The Alternating Knot 7₇

Visit 7₇'s page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 7₇'s page at Knotilus!

Acknowledgement

KnotPlot

PD Presentation: X₁₄₂₅ X_5,10,6,11 X₃₉₄₈ X_9,3,10,2 X_11,14,12,1 X_7,13,8,12 X_13,7,14,6

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

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

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 + 9 - 5t + t²

Conway Polynomial: 1 - z² + z⁴

Other knots with the same Alexander/Conway Polynomial: {K11n28, ...}

Determinant and Signature: {21, 0}

Jones Polynomial: - q^-3 + 3q^-2 - 3q^-1 + 4 - 4q + 3q² - 2q³ + q⁴

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

A2 (sl(3)) Invariant: - q^-10 + q^-8 + q^-6 + 2q^-2 + q² - q⁴ - q⁶ - q¹⁰ + q¹² + q¹⁴

HOMFLY-PT Polynomial: a^-4 - 2a^-2 - 2a^-2z² + 2 + 2z² + z⁴ - a²z²

Kauffman Polynomial: a^-4 - 2a^-4z² + a^-4z⁴ + 2a^-3z - 4a^-3z³ + 2a^-3z⁵ + 2a^-2 - 6a^-2z² + 2a^-2z⁴ + a^-2z⁶ + 3a^-1z - 8a^-1z³ + 5a^-1z⁵ + 2 - 7z² + 4z⁴ + z⁶ + az - 3az³ + 3az⁵ - 3a²z² + 3a²z⁴ + a³z³

V₂ and V₃, the type 2 and 3 Vassiliev invariants: {-1, -1}

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=0 is the signature of 7₇. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

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

j = 9 1

j = 7 1

j = 5 2 1

j = 3 2 1

j = 1 2 2

j = -1 2 3

j = -3 1 1

j = -5 2

j = -7 1

n Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2 q^-9 - 3q^-8 + 8q^-6 - 9q^-5 - 2q^-4 + 16q^-3 - 14q^-2 - 5q^-1 + 21 - 15q - 7q² + 20q³ - 11q⁴ - 7q⁵ + 14q⁶ - 5q⁷ - 5q⁸ + 6q⁹ - q¹⁰ - 2q¹¹ + q¹²

3 - q^-18 + 3q^-17 - 5q^-15 - 3q^-14 + 9q^-13 + 9q^-12 - 17q^-11 - 12q^-10 + 20q^-9 + 23q^-8 - 28q^-7 - 28q^-6 + 28q^-5 + 40q^-4 - 34q^-3 - 43q^-2 + 32q^-1 + 49 - 32q - 50q² + 28q³ + 50q⁴ - 23q⁵ - 48q⁶ + 19q⁷ + 43q⁸ - 11q⁹ - 38q¹⁰ + 5q¹¹ + 31q¹² - q¹³ - 23q¹⁴ - 3q¹⁵ + 16q¹⁶ + 4q¹⁷ - 9q¹⁸ - 4q¹⁹ + 5q²⁰ + 2q²¹ - q²² - 2q²³ + q²⁴

4 q^-30 - 3q^-29 + 5q^-27 + 3q^-25 - 16q^-24 - 2q^-23 + 17q^-22 + 6q^-21 + 15q^-20 - 47q^-19 - 17q^-18 + 36q^-17 + 28q^-16 + 40q^-15 - 90q^-14 - 50q^-13 + 47q^-12 + 61q^-11 + 82q^-10 - 127q^-9 - 93q^-8 + 44q^-7 + 90q^-6 + 125q^-5 - 147q^-4 - 125q^-3 + 31q^-2 + 103q^-1 + 156 - 149q - 136q² + 16q³ + 98q⁴ + 167q⁵ - 131q⁶ - 129q⁷ - 4q⁸ + 79q⁹ + 162q¹⁰ - 98q¹¹ - 109q¹² - 24q¹³ + 50q¹⁴ + 141q¹⁵ - 54q¹⁶ - 76q¹⁷ - 39q¹⁸ + 17q¹⁹ + 106q²⁰ - 16q²¹ - 39q²² - 38q²³ - 7q²⁴ + 62q²⁵ + 3q²⁶ - 10q²⁷ - 23q²⁸ - 14q²⁹ + 25q³⁰ + 4q³¹ + 2q³² - 7q³³ - 8q³⁴ + 6q³⁵ + q³⁶ + 2q³⁷ - q³⁸ - 2q³⁹ + q⁴⁰

5 - q^-45 + 3q^-44 - 5q^-42 + 4q^-39 + 9q^-38 + 2q^-37 - 17q^-36 - 15q^-35 + 21q^-33 + 32q^-32 + 9q^-31 - 33q^-30 - 63q^-29 - 24q^-28 + 56q^-27 + 91q^-26 + 50q^-25 - 55q^-24 - 149q^-23 - 95q^-22 + 76q^-21 + 190q^-20 + 141q^-19 - 50q^-18 - 253q^-17 - 211q^-16 + 55q^-15 + 283q^-14 + 269q^-13 - 11q^-12 - 330q^-11 - 335q^-10 - q^-9 + 341q^-8 + 378q^-7 + 49q^-6 - 363q^-5 - 420q^-4 - 62q^-3 + 354q^-2 + 441q^-1 + 98 - 353q - 455q² - 111q³ + 335q⁴ + 452q⁵ + 132q⁶ - 311q⁷ - 446q⁸ - 147q⁹ + 283q¹⁰ + 427q¹¹ + 159q¹² - 242q¹³ - 400q¹⁴ - 180q¹⁵ + 202q¹⁶ + 368q¹⁷ + 187q¹⁸ - 149q¹⁹ - 324q²⁰ - 198q²¹ + 96q²² + 278q²³ + 197q²⁴ - 45q²⁵ - 221q²⁶ - 189q²⁷ + 162q²⁹ + 173q³⁰ + 33q³¹ - 110q³² - 141q³³ - 55q³⁴ + 61q³⁵ + 109q³⁶ + 60q³⁷ - 26q³⁸ - 72q³⁹ - 56q⁴⁰ + 3q⁴¹ + 45q⁴² + 39q⁴³ + 9q⁴⁴ - 19q⁴⁵ - 29q⁴⁶ - 11q⁴⁷ + 11q⁴⁸ + 12q⁴⁹ + 7q⁵⁰ + 2q⁵¹ - 9q⁵² - 6q⁵³ + 2q⁵⁴ + 2q⁵⁵ + q⁵⁶ + 2q⁵⁷ - q⁵⁸ - 2q⁵⁹ + q⁶⁰

6 q^-63 - 3q^-62 + 5q^-60 - 7q^-57 + 3q^-56 - 9q^-55 - 2q^-54 + 26q^-53 + 6q^-52 - 27q^-50 - 6q^-49 - 35q^-48 - q^-47 + 79q^-46 + 44q^-45 + 14q^-44 - 74q^-43 - 44q^-42 - 124q^-41 - 22q^-40 + 183q^-39 + 161q^-38 + 100q^-37 - 122q^-36 - 145q^-35 - 331q^-34 - 127q^-33 + 299q^-32 + 383q^-31 + 321q^-30 - 90q^-29 - 264q^-28 - 667q^-27 - 373q^-26 + 335q^-25 + 640q^-24 + 669q^-23 + 76q^-22 - 312q^-21 - 1034q^-20 - 724q^-19 + 241q^-18 + 825q^-17 + 1031q^-16 + 332q^-15 - 249q^-14 - 1311q^-13 - 1058q^-12 + 66q^-11 + 890q^-10 + 1288q^-9 + 572q^-8 - 118q^-7 - 1449q^-6 - 1279q^-5 - 102q^-4 + 862q^-3 + 1400q^-2 + 729q^-1 + 18 - 1467q - 1371q² - 225q³ + 787q⁴ + 1395q⁵ + 804q⁶ + 133q⁷ - 1398q⁸ - 1362q⁹ - 315q¹⁰ + 674q¹¹ + 1305q¹² + 828q¹³ + 243q¹⁴ - 1245q¹⁵ - 1281q¹⁶ - 404q¹⁷ + 507q¹⁸ + 1137q¹⁹ + 823q²⁰ + 373q²¹ - 1000q²² - 1130q²³ - 498q²⁴ + 275q²⁵ + 886q²⁶ + 775q²⁷ + 511q²⁸ - 672q²⁹ - 899q³⁰ - 556q³¹ + 16q³² + 561q³³ + 650q³⁴ + 595q³⁵ - 318q³⁶ - 591q³⁷ - 517q³⁸ - 184q³⁹ + 224q⁴⁰ + 435q⁴¹ + 560q⁴² - 38q⁴³ - 271q⁴⁴ - 364q⁴⁵ - 249q⁴⁶ - 23q⁴⁷ + 192q⁴⁸ + 398q⁴⁹ + 90q⁵⁰ - 41q⁵¹ - 170q⁵² - 179q⁵³ - 116q⁵⁴ + 22q⁵⁵ + 200q⁵⁶ + 79q⁵⁷ + 47q⁵⁸ - 36q⁵⁹ - 70q⁶⁰ - 88q⁶¹ - 35q⁶² + 67q⁶³ + 27q⁶⁴ + 37q⁶⁵ + 8q⁶⁶ - 8q⁶⁷ - 35q⁶⁸ - 24q⁶⁹ + 15q⁷⁰ + q⁷¹ + 12q⁷² + 6q⁷³ + 5q⁷⁴ - 9q⁷⁵ - 8q⁷⁶ + 4q⁷⁷ - 2q⁷⁸ + 2q⁷⁹ + q⁸⁰ + 2q⁸¹ - q⁸² - 2q⁸³ + q⁸⁴

7 - q^-84 + 3q^-83 - 5q^-81 + 7q^-78 - 3q^-76 + 9q^-75 - 7q^-74 - 17q^-73 - 6q^-72 + 27q^-70 + 23q^-69 - 2q^-68 + 16q^-67 - 34q^-66 - 60q^-65 - 34q^-64 - 17q^-63 + 84q^-62 + 108q^-61 + 57q^-60 + 41q^-59 - 108q^-58 - 195q^-57 - 149q^-56 - 103q^-55 + 160q^-54 + 334q^-53 + 287q^-52 + 201q^-51 - 196q^-50 - 478q^-49 - 495q^-48 - 414q^-47 + 161q^-46 + 686q^-45 + 818q^-44 + 682q^-43 - 98q^-42 - 838q^-41 - 1131q^-40 - 1116q^-39 - 138q^-38 + 988q^-37 + 1567q^-36 + 1587q^-35 + 393q^-34 - 1008q^-33 - 1892q^-32 - 2162q^-31 - 853q^-30 + 983q^-29 + 2262q^-28 + 2701q^-27 + 1279q^-26 - 819q^-25 - 2449q^-24 - 3250q^-23 - 1817q^-22 + 618q^-21 + 2636q^-20 + 3675q^-19 + 2251q^-18 - 333q^-17 - 2651q^-16 - 4040q^-15 - 2708q^-14 + 81q^-13 + 2666q^-12 + 4263q^-11 + 3011q^-10 + 194q^-9 - 2556q^-8 - 4431q^-7 - 3297q^-6 - 398q^-5 + 2486q^-4 + 4481q^-3 + 3437q^-2 + 596q^-1 - 2339 - 4497q - 3566q² - 730q³ + 2233q⁴ + 4443q⁵ + 3599q⁶ + 849q⁷ - 2082q⁸ - 4354q⁹ - 3618q¹⁰ - 955q¹¹ + 1934q¹² + 4235q¹³ + 3592q¹⁴ + 1044q¹⁵ - 1755q¹⁶ - 4050q¹⁷ - 3540q¹⁸ - 1168q¹⁹ + 1535q²⁰ + 3840q²¹ + 3479q²² + 1274q²³ - 1278q²⁴ - 3554q²⁵ - 3361q²⁶ - 1422q²⁷ + 954q²⁸ + 3224q²⁹ + 3231q³⁰ + 1557q³¹ - 617q³² - 2818q³³ - 3024q³⁴ - 1684q³⁵ + 228q³⁶ + 2352q³⁷ + 2782q³⁸ + 1772q³⁹ + 147q⁴⁰ - 1857q⁴¹ - 2451q⁴² - 1789q⁴³ - 498q⁴⁴ + 1329q⁴⁵ + 2061q⁴⁶ + 1739q⁴⁷ + 776q⁴⁸ - 821q⁴⁹ - 1633q⁵⁰ - 1583q⁵¹ - 956q⁵² + 369q⁵³ + 1174q⁵⁴ + 1344q⁵⁵ + 1036q⁵⁶ - 4q⁵⁷ - 750q⁵⁸ - 1057q⁵⁹ - 984q⁶⁰ - 239q⁶¹ + 372q⁶² + 741q⁶³ + 854q⁶⁴ + 368q⁶⁵ - 101q⁶⁶ - 446q⁶⁷ - 661q⁶⁸ - 391q⁶⁹ - 77q⁷⁰ + 217q⁷¹ + 461q⁷² + 325q⁷³ + 154q⁷⁴ - 44q⁷⁵ - 275q⁷⁶ - 250q⁷⁷ - 165q⁷⁸ - 30q⁷⁹ + 149q⁸⁰ + 140q⁸¹ + 123q⁸² + 76q⁸³ - 53q⁸⁴ - 81q⁸⁵ - 93q⁸⁶ - 58q⁸⁷ + 25q⁸⁸ + 25q⁸⁹ + 37q⁹⁰ + 49q⁹¹ + 8q⁹² - 9q⁹³ - 28q⁹⁴ - 27q⁹⁵ + 5q⁹⁶ - 2q⁹⁷ + 2q⁹⁸ + 14q⁹⁹ + 6q¹⁰⁰ + 4q¹⁰¹ - 6q¹⁰² - 8q¹⁰³ + 2q¹⁰⁴ - 2q¹⁰⁶ + 2q¹⁰⁷ + q¹⁰⁸ + 2q¹⁰⁹ - q¹¹⁰ - 2q¹¹¹ + q¹¹²

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, 7]]

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

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

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

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

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

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

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

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

Out[6]=
{4, 7}

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

Out[7]=
4

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

Out[8]=
-Graphics-

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

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

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

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

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

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

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

Out[12]=
{Knot[7, 7], Knot[11, NonAlternating, 28]}

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

Out[13]=
{21, 0}

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

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

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

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

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

Out[16]=
-10 -8 -6 2 2 4 6 10 12 14 -q + q + q + -- + q - q - q - q + q + q 2 q

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

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

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

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

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

Out[19]=
{-1, -1}

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

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

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

Out[21]=
-9 3 8 9 2 16 14 5 2 3 4 21 + q - -- + -- - -- - -- + -- - -- - - - 15 q - 7 q + 20 q - 11 q - 8 6 5 4 3 2 q q q q q q q 5 6 7 8 9 10 11 12 > 7 q + 14 q - 5 q - 5 q + 6 q - q - 2 q + q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 7₇

7₆
8₁

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q^-9 - 3q^-8 + 8q^-6 - 9q^-5 - 2q^-4 + 16q^-3 - 14q^-2 - 5q^-1 + 21 - 15q - 7q² + 20q³ - 11q⁴ - 7q⁵ + 14q⁶ - 5q⁷ - 5q⁸ + 6q⁹ - q¹⁰ - 2q¹¹ + q¹²
3	- q^-18 + 3q^-17 - 5q^-15 - 3q^-14 + 9q^-13 + 9q^-12 - 17q^-11 - 12q^-10 + 20q^-9 + 23q^-8 - 28q^-7 - 28q^-6 + 28q^-5 + 40q^-4 - 34q^-3 - 43q^-2 + 32q^-1 + 49 - 32q - 50q² + 28q³ + 50q⁴ - 23q⁵ - 48q⁶ + 19q⁷ + 43q⁸ - 11q⁹ - 38q¹⁰ + 5q¹¹ + 31q¹² - q¹³ - 23q¹⁴ - 3q¹⁵ + 16q¹⁶ + 4q¹⁷ - 9q¹⁸ - 4q¹⁹ + 5q²⁰ + 2q²¹ - q²² - 2q²³ + q²⁴
4	q^-30 - 3q^-29 + 5q^-27 + 3q^-25 - 16q^-24 - 2q^-23 + 17q^-22 + 6q^-21 + 15q^-20 - 47q^-19 - 17q^-18 + 36q^-17 + 28q^-16 + 40q^-15 - 90q^-14 - 50q^-13 + 47q^-12 + 61q^-11 + 82q^-10 - 127q^-9 - 93q^-8 + 44q^-7 + 90q^-6 + 125q^-5 - 147q^-4 - 125q^-3 + 31q^-2 + 103q^-1 + 156 - 149q - 136q² + 16q³ + 98q⁴ + 167q⁵ - 131q⁶ - 129q⁷ - 4q⁸ + 79q⁹ + 162q¹⁰ - 98q¹¹ - 109q¹² - 24q¹³ + 50q¹⁴ + 141q¹⁵ - 54q¹⁶ - 76q¹⁷ - 39q¹⁸ + 17q¹⁹ + 106q²⁰ - 16q²¹ - 39q²² - 38q²³ - 7q²⁴ + 62q²⁵ + 3q²⁶ - 10q²⁷ - 23q²⁸ - 14q²⁹ + 25q³⁰ + 4q³¹ + 2q³² - 7q³³ - 8q³⁴ + 6q³⁵ + q³⁶ + 2q³⁷ - q³⁸ - 2q³⁹ + q⁴⁰
5	- q^-45 + 3q^-44 - 5q^-42 + 4q^-39 + 9q^-38 + 2q^-37 - 17q^-36 - 15q^-35 + 21q^-33 + 32q^-32 + 9q^-31 - 33q^-30 - 63q^-29 - 24q^-28 + 56q^-27 + 91q^-26 + 50q^-25 - 55q^-24 - 149q^-23 - 95q^-22 + 76q^-21 + 190q^-20 + 141q^-19 - 50q^-18 - 253q^-17 - 211q^-16 + 55q^-15 + 283q^-14 + 269q^-13 - 11q^-12 - 330q^-11 - 335q^-10 - q^-9 + 341q^-8 + 378q^-7 + 49q^-6 - 363q^-5 - 420q^-4 - 62q^-3 + 354q^-2 + 441q^-1 + 98 - 353q - 455q² - 111q³ + 335q⁴ + 452q⁵ + 132q⁶ - 311q⁷ - 446q⁸ - 147q⁹ + 283q¹⁰ + 427q¹¹ + 159q¹² - 242q¹³ - 400q¹⁴ - 180q¹⁵ + 202q¹⁶ + 368q¹⁷ + 187q¹⁸ - 149q¹⁹ - 324q²⁰ - 198q²¹ + 96q²² + 278q²³ + 197q²⁴ - 45q²⁵ - 221q²⁶ - 189q²⁷ + 162q²⁹ + 173q³⁰ + 33q³¹ - 110q³² - 141q³³ - 55q³⁴ + 61q³⁵ + 109q³⁶ + 60q³⁷ - 26q³⁸ - 72q³⁹ - 56q⁴⁰ + 3q⁴¹ + 45q⁴² + 39q⁴³ + 9q⁴⁴ - 19q⁴⁵ - 29q⁴⁶ - 11q⁴⁷ + 11q⁴⁸ + 12q⁴⁹ + 7q⁵⁰ + 2q⁵¹ - 9q⁵² - 6q⁵³ + 2q⁵⁴ + 2q⁵⁵ + q⁵⁶ + 2q⁵⁷ - q⁵⁸ - 2q⁵⁹ + q⁶⁰
6	q^-63 - 3q^-62 + 5q^-60 - 7q^-57 + 3q^-56 - 9q^-55 - 2q^-54 + 26q^-53 + 6q^-52 - 27q^-50 - 6q^-49 - 35q^-48 - q^-47 + 79q^-46 + 44q^-45 + 14q^-44 - 74q^-43 - 44q^-42 - 124q^-41 - 22q^-40 + 183q^-39 + 161q^-38 + 100q^-37 - 122q^-36 - 145q^-35 - 331q^-34 - 127q^-33 + 299q^-32 + 383q^-31 + 321q^-30 - 90q^-29 - 264q^-28 - 667q^-27 - 373q^-26 + 335q^-25 + 640q^-24 + 669q^-23 + 76q^-22 - 312q^-21 - 1034q^-20 - 724q^-19 + 241q^-18 + 825q^-17 + 1031q^-16 + 332q^-15 - 249q^-14 - 1311q^-13 - 1058q^-12 + 66q^-11 + 890q^-10 + 1288q^-9 + 572q^-8 - 118q^-7 - 1449q^-6 - 1279q^-5 - 102q^-4 + 862q^-3 + 1400q^-2 + 729q^-1 + 18 - 1467q - 1371q² - 225q³ + 787q⁴ + 1395q⁵ + 804q⁶ + 133q⁷ - 1398q⁸ - 1362q⁹ - 315q¹⁰ + 674q¹¹ + 1305q¹² + 828q¹³ + 243q¹⁴ - 1245q¹⁵ - 1281q¹⁶ - 404q¹⁷ + 507q¹⁸ + 1137q¹⁹ + 823q²⁰ + 373q²¹ - 1000q²² - 1130q²³ - 498q²⁴ + 275q²⁵ + 886q²⁶ + 775q²⁷ + 511q²⁸ - 672q²⁹ - 899q³⁰ - 556q³¹ + 16q³² + 561q³³ + 650q³⁴ + 595q³⁵ - 318q³⁶ - 591q³⁷ - 517q³⁸ - 184q³⁹ + 224q⁴⁰ + 435q⁴¹ + 560q⁴² - 38q⁴³ - 271q⁴⁴ - 364q⁴⁵ - 249q⁴⁶ - 23q⁴⁷ + 192q⁴⁸ + 398q⁴⁹ + 90q⁵⁰ - 41q⁵¹ - 170q⁵² - 179q⁵³ - 116q⁵⁴ + 22q⁵⁵ + 200q⁵⁶ + 79q⁵⁷ + 47q⁵⁸ - 36q⁵⁹ - 70q⁶⁰ - 88q⁶¹ - 35q⁶² + 67q⁶³ + 27q⁶⁴ + 37q⁶⁵ + 8q⁶⁶ - 8q⁶⁷ - 35q⁶⁸ - 24q⁶⁹ + 15q⁷⁰ + q⁷¹ + 12q⁷² + 6q⁷³ + 5q⁷⁴ - 9q⁷⁵ - 8q⁷⁶ + 4q⁷⁷ - 2q⁷⁸ + 2q⁷⁹ + q⁸⁰ + 2q⁸¹ - q⁸² - 2q⁸³ + q⁸⁴
7	- q^-84 + 3q^-83 - 5q^-81 + 7q^-78 - 3q^-76 + 9q^-75 - 7q^-74 - 17q^-73 - 6q^-72 + 27q^-70 + 23q^-69 - 2q^-68 + 16q^-67 - 34q^-66 - 60q^-65 - 34q^-64 - 17q^-63 + 84q^-62 + 108q^-61 + 57q^-60 + 41q^-59 - 108q^-58 - 195q^-57 - 149q^-56 - 103q^-55 + 160q^-54 + 334q^-53 + 287q^-52 + 201q^-51 - 196q^-50 - 478q^-49 - 495q^-48 - 414q^-47 + 161q^-46 + 686q^-45 + 818q^-44 + 682q^-43 - 98q^-42 - 838q^-41 - 1131q^-40 - 1116q^-39 - 138q^-38 + 988q^-37 + 1567q^-36 + 1587q^-35 + 393q^-34 - 1008q^-33 - 1892q^-32 - 2162q^-31 - 853q^-30 + 983q^-29 + 2262q^-28 + 2701q^-27 + 1279q^-26 - 819q^-25 - 2449q^-24 - 3250q^-23 - 1817q^-22 + 618q^-21 + 2636q^-20 + 3675q^-19 + 2251q^-18 - 333q^-17 - 2651q^-16 - 4040q^-15 - 2708q^-14 + 81q^-13 + 2666q^-12 + 4263q^-11 + 3011q^-10 + 194q^-9 - 2556q^-8 - 4431q^-7 - 3297q^-6 - 398q^-5 + 2486q^-4 + 4481q^-3 + 3437q^-2 + 596q^-1 - 2339 - 4497q - 3566q² - 730q³ + 2233q⁴ + 4443q⁵ + 3599q⁶ + 849q⁷ - 2082q⁸ - 4354q⁹ - 3618q¹⁰ - 955q¹¹ + 1934q¹² + 4235q¹³ + 3592q¹⁴ + 1044q¹⁵ - 1755q¹⁶ - 4050q¹⁷ - 3540q¹⁸ - 1168q¹⁹ + 1535q²⁰ + 3840q²¹ + 3479q²² + 1274q²³ - 1278q²⁴ - 3554q²⁵ - 3361q²⁶ - 1422q²⁷ + 954q²⁸ + 3224q²⁹ + 3231q³⁰ + 1557q³¹ - 617q³² - 2818q³³ - 3024q³⁴ - 1684q³⁵ + 228q³⁶ + 2352q³⁷ + 2782q³⁸ + 1772q³⁹ + 147q⁴⁰ - 1857q⁴¹ - 2451q⁴² - 1789q⁴³ - 498q⁴⁴ + 1329q⁴⁵ + 2061q⁴⁶ + 1739q⁴⁷ + 776q⁴⁸ - 821q⁴⁹ - 1633q⁵⁰ - 1583q⁵¹ - 956q⁵² + 369q⁵³ + 1174q⁵⁴ + 1344q⁵⁵ + 1036q⁵⁶ - 4q⁵⁷ - 750q⁵⁸ - 1057q⁵⁹ - 984q⁶⁰ - 239q⁶¹ + 372q⁶² + 741q⁶³ + 854q⁶⁴ + 368q⁶⁵ - 101q⁶⁶ - 446q⁶⁷ - 661q⁶⁸ - 391q⁶⁹ - 77q⁷⁰ + 217q⁷¹ + 461q⁷² + 325q⁷³ + 154q⁷⁴ - 44q⁷⁵ - 275q⁷⁶ - 250q⁷⁷ - 165q⁷⁸ - 30q⁷⁹ + 149q⁸⁰ + 140q⁸¹ + 123q⁸² + 76q⁸³ - 53q⁸⁴ - 81q⁸⁵ - 93q⁸⁶ - 58q⁸⁷ + 25q⁸⁸ + 25q⁸⁹ + 37q⁹⁰ + 49q⁹¹ + 8q⁹² - 9q⁹³ - 28q⁹⁴ - 27q⁹⁵ + 5q⁹⁶ - 2q⁹⁷ + 2q⁹⁸ + 14q⁹⁹ + 6q¹⁰⁰ + 4q¹⁰¹ - 6q¹⁰² - 8q¹⁰³ + 2q¹⁰⁴ - 2q¹⁰⁶ + 2q¹⁰⁷ + q¹⁰⁸ + 2q¹⁰⁹ - q¹¹⁰ - 2q¹¹¹ + q¹¹²

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

Out[8]=	-Graphics-
In[9]:=	#[Knot[7, 7]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 1, 2, 2, 4, 1}
In[10]:=	alex = Alexander[Knot[7, 7]][t]
Out[10]=	-2 5 2 9 + t - - - 5 t + t t
In[11]:=	Conway[Knot[7, 7]][z]
Out[11]=	2 4 1 - z + z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[7, 7], Knot[11, NonAlternating, 28]}
In[13]:=	{KnotDet[Knot[7, 7]], KnotSignature[Knot[7, 7]]}
Out[13]=	{21, 0}
In[14]:=	Jones[Knot[7, 7]][q]
Out[14]=	-3 3 3 2 3 4 4 - q + -- - - - 4 q + 3 q - 2 q + q 2 q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] \|\| (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[7, 7]}
In[16]:=	A2Invariant[Knot[7, 7]][q]
Out[16]=	-10 -8 -6 2 2 4 6 10 12 14 -q + q + q + -- + q - q - q - q + q + q 2 q
In[17]:=	HOMFLYPT[Knot[7, 7]][a, z]
Out[17]=	2 -4 2 2 2 z 2 2 4 2 + a - -- + 2 z - ---- - a z + z 2 2 a a
In[18]:=	Kauffman[Knot[7, 7]][a, z]
Out[18]=	2 2 3 3 -4 2 2 z 3 z 2 2 z 6 z 2 2 4 z 8 z 2 + a + -- + --- + --- + a z - 7 z - ---- - ---- - 3 a z - ---- - ---- - 2 3 a 4 2 3 a a a a a a 4 4 5 5 6 3 3 3 4 z 2 z 2 4 2 z 5 z 5 6 z > 3 a z + a z + 4 z + -- + ---- + 3 a z + ---- + ---- + 3 a z + z + -- 4 2 3 a 2 a a a a
In[19]:=	{Vassiliev[2][Knot[7, 7]], Vassiliev[3][Knot[7, 7]]}
Out[19]=	{-1, -1}
In[20]:=	Kh[Knot[7, 7]][q, t]
Out[20]=	3 1 2 1 1 2 3 3 2 - + 2 q + ----- + ----- + ----- + ---- + --- + 2 q t + 2 q t + q t + q 7 3 5 2 3 2 3 q t q t q t q t q t 5 2 5 3 7 3 9 4 > 2 q t + q t + q t + q t
In[21]:=	ColouredJones[Knot[7, 7], 2][q]
Out[21]=	-9 3 8 9 2 16 14 5 2 3 4 21 + q - -- + -- - -- - -- + -- - -- - - - 15 q - 7 q + 20 q - 11 q - 8 6 5 4 3 2 q q q q q q q 5 6 7 8 9 10 11 12 > 7 q + 14 q - 5 q - 5 q + 6 q - q - 2 q + q

The Alternating Knot 77

The Alternating Knot 7₇