7 6

7_5

7_7

Visit 7 6's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 7 6's page at Knotilus! Visit 7 6's page at the original Knot Atlas!

7 6 Quick Notes

7 6 Further Notes and Views

Knot presentations

Planar diagram presentation	X1425 X3849 X5,12,6,13 X9,1,10,14 X13,11,14,10 X11,6,12,7 X7283
Gauss code	-1, 7, -2, 1, -3, 6, -7, 2, -4, 5, -6, 3, -5, 4
Dowker-Thistlethwaite code	4 8 12 2 14 6 10
Conway Notation	[2212]

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 [-8][-1] Hyperbolic Volume 7.08493 A-Polynomial See Data:7 6/A-polynomial

[edit Notes for 7 6's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus $1$ Topological 4 genus $1$ Concordance genus ${\textrm {ConcordanceGenus}}({\textrm {Knot}}(7,6))$ Rasmussen s-Invariant -2

[edit Notes for 7 6's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{2}-t^{-2}+5t+5t^{-1}-7$
Conway polynomial	$-z^{4}+z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 19, -2 }
Jones polynomial	$q-2+3q^{-1}-3q^{-2}+4q^{-3}-3q^{-4}+2q^{-5}-q^{-6}$
HOMFLY-PT polynomial (db, data sources)	$-a^{6}+2a^{4}z^{2}+2a^{4}-a^{2}z^{4}-2a^{2}z^{2}-a^{2}+z^{2}+1$
Kauffman polynomial (db, data sources)	$a^{7}z^{3}-a^{7}z+2a^{6}z^{4}-2a^{6}z^{2}+a^{6}+2a^{5}z^{5}-a^{5}z^{3}+a^{4}z^{6}+2a^{4}z^{4}-4a^{4}z^{2}+2a^{4}+4a^{3}z^{5}-6a^{3}z^{3}+2a^{3}z+a^{2}z^{6}+a^{2}z^{4}-4a^{2}z^{2}+a^{2}+2az^{5}-4az^{3}+az+z^{4}-2z^{2}+1$
The A2 invariant	$-q^{20}-q^{18}+q^{16}+q^{12}+q^{10}+q^{6}-q^{4}+q^{2}+q^{-4}$
The G2 invariant	$q^{100}-q^{98}+2q^{96}-2q^{94}-3q^{88}+5q^{86}-6q^{84}+4q^{82}-4q^{80}-q^{78}+5q^{76}-8q^{74}+8q^{72}-5q^{70}+q^{68}+2q^{66}-5q^{64}+4q^{62}-2q^{58}+6q^{56}-5q^{54}+2q^{52}+6q^{50}-9q^{48}+11q^{46}-9q^{44}+5q^{42}+2q^{40}-6q^{38}+10q^{36}-9q^{34}+8q^{32}-3q^{30}-3q^{28}+5q^{26}-5q^{24}+3q^{22}-3q^{18}+5q^{16}-3q^{14}-q^{12}+5q^{10}-8q^{8}+8q^{6}-5q^{4}-q^{2}+5-6q^{-2}+8q^{-4}-4q^{-6}+2q^{-8}+q^{-10}-3q^{-12}+3q^{-14}-q^{-16}+q^{-18}$

Further Quantum Invariants

Further quantum knot invariants for 7_6.

A1 Invariants.

Weight	Invariant
1	$-q^{13}+q^{11}-q^{9}+q^{7}+q^{5}+q-q^{-1}+q^{-3}$
2	$q^{36}-q^{34}-q^{32}+3q^{30}-2q^{28}-3q^{26}+4q^{24}-q^{22}-3q^{20}+2q^{18}+q^{16}-q^{12}+2q^{10}+q^{8}-3q^{6}+2q^{4}+3q^{2}-3+q^{-2}+3q^{-4}-2q^{-6}-q^{-8}+q^{-10}$
3	$-q^{69}+q^{67}+q^{65}-q^{63}-2q^{61}+2q^{59}+5q^{57}-3q^{55}-7q^{53}+2q^{51}+9q^{49}-q^{47}-11q^{45}-q^{43}+11q^{41}+2q^{39}-7q^{37}-4q^{35}+5q^{33}+3q^{31}-2q^{29}-4q^{27}-2q^{25}+3q^{23}+4q^{21}-3q^{19}-6q^{17}+5q^{15}+8q^{13}-2q^{11}-8q^{9}+2q^{7}+10q^{5}+q^{3}-9q-2q^{-1}+7q^{-3}+4q^{-5}-5q^{-7}-5q^{-9}+3q^{-11}+4q^{-13}-2q^{-17}-q^{-19}+q^{-21}$
4	$q^{112}-q^{110}-q^{108}+q^{106}+2q^{102}-4q^{100}-3q^{98}+4q^{96}+4q^{94}+7q^{92}-10q^{90}-12q^{88}+4q^{86}+12q^{84}+18q^{82}-13q^{80}-25q^{78}-6q^{76}+15q^{74}+32q^{72}-5q^{70}-28q^{68}-15q^{66}+8q^{64}+31q^{62}+5q^{60}-16q^{58}-16q^{56}-3q^{54}+17q^{52}+9q^{50}-3q^{48}-10q^{46}-8q^{44}+9q^{40}+9q^{38}-5q^{36}-13q^{34}-10q^{32}+12q^{30}+19q^{28}-q^{26}-16q^{24}-18q^{22}+14q^{20}+26q^{18}+4q^{16}-16q^{14}-26q^{12}+7q^{10}+25q^{8}+13q^{6}-7q^{4}-29q^{2}-4+16q^{-2}+17q^{-4}+6q^{-6}-19q^{-8}-11q^{-10}+q^{-12}+11q^{-14}+12q^{-16}-6q^{-18}-7q^{-20}-5q^{-22}+q^{-24}+6q^{-26}+q^{-28}-2q^{-32}-q^{-34}+q^{-36}$
5	$-q^{165}+q^{163}+q^{161}-q^{159}+2q^{151}+2q^{149}-4q^{147}-6q^{145}-2q^{143}+5q^{141}+11q^{139}+7q^{137}-6q^{135}-21q^{133}-17q^{131}+9q^{129}+31q^{127}+31q^{125}-q^{123}-41q^{121}-52q^{119}-11q^{117}+47q^{115}+68q^{113}+30q^{111}-43q^{109}-81q^{107}-49q^{105}+31q^{103}+85q^{101}+61q^{99}-16q^{97}-78q^{95}-65q^{93}-q^{91}+60q^{89}+68q^{87}+12q^{85}-42q^{83}-52q^{81}-21q^{79}+21q^{77}+43q^{75}+25q^{73}-8q^{71}-27q^{69}-26q^{67}-8q^{65}+15q^{63}+26q^{61}+18q^{59}-7q^{57}-29q^{55}-29q^{53}-2q^{51}+34q^{49}+39q^{47}+4q^{45}-39q^{43}-48q^{41}-10q^{39}+44q^{37}+61q^{35}+15q^{33}-47q^{31}-67q^{29}-26q^{27}+46q^{25}+75q^{23}+36q^{21}-35q^{19}-77q^{17}-47q^{15}+23q^{13}+73q^{11}+58q^{9}-3q^{7}-58q^{5}-64q^{3}-14q+42q^{-1}+60q^{-3}+28q^{-5}-19q^{-7}-48q^{-9}-38q^{-11}+q^{-13}+32q^{-15}+35q^{-17}+12q^{-19}-14q^{-21}-26q^{-23}-19q^{-25}+3q^{-27}+16q^{-29}+15q^{-31}+4q^{-33}-5q^{-35}-9q^{-37}-7q^{-39}+q^{-41}+4q^{-43}+3q^{-45}+q^{-47}-2q^{-51}-q^{-53}+q^{-55}$
6	$q^{228}-q^{226}-q^{224}+q^{222}-2q^{216}+2q^{214}-q^{212}-2q^{210}+6q^{208}+4q^{206}-8q^{202}-4q^{200}-8q^{198}-3q^{196}+19q^{194}+21q^{192}+9q^{190}-19q^{188}-24q^{186}-38q^{184}-19q^{182}+41q^{180}+67q^{178}+57q^{176}-9q^{174}-58q^{172}-110q^{170}-86q^{168}+32q^{166}+126q^{164}+155q^{162}+67q^{160}-49q^{158}-188q^{156}-200q^{154}-51q^{152}+124q^{150}+237q^{148}+178q^{146}+35q^{144}-185q^{142}-273q^{140}-156q^{138}+43q^{136}+222q^{134}+233q^{132}+128q^{130}-98q^{128}-235q^{126}-194q^{124}-46q^{122}+123q^{120}+186q^{118}+156q^{116}-4q^{114}-128q^{112}-149q^{110}-83q^{108}+25q^{106}+93q^{104}+116q^{102}+47q^{100}-29q^{98}-77q^{96}-75q^{94}-33q^{92}+20q^{90}+65q^{88}+65q^{86}+35q^{84}-26q^{82}-66q^{80}-71q^{78}-24q^{76}+39q^{74}+88q^{72}+80q^{70}-6q^{68}-83q^{66}-118q^{64}-54q^{62}+43q^{60}+133q^{58}+132q^{56}+7q^{54}-115q^{52}-173q^{50}-92q^{48}+42q^{46}+178q^{44}+191q^{42}+43q^{40}-123q^{38}-220q^{36}-149q^{34}-q^{32}+184q^{30}+239q^{28}+111q^{26}-69q^{24}-213q^{22}-200q^{20}-88q^{18}+116q^{16}+230q^{14}+175q^{12}+32q^{10}-130q^{8}-192q^{6}-166q^{4}-q^{2}+137+170q^{-2}+114q^{-4}-6q^{-6}-101q^{-8}-160q^{-10}-84q^{-12}+16q^{-14}+86q^{-16}+108q^{-18}+69q^{-20}+3q^{-22}-78q^{-24}-76q^{-26}-47q^{-28}+39q^{-32}+56q^{-34}+43q^{-36}-8q^{-38}-23q^{-40}-32q^{-42}-23q^{-44}-6q^{-46}+14q^{-48}+23q^{-50}+9q^{-52}+4q^{-54}-5q^{-56}-8q^{-58}-9q^{-60}-q^{-62}+4q^{-64}+q^{-66}+3q^{-68}+q^{-70}-2q^{-74}-q^{-76}+q^{-78}$

A2 Invariants.

Weight	Invariant
1,0	$-q^{20}-q^{18}+q^{16}+q^{12}+q^{10}+q^{6}-q^{4}+q^{2}+q^{-4}$
1,1	$q^{52}-2q^{50}+4q^{48}-6q^{46}+11q^{44}-14q^{42}+16q^{40}-22q^{38}+19q^{36}-18q^{34}+10q^{32}-2q^{30}-7q^{28}+22q^{26}-28q^{24}+36q^{22}-39q^{20}+36q^{18}-34q^{16}+26q^{14}-16q^{12}+8q^{10}+6q^{8}-10q^{6}+19q^{4}-20q^{2}+22-18q^{-2}+13q^{-4}-10q^{-6}+6q^{-8}-2q^{-10}+q^{-12}$
2,0	$q^{50}+q^{48}-2q^{44}-q^{42}+q^{40}-q^{38}-2q^{36}+2q^{32}-2q^{28}+q^{26}+q^{24}-q^{22}+q^{20}+q^{18}+2q^{12}-q^{8}+q^{6}+3q^{4}-1+2q^{-2}+q^{-4}-q^{-6}-q^{-8}+q^{-12}$
3,0	$-q^{90}-q^{88}+q^{84}+3q^{82}-q^{78}-q^{76}+2q^{74}+5q^{72}-2q^{70}-5q^{68}-5q^{66}+3q^{64}+7q^{62}-q^{60}-7q^{58}-6q^{56}+4q^{54}+8q^{52}-5q^{48}-2q^{46}+5q^{44}+3q^{42}-3q^{40}-4q^{38}-q^{36}-4q^{32}-2q^{30}+q^{28}+4q^{26}+q^{24}-2q^{22}+3q^{20}+8q^{18}+5q^{16}-3q^{14}-6q^{12}+3q^{10}+7q^{8}+2q^{6}-6q^{4}-6q^{2}+4+6q^{-2}+2q^{-4}-4q^{-6}-4q^{-8}+2q^{-10}+3q^{-12}+3q^{-14}-q^{-16}-2q^{-18}-q^{-20}+q^{-24}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{42}-q^{40}+2q^{36}-3q^{34}-2q^{32}+2q^{30}-3q^{28}-q^{26}+3q^{24}+q^{20}+q^{18}+2q^{16}-q^{12}+2q^{10}+q^{8}-3q^{6}+2q^{4}+2q^{2}-2+2q^{-2}+q^{-4}-q^{-6}+q^{-8}$
1,0,0	$-q^{27}-q^{25}-q^{23}+q^{21}+2q^{17}+q^{15}+q^{13}-q^{5}+q^{3}+q^{-1}+q^{-5}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{56}+q^{54}-q^{50}-3q^{44}-4q^{42}-2q^{36}+q^{34}+4q^{32}+2q^{30}+2q^{26}+q^{24}-2q^{22}-q^{20}+3q^{18}-q^{16}+4q^{12}+2q^{10}-2q^{8}+2q^{4}-1+q^{-2}+2q^{-4}+q^{-10}$
1,0,0,0	$-q^{34}-q^{32}-q^{30}-q^{28}+q^{26}+2q^{22}+2q^{20}+q^{18}+q^{16}-q^{10}-q^{6}+q^{4}+1+q^{-2}+q^{-6}$

B2 Invariants.

Weight	Invariant
0,1	$-q^{42}+q^{40}-2q^{38}+2q^{36}-3q^{34}+2q^{32}-2q^{30}+q^{28}+q^{26}-q^{24}+4q^{22}-3q^{20}+5q^{18}-4q^{16}+4q^{14}-3q^{12}+2q^{10}-q^{8}-q^{6}+2q^{4}-2q^{2}+2-2q^{-2}+3q^{-4}-q^{-6}+q^{-8}$
1,0	$q^{68}-q^{64}-q^{62}+q^{60}+2q^{58}-3q^{54}-2q^{52}+q^{50}+2q^{48}-q^{46}-3q^{44}+2q^{40}+q^{38}-2q^{36}+2q^{32}+2q^{30}-q^{28}-q^{26}+q^{24}+2q^{22}-q^{18}+2q^{14}+q^{12}-2q^{10}-2q^{8}+2q^{6}+3q^{4}-2-q^{-2}+2q^{-4}+2q^{-6}-q^{-8}-q^{-10}+q^{-14}$

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{58}-q^{56}+q^{54}-q^{52}+2q^{50}-3q^{48}-3q^{44}+q^{42}-2q^{40}-q^{38}+4q^{32}-q^{30}+4q^{28}-2q^{26}+5q^{24}-2q^{22}+3q^{20}-3q^{18}+2q^{16}-q^{14}+q^{12}-q^{10}-q^{8}+2q^{6}-q^{4}+2q^{2}-1+3q^{-2}-q^{-4}+2q^{-6}-q^{-8}+q^{-10}$

G2 Invariants.

Weight

Invariant

1,0

$q^{100}-q^{98}+2q^{96}-2q^{94}-3q^{88}+5q^{86}-6q^{84}+4q^{82}-4q^{80}-q^{78}+5q^{76}-8q^{74}+8q^{72}-5q^{70}+q^{68}+2q^{66}-5q^{64}+4q^{62}-2q^{58}+6q^{56}-5q^{54}+2q^{52}+6q^{50}-9q^{48}+11q^{46}-9q^{44}+5q^{42}+2q^{40}-6q^{38}+10q^{36}-9q^{34}+8q^{32}-3q^{30}-3q^{28}+5q^{26}-5q^{24}+3q^{22}-3q^{18}+5q^{16}-3q^{14}-q^{12}+5q^{10}-8q^{8}+8q^{6}-5q^{4}-q^{2}+5-6q^{-2}+8q^{-4}-4q^{-6}+2q^{-8}+q^{-10}-3q^{-12}+3q^{-14}-q^{-16}+q^{-18}$

.

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 6"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t^{2}-t^{-2}+5t+5t^{-1}-7$

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]=

{ 19, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q-2+3q^{-1}-3q^{-2}+4q^{-3}-3q^{-4}+2q^{-5}-q^{-6}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-a^{6}+2a^{4}z^{2}+2a^{4}-a^{2}z^{4}-2a^{2}z^{2}-a^{2}+z^{2}+1$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

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

Vassiliev invariants

V2 and V3:

(1, -2)

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 $4$ $-16$ $8$ ${\frac {158}{3}}$ ${\frac {34}{3}}$ $-64$ $-{\frac {544}{3}}$ $-{\frac {64}{3}}$ $-48$ ${\frac {32}{3}}$ $128$ ${\frac {632}{3}}$ ${\frac {136}{3}}$ ${\frac {19471}{30}}$ $-{\frac {474}{5}}$ ${\frac {17342}{45}}$ ${\frac {401}{18}}$ ${\frac {1711}{30}}$

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 $s=$-2 is the signature of 7 6. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \	-5	-4	-3	-2	-1	0	1	2	χ
3								1	1
1							1		-1
-1						2	1		1
-3					2	2			0
-5				2	1				1
-7			1	2					1
-9		1	2						-1
-11		1							1
-13	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[7, 6]]
Out[2]=	7
In[3]:=	PD[Knot[7, 6]]
Out[3]=	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[4]:=	GaussCode[Knot[7, 6]]
Out[4]=	GaussCode[-1, 7, -2, 1, -3, 6, -7, 2, -4, 5, -6, 3, -5, 4]
In[5]:=	BR[Knot[7, 6]]
Out[5]=	BR[4, {-1, -1, 2, -1, -3, 2, -3}]
In[6]:=	alex = Alexander[Knot[7, 6]][t]
Out[6]=	-2 5 2 -7 - t + - + 5 t - t t
In[7]:=	Conway[Knot[7, 6]][z]
Out[7]=	2 4 1 + z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[7, 6], Knot[10, 133]}
In[9]:=	{KnotDet[Knot[7, 6]], KnotSignature[Knot[7, 6]]}
Out[9]=	{19, -2}
In[10]:=	J=Jones[Knot[7, 6]][q]
Out[10]=	-6 2 3 4 3 3 -2 - q + -- - -- + -- - -- + - + q 5 4 3 2 q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[7, 6]}
In[12]:=	A2Invariant[Knot[7, 6]][q]
Out[12]=	-20 -18 -16 -12 -10 -6 -4 -2 4 -q - q + q + q + q + q - q + q + q
In[13]:=	Kauffman[Knot[7, 6]][a, z]
Out[13]=	2 4 6 3 7 2 2 2 4 2 1 + a + 2 a + a + a z + 2 a z - a z - 2 z - 4 a z - 4 a z - 6 2 3 3 3 5 3 7 3 4 2 4 4 4 2 a z - 4 a z - 6 a z - a z + a z + z + a z + 2 a z + 6 4 5 3 5 5 5 2 6 4 6 2 a z + 2 a z + 4 a z + 2 a z + a z + a z
In[14]:=	{Vassiliev[2][Knot[7, 6]], Vassiliev[3][Knot[7, 6]]}
Out[14]=	{0, -2}
In[15]:=	Kh[Knot[7, 6]][q, t]
Out[15]=	2 2 1 1 1 2 1 2 2 -- + - + ------ + ------ + ----- + ----- + ----- + ----- + ----- + 3 q 13 5 11 4 9 4 9 3 7 3 7 2 5 2 q q t q t q t q t q t q t q t 1 2 t 3 2 ---- + ---- + - + q t + q t 5 3 q q t q t