, -2}
In[14]:=	Jones[Knot[7, 6]][q]
Out[14]=	-6 2 3 4 3 3 -2 - q + -- - -- + -- - -- + - + q 5 4 3 2 q q q q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] \|\| (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[7, 6]}
In[16]:=	A2Invariant[Knot[7, 6]][q]
Out[16]=	-20 -18 -16 -12 -10 -6 -4 -2 4 -q - q + q + q + q + q - q + q + q
In[17]:=	HOMFLYPT[Knot[7, 6]][a, z]
Out[17]=	2 4 6 2 2 2 4 2 2 4 1 - a + 2 a - a + z - 2 a z + 2 a z - a z
In[18]:=	Kauffman[Knot[7, 6]][a, z]
Out[18]=	2 4 6 3 7 2 2 2 4 2 6 2 1 + a + 2 a + a + a z + 2 a z - a z - 2 z - 4 a z - 4 a z - 2 a z - 3 3 3 5 3 7 3 4 2 4 4 4 6 4 > 4 a z - 6 a z - a z + a z + z + a z + 2 a z + 2 a z + 5 3 5 5 5 2 6 4 6 > 2 a z + 4 a z + 2 a z + a z + a z
In[19]:=	{Vassiliev[2][Knot[7, 6]], Vassiliev[3][Knot[7, 6]]}
Out[19]=	{1, -2}
In[20]:=	Kh[Knot[7, 6]][q, t]
Out[20]=	2 2 1 1 1 2 1 2 2 1 -- + - + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ---- + 3 q 13 5 11 4 9 4 9 3 7 3 7 2 5 2 5 q q t q t q t q t q t q t q t q t 2 t 3 2 > ---- + - + q t + q t 3 q q t
In[21]:=	ColouredJones[Knot[7, 6], 2][q]
Out[21]=	-17 2 5 7 -12 12 12 3 17 13 4 16 10 -4 + q - --- + --- - --- - q + --- - --- - -- + -- - -- - -- + -- - -- - 16 14 13 11 10 9 8 7 6 5 4 q q q q q q q q q q q 5 12 5 2 3 4 > -- + -- - - + 6 q - q - 2 q + q 3 2 q q q

The Alternating Knot 76

The Alternating Knot 7₆