7 7

7_6

8_1

(KnotPlot image)	See the full Rolfsen Knot Table. Visit 7 7's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 7 7 at Knotilus!

Ornamental knot

Mongolian ornament ; sum of two 7.7

Depiction with three loops

Sum of 4.1 and 7.7

Knot presentations

Planar diagram presentation	X1425 X5,10,6,11 X3948 X9,3,10,2 X11,14,12,1 X7,13,8,12 X13,7,14,6
Gauss code	-1, 4, -3, 1, -2, 7, -6, 3, -4, 2, -5, 6, -7, 5
Dowker-Thistlethwaite code	4 8 10 12 2 14 6
Conway Notation	[21112]

Minimum Braid Representative	A Morse Link Presentation	An Arc Presentation
Length is 7, width is 4, Braid index is 4		[{9, 3}, {2, 7}, {8, 4}, {3, 5}, {7, 9}, {4, 1}, {6, 2}, {5, 8}, {1, 6}]

[edit Notes on presentations of 7 7]

Knot 7_7.

A graph, knot 7_7.

Computer Talk

The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of May 31, 2006, 14:15:20.091. Read more at http://katlas.org/wiki/KnotTheory.

In[3]:=

K = Knot["7 7"];

In[4]:=

PD[K]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]=

X1425 X5,10,6,11 X3948 X9,3,10,2 X11,14,12,1 X7,13,8,12 X13,7,14,6

In[5]:=

GaussCode[K]

Out[5]=

-1, 4, -3, 1, -2, 7, -6, 3, -4, 2, -5, 6, -7, 5

In[6]:=

DTCode[K]

Out[6]=

4 8 10 12 2 14 6

(The path below may be different on your system)

In[7]:=

AppendTo[$Path, "C:/bin/LinKnot/"];

In[8]:=

ConwayNotation[K]

Out[8]=

[21112]

In[9]:=

br = BR[K]

KnotTheory::credits: The minimum braids representing the knots with up to 10 crossings were provided by Thomas Gittings. See arXiv:math.GT/0401051.

Out[9]=

${\textrm {BR}}(4,\{1,-2,1,-2,3,-2,3\})$

In[10]:=

{First[br], Crossings[br], BraidIndex[K]}

KnotTheory::credits: The braid index data known to KnotTheory` is taken from Charles Livingston's http://www.indiana.edu/~knotinfo/.

KnotTheory::loading: Loading precomputed data in IndianaData`.

Out[10]=

{ 4, 7, 4 }

In[11]:=

Show[BraidPlot[br]]

Out[11]=

-Graphics-

In[12]:=

Show[DrawMorseLink[K]]

KnotTheory::credits: "MorseLink was added to KnotTheory` by Siddarth Sankaran at the University of Toronto in the summer of 2005."

KnotTheory::credits: "DrawMorseLink was written by Siddarth Sankaran at the University of Toronto in the summer of 2005."

Out[12]=

-Graphics-

In[13]:=

ap = ArcPresentation[K]

Out[13]=

ArcPresentation[{9, 3}, {2, 7}, {8, 4}, {3, 5}, {7, 9}, {4, 1}, {6, 2}, {5, 8}, {1, 6}]

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 2 Bridge index 2 Super bridge index 4 Nakanishi index 1 Maximal Thurston-Bennequin number Failed to parse (syntax error): {\displaystyle \text{$\$$Failed}} Hyperbolic Volume 7.64338 A-Polynomial See Data:7 7/A-polynomial

[edit Notes for 7 7's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus $1$ Topological 4 genus $1$ Concordance genus $2$ Rasmussen s-Invariant 0

[edit Notes for 7 7's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{2}+t^{-2}-5t-5t^{-1}+9$
Conway polynomial	$z^{4}-z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 21, 0 }
Jones polynomial	$q^{4}-2q^{3}-q^{-3}+3q^{2}+3q^{-2}-4q-3q^{-1}+4$
HOMFLY-PT polynomial (db, data sources)	$a^{-4}-a^{2}z^{2}-2z^{2}a^{-2}-2a^{-2}+z^{4}+2z^{2}+2$
Kauffman polynomial (db, data sources)	$z^{4}a^{-4}-2z^{2}a^{-4}+a^{-4}+2z^{5}a^{-3}+a^{3}z^{3}-4z^{3}a^{-3}+2za^{-3}+z^{6}a^{-2}+3a^{2}z^{4}+2z^{4}a^{-2}-3a^{2}z^{2}-6z^{2}a^{-2}+2a^{-2}+3az^{5}+5z^{5}a^{-1}-3az^{3}-8z^{3}a^{-1}+az+3za^{-1}+z^{6}+4z^{4}-7z^{2}+2$
The A2 invariant	$-q^{10}+q^{8}+q^{6}+2q^{2}+q^{-2}-q^{-4}-q^{-6}-q^{-10}+q^{-12}+q^{-14}$
The G2 invariant	$q^{52}-2q^{50}+3q^{48}-4q^{46}+q^{42}-4q^{40}+9q^{38}-9q^{36}+9q^{34}-3q^{32}-4q^{30}+9q^{28}-10q^{26}+9q^{24}-5q^{22}-q^{20}+5q^{18}-4q^{16}+4q^{14}+2q^{12}-7q^{10}+10q^{8}-5q^{6}-2q^{4}+8q^{2}-12+17q^{-2}-11q^{-4}+5q^{-6}+3q^{-8}-9q^{-10}+15q^{-12}-14q^{-14}+6q^{-16}-q^{-18}-4q^{-20}+6q^{-22}-6q^{-24}+q^{-26}+3q^{-28}-7q^{-30}+5q^{-32}-4q^{-34}-5q^{-36}+10q^{-38}-11q^{-40}+9q^{-42}-4q^{-44}-q^{-46}+7q^{-48}-8q^{-50}+9q^{-52}-4q^{-54}+q^{-56}+q^{-58}-3q^{-60}+3q^{-62}-q^{-64}+q^{-66}$

Further Quantum Invariants

Further quantum knot invariants for 7_7.

A1 Invariants.

Weight	Invariant
1	$-q^{7}+2q^{5}+q-q^{-3}+q^{-5}-q^{-7}+q^{-9}$
2	$q^{20}-2q^{18}-2q^{16}+5q^{14}-q^{12}-3q^{10}+5q^{8}-3q^{4}+2q^{2}+1-q^{-2}-2q^{-4}+2q^{-6}+2q^{-8}-4q^{-10}+2q^{-12}+4q^{-14}-4q^{-16}+3q^{-20}-2q^{-22}-q^{-24}+q^{-26}$
3	$-q^{39}+2q^{37}+2q^{35}-3q^{33}-5q^{31}+q^{29}+10q^{27}-2q^{25}-11q^{23}+14q^{19}+3q^{17}-13q^{15}-5q^{13}+12q^{11}+6q^{9}-9q^{7}-5q^{5}+4q^{3}+6q-q^{-1}-5q^{-3}-4q^{-5}+5q^{-7}+7q^{-9}-2q^{-11}-9q^{-13}+3q^{-15}+13q^{-17}-q^{-19}-13q^{-21}-3q^{-23}+12q^{-25}+4q^{-27}-11q^{-29}-6q^{-31}+8q^{-33}+7q^{-35}-4q^{-37}-6q^{-39}+2q^{-41}+4q^{-43}-2q^{-47}-q^{-49}+q^{-51}$
4	$q^{64}-2q^{62}-2q^{60}+3q^{58}+3q^{56}+5q^{54}-8q^{52}-10q^{50}+2q^{48}+8q^{46}+20q^{44}-11q^{42}-26q^{40}-7q^{38}+15q^{36}+40q^{34}-3q^{32}-36q^{30}-25q^{28}+8q^{26}+50q^{24}+13q^{22}-30q^{20}-33q^{18}-4q^{16}+39q^{14}+19q^{12}-13q^{10}-26q^{8}-13q^{6}+18q^{4}+16q^{2}+5-10q^{-2}-15q^{-4}-4q^{-6}+14q^{-8}+21q^{-10}+q^{-12}-18q^{-14}-23q^{-16}+10q^{-18}+30q^{-20}+10q^{-22}-19q^{-24}-40q^{-26}+4q^{-28}+37q^{-30}+22q^{-32}-11q^{-34}-46q^{-36}-8q^{-38}+29q^{-40}+30q^{-42}+6q^{-44}-38q^{-46}-19q^{-48}+10q^{-50}+25q^{-52}+18q^{-54}-19q^{-56}-18q^{-58}-6q^{-60}+10q^{-62}+16q^{-64}-3q^{-66}-6q^{-68}-6q^{-70}+6q^{-74}+q^{-76}-2q^{-80}-q^{-82}+q^{-84}$
5	$-q^{95}+2q^{93}+2q^{91}-3q^{89}-3q^{87}-3q^{85}+2q^{83}+8q^{81}+10q^{79}-2q^{77}-17q^{75}-17q^{73}+23q^{69}+30q^{67}+14q^{65}-34q^{63}-58q^{61}-23q^{59}+36q^{57}+77q^{55}+55q^{53}-31q^{51}-102q^{49}-82q^{47}+17q^{45}+108q^{43}+113q^{41}+9q^{39}-107q^{37}-128q^{35}-35q^{33}+93q^{31}+132q^{29}+55q^{27}-69q^{25}-125q^{23}-67q^{21}+42q^{19}+102q^{17}+69q^{15}-16q^{13}-77q^{11}-64q^{9}-q^{7}+48q^{5}+58q^{3}+23q-26q^{-1}-45q^{-3}-34q^{-5}+42q^{-9}+51q^{-11}+15q^{-13}-37q^{-15}-62q^{-17}-35q^{-19}+34q^{-21}+80q^{-23}+47q^{-25}-34q^{-27}-93q^{-29}-65q^{-31}+28q^{-33}+104q^{-35}+86q^{-37}-20q^{-39}-110q^{-41}-100q^{-43}+4q^{-45}+107q^{-47}+116q^{-49}+20q^{-51}-96q^{-53}-120q^{-55}-42q^{-57}+69q^{-59}+117q^{-61}+62q^{-63}-39q^{-65}-103q^{-67}-76q^{-69}+8q^{-71}+77q^{-73}+76q^{-75}+18q^{-77}-46q^{-79}-67q^{-81}-32q^{-83}+21q^{-85}+48q^{-87}+34q^{-89}-27q^{-93}-29q^{-95}-8q^{-97}+12q^{-99}+17q^{-101}+8q^{-103}-2q^{-105}-8q^{-107}-8q^{-109}+4q^{-113}+3q^{-115}+q^{-117}-2q^{-121}-q^{-123}+q^{-125}$
6	$q^{132}-2q^{130}-2q^{128}+3q^{126}+3q^{124}+3q^{122}-4q^{120}-2q^{118}-8q^{116}-10q^{114}+11q^{112}+17q^{110}+17q^{108}-3q^{106}-12q^{104}-38q^{102}-37q^{100}+16q^{98}+54q^{96}+68q^{94}+21q^{92}-17q^{90}-106q^{88}-127q^{86}-23q^{84}+94q^{82}+180q^{80}+132q^{78}+31q^{76}-176q^{74}-281q^{72}-165q^{70}+57q^{68}+278q^{66}+310q^{64}+191q^{62}-145q^{60}-391q^{58}-355q^{56}-98q^{54}+250q^{52}+416q^{50}+368q^{48}+q^{46}-350q^{44}-444q^{42}-259q^{40}+103q^{38}+359q^{36}+422q^{34}+145q^{32}-189q^{30}-364q^{28}-299q^{26}-42q^{24}+198q^{22}+329q^{20}+191q^{18}-30q^{16}-198q^{14}-226q^{12}-114q^{10}+43q^{8}+179q^{6}+161q^{4}+69q^{2}-54-129q^{-2}-134q^{-4}-59q^{-6}+56q^{-8}+125q^{-10}+134q^{-12}+44q^{-14}-69q^{-16}-159q^{-18}-135q^{-20}-25q^{-22}+128q^{-24}+209q^{-26}+120q^{-28}-47q^{-30}-215q^{-32}-220q^{-34}-90q^{-36}+155q^{-38}+306q^{-40}+212q^{-42}-20q^{-44}-271q^{-46}-319q^{-48}-181q^{-50}+147q^{-52}+378q^{-54}+320q^{-56}+61q^{-58}-264q^{-60}-389q^{-62}-305q^{-64}+49q^{-66}+357q^{-68}+396q^{-70}+196q^{-72}-141q^{-74}-356q^{-76}-391q^{-78}-111q^{-80}+209q^{-82}+362q^{-84}+297q^{-86}+50q^{-88}-193q^{-90}-355q^{-92}-227q^{-94}+3q^{-96}+197q^{-98}+267q^{-100}+174q^{-102}+4q^{-104}-194q^{-106}-205q^{-108}-117q^{-110}+17q^{-112}+126q^{-114}+154q^{-116}+97q^{-118}-36q^{-120}-88q^{-122}-98q^{-124}-54q^{-126}+8q^{-128}+61q^{-130}+72q^{-132}+20q^{-134}-6q^{-136}-31q^{-138}-33q^{-140}-20q^{-142}+6q^{-144}+22q^{-146}+11q^{-148}+8q^{-150}-2q^{-152}-7q^{-154}-10q^{-156}-2q^{-158}+4q^{-160}+q^{-162}+3q^{-164}+q^{-166}-2q^{-170}-q^{-172}+q^{-174}$

A2 Invariants.

Weight	Invariant
1,0	$-q^{10}+q^{8}+q^{6}+2q^{2}+q^{-2}-q^{-4}-q^{-6}-q^{-10}+q^{-12}+q^{-14}$
1,1	$q^{28}-4q^{26}+8q^{24}-12q^{22}+18q^{20}-28q^{18}+30q^{16}-30q^{14}+30q^{12}-18q^{10}+12q^{8}+10q^{6}-23q^{4}+38q^{2}-52+54q^{-2}-60q^{-4}+52q^{-6}-44q^{-8}+30q^{-10}-11q^{-12}+2q^{-14}+16q^{-16}-24q^{-18}+31q^{-20}-30q^{-22}+26q^{-24}-24q^{-26}+15q^{-28}-10q^{-30}+6q^{-32}-2q^{-34}+q^{-36}$
2,0	$q^{26}-q^{24}-2q^{22}+2q^{18}+q^{16}-3q^{14}+2q^{12}+4q^{10}+q^{8}-q^{6}+2q^{4}+q^{2}-1-q^{-2}-q^{-4}-2q^{-6}-2q^{-8}+2q^{-10}-q^{-14}+3q^{-16}+4q^{-18}-q^{-22}+q^{-24}+q^{-26}-2q^{-28}-3q^{-30}+q^{-34}+q^{-36}$
3,0	$-q^{48}+q^{46}+2q^{44}+q^{42}-2q^{40}-6q^{38}+q^{36}+4q^{34}+5q^{32}-4q^{30}-11q^{28}+10q^{24}+12q^{22}-4q^{20}-11q^{18}+11q^{14}+8q^{12}-4q^{10}-8q^{8}+q^{6}+4q^{4}+q^{2}-5-4q^{-2}-q^{-4}-2q^{-6}-3q^{-8}+8q^{-12}+5q^{-14}-q^{-18}+8q^{-20}+11q^{-22}-2q^{-24}-11q^{-26}-7q^{-28}+5q^{-30}+7q^{-32}-6q^{-34}-11q^{-36}-5q^{-38}+6q^{-40}+10q^{-42}-4q^{-46}-4q^{-48}+4q^{-50}+7q^{-52}+2q^{-54}-2q^{-56}-5q^{-58}-2q^{-60}+q^{-64}+q^{-66}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{22}-2q^{20}-q^{18}+3q^{16}-3q^{14}+q^{12}+5q^{10}-q^{8}+3q^{4}-q^{2}-1+q^{-4}-2q^{-8}+3q^{-10}+q^{-12}-4q^{-14}+2q^{-16}+q^{-18}-3q^{-20}+2q^{-22}+q^{-24}-q^{-26}+q^{-28}$
1,0,0	$-q^{13}+q^{11}+q^{7}+2q^{3}+q+q^{-1}+q^{-3}-q^{-5}-q^{-7}-2q^{-9}-q^{-13}+q^{-15}+q^{-17}+q^{-19}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{28}-q^{26}-2q^{24}+q^{22}+q^{20}-2q^{18}-q^{16}+4q^{14}+3q^{12}+2q^{8}+6q^{6}-2q^{2}+1-2q^{-2}-5q^{-4}-q^{-6}+q^{-8}-q^{-10}+q^{-12}+5q^{-14}+3q^{-16}-2q^{-18}+2q^{-22}-2q^{-24}-3q^{-26}+q^{-30}+q^{-36}+q^{-38}$
1,0,0,0	$-q^{16}+q^{14}+q^{8}+2q^{4}+q^{2}+2+q^{-2}+q^{-4}-q^{-6}-q^{-8}-2q^{-10}-2q^{-12}-q^{-16}+q^{-18}+q^{-20}+q^{-22}+q^{-24}$

B2 Invariants.

Weight	Invariant
0,1	$-q^{22}+2q^{20}-3q^{18}+3q^{16}-3q^{14}+3q^{12}-q^{10}+q^{8}+2q^{6}-q^{4}+5q^{2}-5+6q^{-2}-5q^{-4}+4q^{-6}-4q^{-8}+q^{-10}-q^{-12}-2q^{-14}+2q^{-16}-3q^{-18}+3q^{-20}-2q^{-22}+3q^{-24}-q^{-26}+q^{-28}$
1,0	$q^{36}-2q^{32}-2q^{30}+q^{28}+3q^{26}-3q^{22}-q^{20}+4q^{18}+3q^{16}-2q^{12}+q^{10}+2q^{8}+q^{6}-3q^{4}-q^{2}+2+q^{-2}-2q^{-4}-2q^{-6}+q^{-8}+2q^{-10}-2q^{-14}+q^{-16}+3q^{-18}+q^{-20}-3q^{-22}-2q^{-24}+2q^{-26}+3q^{-28}-q^{-30}-3q^{-32}-q^{-34}+2q^{-36}+2q^{-38}-q^{-40}-q^{-42}+q^{-46}$

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{30}-2q^{28}+q^{26}-2q^{24}+3q^{22}-3q^{20}+2q^{18}-q^{16}+3q^{14}+q^{12}+q^{8}+4q^{4}-3q^{2}+4-4q^{-2}+5q^{-4}-3q^{-6}+3q^{-8}-4q^{-10}+2q^{-12}-q^{-14}-q^{-18}-2q^{-20}+2q^{-22}-2q^{-24}+2q^{-26}-2q^{-28}+3q^{-30}-q^{-32}+2q^{-34}-q^{-36}+q^{-38}$

G2 Invariants.

Weight

Invariant

1,0

$q^{52}-2q^{50}+3q^{48}-4q^{46}+q^{42}-4q^{40}+9q^{38}-9q^{36}+9q^{34}-3q^{32}-4q^{30}+9q^{28}-10q^{26}+9q^{24}-5q^{22}-q^{20}+5q^{18}-4q^{16}+4q^{14}+2q^{12}-7q^{10}+10q^{8}-5q^{6}-2q^{4}+8q^{2}-12+17q^{-2}-11q^{-4}+5q^{-6}+3q^{-8}-9q^{-10}+15q^{-12}-14q^{-14}+6q^{-16}-q^{-18}-4q^{-20}+6q^{-22}-6q^{-24}+q^{-26}+3q^{-28}-7q^{-30}+5q^{-32}-4q^{-34}-5q^{-36}+10q^{-38}-11q^{-40}+9q^{-42}-4q^{-44}-q^{-46}+7q^{-48}-8q^{-50}+9q^{-52}-4q^{-54}+q^{-56}+q^{-58}-3q^{-60}+3q^{-62}-q^{-64}+q^{-66}$

.

Computer Talk

The above data is available with the Mathematica package KnotTheory`, as shown in the (simulated) Mathematica session below. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot 5_2) as the notebook PolynomialInvariantsSession.nb.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of August 31, 2006, 11:25:27.5625. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["7 7"];

In[4]:=

Alexander[K][t]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]=

$t^{2}+t^{-2}-5t-5t^{-1}+9$

In[5]:=

Conway[K][z]

Out[5]=

$z^{4}-z^{2}+1$

In[6]:=

Alexander[K, 2][t]

KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.

Out[6]=

$\{1\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 21, 0 }

In[8]:=

Jones[K][q]

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[8]=

$q^{4}-2q^{3}-q^{-3}+3q^{2}+3q^{-2}-4q-3q^{-1}+4$

In[9]:=

HOMFLYPT[K][a, z]

KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.

Out[9]=

$a^{-4}-a^{2}z^{2}-2z^{2}a^{-2}-2a^{-2}+z^{4}+2z^{2}+2$

In[10]:=

Kauffman[K][a, z]

KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.

Out[10]=

$z^{4}a^{-4}-2z^{2}a^{-4}+a^{-4}+2z^{5}a^{-3}+a^{3}z^{3}-4z^{3}a^{-3}+2za^{-3}+z^{6}a^{-2}+3a^{2}z^{4}+2z^{4}a^{-2}-3a^{2}z^{2}-6z^{2}a^{-2}+2a^{-2}+3az^{5}+5z^{5}a^{-1}-3az^{3}-8z^{3}a^{-1}+az+3za^{-1}+z^{6}+4z^{4}-7z^{2}+2$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11n28,}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {}

Computer Talk

The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of May 31, 2006, 14:15:20.091. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["7 7"];

In[4]:=

{A = Alexander[K][t], J = Jones[K][q]}

KnotTheory::loading: Loading precomputed data in PD4Knots`.

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[4]=

{ $t^{2}+t^{-2}-5t-5t^{-1}+9$, $q^{4}-2q^{3}-q^{-3}+3q^{2}+3q^{-2}-4q-3q^{-1}+4$ }

In[5]:=

DeleteCases[Select[AllKnots[], (A === Alexander[#][t]) &], K]

KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`.

KnotTheory::credits: The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.

Out[5]=

{K11n28,}

In[6]:=

DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]

KnotTheory::loading: Loading precomputed data in Jones4Knots11`.

Out[6]=

{}

Vassiliev invariants

V2 and V3:

(-1, -1)

V2,1 through V6,9:

V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9 $-4$ $-8$ $8$ $-{\frac {14}{3}}$ $-{\frac {10}{3}}$ $32$ ${\frac {112}{3}}$ ${\frac {64}{3}}$ $-8$ $-{\frac {32}{3}}$ $32$ ${\frac {56}{3}}$ ${\frac {40}{3}}$ ${\frac {2849}{30}}$ $-{\frac {218}{15}}$ ${\frac {3418}{45}}$ $-{\frac {161}{18}}$ ${\frac {449}{30}}$

V_2,1 through V_6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V₂ and V₃.

Khovanov Homology

The coefficients of the monomials $t^{r}q^{j}$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$). 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 7 7. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -3 -2 -1 0 1 2 3 4 χ 9               1 1 7             1   -1 5           2 1   1 3         2 1     -1 1       2 2       0 -1     2 3         1 -3   1 1           0 -5   2             2 -7 1               -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-1$ $i=1$ $r=-3$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-1$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=0$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }^{2}$ $r=1$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=2$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=3$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=4$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

The Coloured Jones Polynomials

The Coloured Jones Polynomials (in the $n+1$-dimensional representation of $sl(2)$)

$n$	$J_{n}$
2	$q^{12}-2q^{11}-q^{10}+6q^{9}-5q^{8}-5q^{7}+14q^{6}-7q^{5}-11q^{4}+20q^{3}-7q^{2}-15q+21-5q^{-1}-14q^{-2}+16q^{-3}-2q^{-4}-9q^{-5}+8q^{-6}-3q^{-8}+q^{-9}$
3	$q^{24}-2q^{23}-q^{22}+2q^{21}+5q^{20}-4q^{19}-9q^{18}+4q^{17}+16q^{16}-3q^{15}-23q^{14}-q^{13}+31q^{12}+5q^{11}-38q^{10}-11q^{9}+43q^{8}+19q^{7}-48q^{6}-23q^{5}+50q^{4}+28q^{3}-50q^{2}-32q+49+32q^{-1}-43q^{-2}-34q^{-3}+40q^{-4}+28q^{-5}-28q^{-6}-28q^{-7}+23q^{-8}+20q^{-9}-12q^{-10}-17q^{-11}+9q^{-12}+9q^{-13}-3q^{-14}-5q^{-15}+3q^{-17}-q^{-18}$
4	$q^{40}-2q^{39}-q^{38}+2q^{37}+q^{36}+6q^{35}-8q^{34}-7q^{33}+2q^{32}+4q^{31}+25q^{30}-14q^{29}-23q^{28}-10q^{27}+3q^{26}+62q^{25}-7q^{24}-38q^{23}-39q^{22}-16q^{21}+106q^{20}+17q^{19}-39q^{18}-76q^{17}-54q^{16}+141q^{15}+50q^{14}-24q^{13}-109q^{12}-98q^{11}+162q^{10}+79q^{9}-4q^{8}-129q^{7}-131q^{6}+167q^{5}+98q^{4}+16q^{3}-136q^{2}-149q+156+103q^{-1}+31q^{-2}-125q^{-3}-147q^{-4}+125q^{-5}+90q^{-6}+44q^{-7}-93q^{-8}-127q^{-9}+82q^{-10}+61q^{-11}+47q^{-12}-50q^{-13}-90q^{-14}+40q^{-15}+28q^{-16}+36q^{-17}-17q^{-18}-47q^{-19}+15q^{-20}+6q^{-21}+17q^{-22}-2q^{-23}-16q^{-24}+3q^{-25}+5q^{-27}-3q^{-29}+q^{-30}$
5	$q^{60}-2q^{59}-q^{58}+2q^{57}+q^{56}+2q^{55}+2q^{54}-6q^{53}-9q^{52}+2q^{51}+7q^{50}+12q^{49}+11q^{48}-11q^{47}-29q^{46}-19q^{45}+9q^{44}+39q^{43}+45q^{42}+3q^{41}-56q^{40}-72q^{39}-26q^{38}+60q^{37}+109q^{36}+61q^{35}-55q^{34}-141q^{33}-110q^{32}+33q^{31}+173q^{30}+162q^{29}-189q^{27}-221q^{26}-45q^{25}+197q^{24}+278q^{23}+96q^{22}-198q^{21}-324q^{20}-149q^{19}+187q^{18}+368q^{17}+202q^{16}-180q^{15}-400q^{14}-242q^{13}+159q^{12}+427q^{11}+283q^{10}-147q^{9}-446q^{8}-311q^{7}+132q^{6}+452q^{5}+335q^{4}-111q^{3}-455q^{2}-353q+98+441q^{-1}+354q^{-2}-62q^{-3}-420q^{-4}-363q^{-5}+49q^{-6}+378q^{-7}+341q^{-8}-q^{-9}-335q^{-10}-330q^{-11}-11q^{-12}+269q^{-13}+283q^{-14}+55q^{-15}-211q^{-16}-253q^{-17}-50q^{-18}+141q^{-19}+190q^{-20}+76q^{-21}-95q^{-22}-149q^{-23}-55q^{-24}+50q^{-25}+91q^{-26}+56q^{-27}-24q^{-28}-63q^{-29}-33q^{-30}+9q^{-31}+32q^{-32}+21q^{-33}-15q^{-35}-17q^{-36}+2q^{-37}+9q^{-38}+4q^{-39}-5q^{-42}+3q^{-44}-q^{-45}$
6	$q^{84}-2q^{83}-q^{82}+2q^{81}+q^{80}+2q^{79}-2q^{78}+4q^{77}-8q^{76}-9q^{75}+5q^{74}+6q^{73}+12q^{72}+q^{71}+15q^{70}-24q^{69}-35q^{68}-8q^{67}+8q^{66}+37q^{65}+27q^{64}+67q^{63}-35q^{62}-88q^{61}-70q^{60}-36q^{59}+47q^{58}+79q^{57}+200q^{56}+22q^{55}-116q^{54}-179q^{53}-170q^{52}-41q^{51}+90q^{50}+398q^{49}+192q^{48}-23q^{47}-249q^{46}-364q^{45}-271q^{44}-38q^{43}+560q^{42}+435q^{41}+224q^{40}-184q^{39}-517q^{38}-591q^{37}-318q^{36}+595q^{35}+650q^{34}+561q^{33}+16q^{32}-556q^{31}-899q^{30}-672q^{29}+511q^{28}+775q^{27}+886q^{26}+275q^{25}-498q^{24}-1130q^{23}-1000q^{22}+373q^{21}+823q^{20}+1137q^{19}+507q^{18}-404q^{17}-1281q^{16}-1245q^{15}+243q^{14}+828q^{13}+1305q^{12}+674q^{11}-315q^{10}-1362q^{9}-1398q^{8}+133q^{7}+804q^{6}+1395q^{5}+787q^{4}-225q^{3}-1371q^{2}-1467q+18+729q^{-1}+1400q^{-2}+862q^{-3}-102q^{-4}-1279q^{-5}-1449q^{-6}-118q^{-7}+572q^{-8}+1288q^{-9}+890q^{-10}+66q^{-11}-1058q^{-12}-1311q^{-13}-249q^{-14}+332q^{-15}+1031q^{-16}+825q^{-17}+241q^{-18}-724q^{-19}-1034q^{-20}-312q^{-21}+76q^{-22}+669q^{-23}+640q^{-24}+335q^{-25}-373q^{-26}-667q^{-27}-264q^{-28}-90q^{-29}+321q^{-30}+383q^{-31}+299q^{-32}-127q^{-33}-331q^{-34}-145q^{-35}-122q^{-36}+100q^{-37}+161q^{-38}+183q^{-39}-22q^{-40}-124q^{-41}-44q^{-42}-74q^{-43}+14q^{-44}+44q^{-45}+79q^{-46}-q^{-47}-35q^{-48}-6q^{-49}-27q^{-50}+6q^{-52}+26q^{-53}-2q^{-54}-9q^{-55}+3q^{-56}-7q^{-57}+5q^{-60}-3q^{-62}+q^{-63}$
7	$q^{112}-2q^{111}-q^{110}+2q^{109}+q^{108}+2q^{107}-2q^{106}+2q^{104}-8q^{103}-6q^{102}+4q^{101}+6q^{100}+14q^{99}+2q^{98}-2q^{97}+5q^{96}-27q^{95}-28q^{94}-9q^{93}+8q^{92}+49q^{91}+37q^{90}+25q^{89}+25q^{88}-58q^{87}-93q^{86}-81q^{85}-53q^{84}+76q^{83}+123q^{82}+140q^{81}+149q^{80}-30q^{79}-165q^{78}-250q^{77}-275q^{76}-44q^{75}+154q^{74}+325q^{73}+461q^{72}+217q^{71}-77q^{70}-391q^{69}-661q^{68}-446q^{67}-101q^{66}+368q^{65}+854q^{64}+741q^{63}+372q^{62}-239q^{61}-984q^{60}-1057q^{59}-750q^{58}-4q^{57}+1036q^{56}+1344q^{55}+1174q^{54}+369q^{53}-956q^{52}-1583q^{51}-1633q^{50}-821q^{49}+776q^{48}+1739q^{47}+2061q^{46}+1329q^{45}-498q^{44}-1789q^{43}-2451q^{42}-1857q^{41}+147q^{40}+1772q^{39}+2782q^{38}+2352q^{37}+228q^{36}-1684q^{35}-3024q^{34}-2818q^{33}-617q^{32}+1557q^{31}+3231q^{30}+3224q^{29}+954q^{28}-1422q^{27}-3361q^{26}-3554q^{25}-1278q^{24}+1274q^{23}+3479q^{22}+3840q^{21}+1535q^{20}-1168q^{19}-3540q^{18}-4050q^{17}-1755q^{16}+1044q^{15}+3592q^{14}+4235q^{13}+1934q^{12}-955q^{11}-3618q^{10}-4354q^{9}-2082q^{8}+849q^{7}+3599q^{6}+4443q^{5}+2233q^{4}-730q^{3}-3566q^{2}-4497q-2339+596q^{-1}+3437q^{-2}+4481q^{-3}+2486q^{-4}-398q^{-5}-3297q^{-6}-4431q^{-7}-2556q^{-8}+194q^{-9}+3011q^{-10}+4263q^{-11}+2666q^{-12}+81q^{-13}-2708q^{-14}-4040q^{-15}-2651q^{-16}-333q^{-17}+2251q^{-18}+3675q^{-19}+2636q^{-20}+618q^{-21}-1817q^{-22}-3250q^{-23}-2449q^{-24}-819q^{-25}+1279q^{-26}+2701q^{-27}+2262q^{-28}+983q^{-29}-853q^{-30}-2162q^{-31}-1892q^{-32}-1008q^{-33}+393q^{-34}+1587q^{-35}+1567q^{-36}+988q^{-37}-138q^{-38}-1116q^{-39}-1131q^{-40}-838q^{-41}-98q^{-42}+682q^{-43}+818q^{-44}+686q^{-45}+161q^{-46}-414q^{-47}-495q^{-48}-478q^{-49}-196q^{-50}+201q^{-51}+287q^{-52}+334q^{-53}+160q^{-54}-103q^{-55}-149q^{-56}-195q^{-57}-108q^{-58}+41q^{-59}+57q^{-60}+108q^{-61}+84q^{-62}-17q^{-63}-34q^{-64}-60q^{-65}-34q^{-66}+16q^{-67}-2q^{-68}+23q^{-69}+27q^{-70}-6q^{-72}-17q^{-73}-7q^{-74}+9q^{-75}-3q^{-76}+7q^{-78}-5q^{-81}+3q^{-83}-q^{-84}$

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session, or any of the Computer Talk sections above.

Modifying This Page

Read me first: Modifying Knot Pages See/edit the Rolfsen Knot Page master template (intermediate). See/edit the Rolfsen_Splice_Base (expert). Back to the top.

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