7 7

7_6

8_1

Visit 7 7's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit [${\textrm {KnotilusURL}}({\textrm {GaussCode}}())$ 7 7's page] at Knotilus! Visit 7 7's page at the original Knot Atlas!

7 7 Quick Notes

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]

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 (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 \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]=

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

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   \	0	χ
1	1	1
-1	1	1

Computer Talk

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

${\textrm {Include}}({\textrm {ColouredJonesM.mhtml}})$

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 17, 2005, 14:44:34)...
In[2]:=	Crossings[Knot[0, 1]]
Out[2]=	0
In[3]:=	PD[Knot[0, 1]]
Out[3]=	PD[Loop[1]]
In[4]:=	GaussCode[Knot[0, 1]]
Out[4]=	GaussCode[]
In[5]:=	BR[Knot[0, 1]]
Out[5]=	BR[1, {}]
In[6]:=	alex = Alexander[Knot[0, 1]][t]
Out[6]=	1
In[7]:=	Conway[Knot[0, 1]][z]
Out[7]=	1
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[0, 1], Knot[11, NonAlternating, 34], Knot[11, NonAlternating, 42]}
In[9]:=	{KnotDet[Knot[0, 1]], KnotSignature[Knot[0, 1]]}
Out[9]=	{1, 0}
In[10]:=	J=Jones[Knot[0, 1]][q]
Out[10]=	1
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[0, 1]}
In[12]:=	A2Invariant[Knot[0, 1]][q]
Out[12]=	-2 2 1 + q + q
In[13]:=	Kauffman[Knot[0, 1]][a, z]
Out[13]=	1
In[14]:=	{Vassiliev[2][Knot[0, 1]], Vassiliev[3][Knot[0, 1]]}
Out[14]=	{0, 0}
In[15]:=	Kh[Knot[0, 1]][q, t]
Out[15]=	1 - + q q