7 7

7_6

8_1

Visit 7 7's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 7 7's page at Knotilus! Visit 7 7's page at the original Knot Atlas! This is the Chinese crown loop of practical knot tying.

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: Length is 7, width is 4. Braid index is 4.

A Morse Link Presentation:

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, $q\leftrightarrow q^{-1}$): {...}

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 $-4$ $-8$ $8$ $-{\frac {14}{3}}$ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{10}{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 32} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{112}{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{64}{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -8} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{32}{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 32} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{56}{3}} ${\frac {40}{3}}$ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{2849}{30}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{218}{15}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{3418}{45}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{161}{18}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t^rq^j} are shown, along with their alternating sums Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \chi} (fixed Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j} , alternation over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} ). The squares with yellow highlighting are those on the "critical diagonals", where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j-2r=s+1} or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j-2r=s-1} , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i=-1} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i=1} $r=-3$ ${\mathbb {Z} }$ $r=-2$ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}\oplus{\mathbb Z}_2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-1} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}\oplus{\mathbb Z}_2^{2}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=0} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}\oplus{\mathbb Z}_2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=1} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}\oplus{\mathbb Z}_2^{2}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=3} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}\oplus{\mathbb Z}_2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=4} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}_2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}}

The Coloured Jones Polynomials

The Coloured Jones Polynomials (in the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n+1} -dimensional representation of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle sl(2)} )

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J_n}
2	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{12}-2 q^{11}-q^{10}+6 q^9-5 q^8-5 q^7+14 q^6-7 q^5-11 q^4+20 q^3-7 q^2-15 q+21-5 q^{-1} -14 q^{-2} +16 q^{-3} -2 q^{-4} -9 q^{-5} +8 q^{-6} -3 q^{-8} + q^{-9} }
3	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{24}-2 q^{23}-q^{22}+2 q^{21}+5 q^{20}-4 q^{19}-9 q^{18}+4 q^{17}+16 q^{16}-3 q^{15}-23 q^{14}-q^{13}+31 q^{12}+5 q^{11}-38 q^{10}-11 q^9+43 q^8+19 q^7-48 q^6-23 q^5+50 q^4+28 q^3-50 q^2-32 q+49+32 q^{-1} -43 q^{-2} -34 q^{-3} +40 q^{-4} +28 q^{-5} -28 q^{-6} -28 q^{-7} +23 q^{-8} +20 q^{-9} -12 q^{-10} -17 q^{-11} +9 q^{-12} +9 q^{-13} -3 q^{-14} -5 q^{-15} +3 q^{-17} - q^{-18} }
4	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{40}-2 q^{39}-q^{38}+2 q^{37}+q^{36}+6 q^{35}-8 q^{34}-7 q^{33}+2 q^{32}+4 q^{31}+25 q^{30}-14 q^{29}-23 q^{28}-10 q^{27}+3 q^{26}+62 q^{25}-7 q^{24}-38 q^{23}-39 q^{22}-16 q^{21}+106 q^{20}+17 q^{19}-39 q^{18}-76 q^{17}-54 q^{16}+141 q^{15}+50 q^{14}-24 q^{13}-109 q^{12}-98 q^{11}+162 q^{10}+79 q^9-4 q^8-129 q^7-131 q^6+167 q^5+98 q^4+16 q^3-136 q^2-149 q+156+103 q^{-1} +31 q^{-2} -125 q^{-3} -147 q^{-4} +125 q^{-5} +90 q^{-6} +44 q^{-7} -93 q^{-8} -127 q^{-9} +82 q^{-10} +61 q^{-11} +47 q^{-12} -50 q^{-13} -90 q^{-14} +40 q^{-15} +28 q^{-16} +36 q^{-17} -17 q^{-18} -47 q^{-19} +15 q^{-20} +6 q^{-21} +17 q^{-22} -2 q^{-23} -16 q^{-24} +3 q^{-25} +5 q^{-27} -3 q^{-29} + q^{-30} }
5	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{60}-2 q^{59}-q^{58}+2 q^{57}+q^{56}+2 q^{55}+2 q^{54}-6 q^{53}-9 q^{52}+2 q^{51}+7 q^{50}+12 q^{49}+11 q^{48}-11 q^{47}-29 q^{46}-19 q^{45}+9 q^{44}+39 q^{43}+45 q^{42}+3 q^{41}-56 q^{40}-72 q^{39}-26 q^{38}+60 q^{37}+109 q^{36}+61 q^{35}-55 q^{34}-141 q^{33}-110 q^{32}+33 q^{31}+173 q^{30}+162 q^{29}-189 q^{27}-221 q^{26}-45 q^{25}+197 q^{24}+278 q^{23}+96 q^{22}-198 q^{21}-324 q^{20}-149 q^{19}+187 q^{18}+368 q^{17}+202 q^{16}-180 q^{15}-400 q^{14}-242 q^{13}+159 q^{12}+427 q^{11}+283 q^{10}-147 q^9-446 q^8-311 q^7+132 q^6+452 q^5+335 q^4-111 q^3-455 q^2-353 q+98+441 q^{-1} +354 q^{-2} -62 q^{-3} -420 q^{-4} -363 q^{-5} +49 q^{-6} +378 q^{-7} +341 q^{-8} - q^{-9} -335 q^{-10} -330 q^{-11} -11 q^{-12} +269 q^{-13} +283 q^{-14} +55 q^{-15} -211 q^{-16} -253 q^{-17} -50 q^{-18} +141 q^{-19} +190 q^{-20} +76 q^{-21} -95 q^{-22} -149 q^{-23} -55 q^{-24} +50 q^{-25} +91 q^{-26} +56 q^{-27} -24 q^{-28} -63 q^{-29} -33 q^{-30} +9 q^{-31} +32 q^{-32} +21 q^{-33} -15 q^{-35} -17 q^{-36} +2 q^{-37} +9 q^{-38} +4 q^{-39} -5 q^{-42} +3 q^{-44} - q^{-45} }
6	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{84}-2 q^{83}-q^{82}+2 q^{81}+q^{80}+2 q^{79}-2 q^{78}+4 q^{77}-8 q^{76}-9 q^{75}+5 q^{74}+6 q^{73}+12 q^{72}+q^{71}+15 q^{70}-24 q^{69}-35 q^{68}-8 q^{67}+8 q^{66}+37 q^{65}+27 q^{64}+67 q^{63}-35 q^{62}-88 q^{61}-70 q^{60}-36 q^{59}+47 q^{58}+79 q^{57}+200 q^{56}+22 q^{55}-116 q^{54}-179 q^{53}-170 q^{52}-41 q^{51}+90 q^{50}+398 q^{49}+192 q^{48}-23 q^{47}-249 q^{46}-364 q^{45}-271 q^{44}-38 q^{43}+560 q^{42}+435 q^{41}+224 q^{40}-184 q^{39}-517 q^{38}-591 q^{37}-318 q^{36}+595 q^{35}+650 q^{34}+561 q^{33}+16 q^{32}-556 q^{31}-899 q^{30}-672 q^{29}+511 q^{28}+775 q^{27}+886 q^{26}+275 q^{25}-498 q^{24}-1130 q^{23}-1000 q^{22}+373 q^{21}+823 q^{20}+1137 q^{19}+507 q^{18}-404 q^{17}-1281 q^{16}-1245 q^{15}+243 q^{14}+828 q^{13}+1305 q^{12}+674 q^{11}-315 q^{10}-1362 q^9-1398 q^8+133 q^7+804 q^6+1395 q^5+787 q^4-225 q^3-1371 q^2-1467 q+18+729 q^{-1} +1400 q^{-2} +862 q^{-3} -102 q^{-4} -1279 q^{-5} -1449 q^{-6} -118 q^{-7} +572 q^{-8} +1288 q^{-9} +890 q^{-10} +66 q^{-11} -1058 q^{-12} -1311 q^{-13} -249 q^{-14} +332 q^{-15} +1031 q^{-16} +825 q^{-17} +241 q^{-18} -724 q^{-19} -1034 q^{-20} -312 q^{-21} +76 q^{-22} +669 q^{-23} +640 q^{-24} +335 q^{-25} -373 q^{-26} -667 q^{-27} -264 q^{-28} -90 q^{-29} +321 q^{-30} +383 q^{-31} +299 q^{-32} -127 q^{-33} -331 q^{-34} -145 q^{-35} -122 q^{-36} +100 q^{-37} +161 q^{-38} +183 q^{-39} -22 q^{-40} -124 q^{-41} -44 q^{-42} -74 q^{-43} +14 q^{-44} +44 q^{-45} +79 q^{-46} - q^{-47} -35 q^{-48} -6 q^{-49} -27 q^{-50} +6 q^{-52} +26 q^{-53} -2 q^{-54} -9 q^{-55} +3 q^{-56} -7 q^{-57} +5 q^{-60} -3 q^{-62} + q^{-63} }
7	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{112}-2 q^{111}-q^{110}+2 q^{109}+q^{108}+2 q^{107}-2 q^{106}+2 q^{104}-8 q^{103}-6 q^{102}+4 q^{101}+6 q^{100}+14 q^{99}+2 q^{98}-2 q^{97}+5 q^{96}-27 q^{95}-28 q^{94}-9 q^{93}+8 q^{92}+49 q^{91}+37 q^{90}+25 q^{89}+25 q^{88}-58 q^{87}-93 q^{86}-81 q^{85}-53 q^{84}+76 q^{83}+123 q^{82}+140 q^{81}+149 q^{80}-30 q^{79}-165 q^{78}-250 q^{77}-275 q^{76}-44 q^{75}+154 q^{74}+325 q^{73}+461 q^{72}+217 q^{71}-77 q^{70}-391 q^{69}-661 q^{68}-446 q^{67}-101 q^{66}+368 q^{65}+854 q^{64}+741 q^{63}+372 q^{62}-239 q^{61}-984 q^{60}-1057 q^{59}-750 q^{58}-4 q^{57}+1036 q^{56}+1344 q^{55}+1174 q^{54}+369 q^{53}-956 q^{52}-1583 q^{51}-1633 q^{50}-821 q^{49}+776 q^{48}+1739 q^{47}+2061 q^{46}+1329 q^{45}-498 q^{44}-1789 q^{43}-2451 q^{42}-1857 q^{41}+147 q^{40}+1772 q^{39}+2782 q^{38}+2352 q^{37}+228 q^{36}-1684 q^{35}-3024 q^{34}-2818 q^{33}-617 q^{32}+1557 q^{31}+3231 q^{30}+3224 q^{29}+954 q^{28}-1422 q^{27}-3361 q^{26}-3554 q^{25}-1278 q^{24}+1274 q^{23}+3479 q^{22}+3840 q^{21}+1535 q^{20}-1168 q^{19}-3540 q^{18}-4050 q^{17}-1755 q^{16}+1044 q^{15}+3592 q^{14}+4235 q^{13}+1934 q^{12}-955 q^{11}-3618 q^{10}-4354 q^9-2082 q^8+849 q^7+3599 q^6+4443 q^5+2233 q^4-730 q^3-3566 q^2-4497 q-2339+596 q^{-1} +3437 q^{-2} +4481 q^{-3} +2486 q^{-4} -398 q^{-5} -3297 q^{-6} -4431 q^{-7} -2556 q^{-8} +194 q^{-9} +3011 q^{-10} +4263 q^{-11} +2666 q^{-12} +81 q^{-13} -2708 q^{-14} -4040 q^{-15} -2651 q^{-16} -333 q^{-17} +2251 q^{-18} +3675 q^{-19} +2636 q^{-20} +618 q^{-21} -1817 q^{-22} -3250 q^{-23} -2449 q^{-24} -819 q^{-25} +1279 q^{-26} +2701 q^{-27} +2262 q^{-28} +983 q^{-29} -853 q^{-30} -2162 q^{-31} -1892 q^{-32} -1008 q^{-33} +393 q^{-34} +1587 q^{-35} +1567 q^{-36} +988 q^{-37} -138 q^{-38} -1116 q^{-39} -1131 q^{-40} -838 q^{-41} -98 q^{-42} +682 q^{-43} +818 q^{-44} +686 q^{-45} +161 q^{-46} -414 q^{-47} -495 q^{-48} -478 q^{-49} -196 q^{-50} +201 q^{-51} +287 q^{-52} +334 q^{-53} +160 q^{-54} -103 q^{-55} -149 q^{-56} -195 q^{-57} -108 q^{-58} +41 q^{-59} +57 q^{-60} +108 q^{-61} +84 q^{-62} -17 q^{-63} -34 q^{-64} -60 q^{-65} -34 q^{-66} +16 q^{-67} -2 q^{-68} +23 q^{-69} +27 q^{-70} -6 q^{-72} -17 q^{-73} -7 q^{-74} +9 q^{-75} -3 q^{-76} +7 q^{-78} -5 q^{-81} +3 q^{-83} - q^{-84} }

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 29, 2005, 15:27:48)...
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 -4 2 2 z 3 z 2 2 z 6 z 2 2 4 z 2 + a + -- + --- + --- + a z - 7 z - ---- - ---- - 3 a z - ---- - 2 3 a 4 2 3 a a a a a 3 4 4 5 5 8 z 3 3 3 4 z 2 z 2 4 2 z 5 z ---- - 3 a z + a z + 4 z + -- + ---- + 3 a z + ---- + ---- + a 4 2 3 a a a a 6 5 6 z 3 a z + z + -- 2 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 21 + q - -- + -- - -- - -- + -- - -- - - - 15 q - 7 q + 20 q - 8 6 5 4 3 2 q q q q q q q 4 5 6 7 8 9 10 11 12 11 q - 7 q + 14 q - 5 q - 5 q + 6 q - q - 2 q + q

See/edit the Rolfsen_Splice_Template. Back to the top.

7_6 8_1