8 6

8_5

8_7

Visit 8 6's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 8 6's page at Knotilus! Visit 8 6's page at the original Knot Atlas! 8 6 Quick Notes

8 6 Further Notes and Views

Knot presentations

Planar diagram presentation	X1425 X9,12,10,13 X3,11,4,10 X11,3,12,2 X5,14,6,15 X7,16,8,1 X15,6,16,7 X13,8,14,9
Gauss code	-1, 4, -3, 1, -5, 7, -6, 8, -2, 3, -4, 2, -8, 5, -7, 6
Dowker-Thistlethwaite code	4 10 14 16 12 2 8 6
Conway Notation	[332]

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 2 Bridge index 2 Super bridge index $\{4,6\}$ Nakanishi index 1 Maximal Thurston-Bennequin number [-9][-1] Hyperbolic Volume 7.47524 A-Polynomial See Data:8 6/A-polynomial

[edit Notes for 8 6's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for 8 6's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-2t^{2}+6t-7+6t^{-1}-2t^{-2}$
Conway polynomial	$-2z^{4}-2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 23, -2 }
Jones polynomial	$q-1+3q^{-1}-4q^{-2}+4q^{-3}-4q^{-4}+3q^{-5}-2q^{-6}+q^{-7}$
HOMFLY-PT polynomial (db, data sources)	$z^{2}a^{6}+a^{6}-z^{4}a^{4}-2z^{2}a^{4}-a^{4}-z^{4}a^{2}-2z^{2}a^{2}-a^{2}+z^{2}+2$
Kauffman polynomial (db, data sources)	$z^{4}a^{8}-2z^{2}a^{8}+2z^{5}a^{7}-4z^{3}a^{7}+za^{7}+2z^{6}a^{6}-4z^{4}a^{6}+3z^{2}a^{6}-a^{6}+z^{7}a^{5}-z^{5}a^{5}+2z^{3}a^{5}-za^{5}+3z^{6}a^{4}-6z^{4}a^{4}+6z^{2}a^{4}-a^{4}+z^{7}a^{3}-2z^{5}a^{3}+5z^{3}a^{3}-3za^{3}+z^{6}a^{2}-2z^{2}a^{2}+a^{2}+z^{5}a-z^{3}a-za+z^{4}-3z^{2}+2$
The A2 invariant	$q^{22}+q^{16}-q^{14}-q^{10}-q^{8}-q^{4}+2q^{2}+1+q^{-2}+q^{-4}$
The G2 invariant	$q^{114}-q^{112}+2q^{110}-3q^{108}+q^{106}-3q^{102}+6q^{100}-7q^{98}+7q^{96}-4q^{94}-2q^{92}+7q^{90}-9q^{88}+11q^{86}-6q^{84}+q^{82}+5q^{80}-7q^{78}+7q^{76}-2q^{74}-3q^{72}+6q^{70}-5q^{68}+2q^{66}+4q^{64}-9q^{62}+12q^{60}-10q^{58}+5q^{56}+q^{54}-10q^{52}+13q^{50}-13q^{48}+10q^{46}-5q^{44}-3q^{42}+7q^{40}-10q^{38}+6q^{36}-3q^{34}-4q^{32}+5q^{30}-5q^{28}-q^{26}+6q^{24}-8q^{22}+8q^{20}-6q^{18}-q^{16}+6q^{14}-8q^{12}+10q^{10}-5q^{8}+3q^{6}+2q^{4}-2q^{2}+4-3q^{-2}+4q^{-4}-q^{-6}+q^{-8}+q^{-10}-q^{-12}+2q^{-14}+q^{-18}$

Further Quantum Invariants

Further quantum knot invariants for 8_6.

A1 Invariants.

Weight	Invariant
1	$q^{15}-q^{13}+q^{11}-q^{9}-q^{3}+2q+q^{-3}$
2	$q^{42}-q^{40}-q^{38}+3q^{36}-q^{34}-4q^{32}+3q^{30}+q^{28}-4q^{26}+3q^{24}+2q^{22}-2q^{20}+q^{16}+q^{14}-3q^{12}+q^{10}+4q^{8}-4q^{6}-q^{4}+4q^{2}-2-q^{-2}+2q^{-4}+q^{-10}$
3	$q^{81}-q^{79}-q^{77}+q^{75}+2q^{73}-q^{71}-5q^{69}+7q^{65}+3q^{63}-7q^{61}-7q^{59}+6q^{57}+10q^{55}-5q^{53}-10q^{51}+3q^{49}+12q^{47}-q^{45}-10q^{43}-q^{41}+7q^{39}+q^{37}-5q^{35}-3q^{33}+q^{31}+5q^{29}+q^{27}-6q^{25}-4q^{23}+8q^{21}+7q^{19}-7q^{17}-10q^{15}+8q^{13}+10q^{11}-3q^{9}-10q^{7}+q^{5}+9q^{3}+q-5q^{-1}-2q^{-3}+3q^{-5}+q^{-7}-q^{-9}-q^{-11}+q^{-13}+q^{-21}$
4	$q^{132}-q^{130}-q^{128}+q^{126}+2q^{122}-3q^{120}-3q^{118}+2q^{116}+2q^{114}+9q^{112}-3q^{110}-11q^{108}-6q^{106}-q^{104}+20q^{102}+10q^{100}-7q^{98}-19q^{96}-19q^{94}+20q^{92}+26q^{90}+9q^{88}-21q^{86}-36q^{84}+6q^{82}+28q^{80}+24q^{78}-11q^{76}-39q^{74}-5q^{72}+20q^{70}+25q^{68}-q^{66}-26q^{64}-9q^{62}+8q^{60}+17q^{58}+5q^{56}-10q^{54}-10q^{52}-2q^{50}+9q^{48}+12q^{46}+5q^{44}-16q^{42}-15q^{40}+3q^{38}+19q^{36}+19q^{34}-20q^{32}-28q^{30}-8q^{28}+24q^{26}+36q^{24}-13q^{22}-30q^{20}-19q^{18}+14q^{16}+38q^{14}+4q^{12}-16q^{10}-23q^{8}-4q^{6}+23q^{4}+10q^{2}-12q^{-2}-9q^{-4}+6q^{-6}+4q^{-8}+5q^{-10}-2q^{-12}-4q^{-14}+q^{-16}-q^{-18}+2q^{-20}-q^{-24}+q^{-26}-q^{-28}+q^{-36}$
5	$q^{195}-q^{193}-q^{191}+q^{189}-q^{181}-2q^{179}+2q^{177}+5q^{175}+2q^{173}-q^{171}-6q^{169}-9q^{167}-5q^{165}+8q^{163}+17q^{161}+13q^{159}-20q^{155}-28q^{153}-17q^{151}+16q^{149}+42q^{147}+37q^{145}+3q^{143}-45q^{141}-63q^{139}-29q^{137}+37q^{135}+81q^{133}+58q^{131}-18q^{129}-87q^{127}-86q^{125}-11q^{123}+82q^{121}+106q^{119}+33q^{117}-65q^{115}-108q^{113}-54q^{111}+47q^{109}+105q^{107}+63q^{105}-30q^{103}-86q^{101}-64q^{99}+13q^{97}+69q^{95}+56q^{93}-2q^{91}-50q^{89}-47q^{87}-6q^{85}+30q^{83}+38q^{81}+12q^{79}-18q^{77}-30q^{75}-21q^{73}+4q^{71}+28q^{69}+32q^{67}+8q^{65}-27q^{63}-44q^{61}-23q^{59}+29q^{57}+61q^{55}+40q^{53}-24q^{51}-78q^{49}-61q^{47}+20q^{45}+90q^{43}+80q^{41}-93q^{37}-105q^{35}-14q^{33}+81q^{31}+106q^{29}+41q^{27}-59q^{25}-106q^{23}-60q^{21}+32q^{19}+85q^{17}+68q^{15}-58q^{11}-64q^{9}-19q^{7}+34q^{5}+49q^{3}+28q-7q^{-1}-31q^{-3}-26q^{-5}-4q^{-7}+14q^{-9}+18q^{-11}+10q^{-13}-5q^{-15}-10q^{-17}-7q^{-19}-2q^{-21}+5q^{-23}+6q^{-25}+q^{-27}-q^{-29}-q^{-31}-3q^{-33}+2q^{-37}+q^{-43}-q^{-45}-q^{-47}+q^{-55}$

A2 Invariants.

Weight	Invariant
1,0	$q^{22}+q^{16}-q^{14}-q^{10}-q^{8}-q^{4}+2q^{2}+1+q^{-2}+q^{-4}$
1,1	$q^{60}-2q^{58}+4q^{56}-8q^{54}+13q^{52}-16q^{50}+22q^{48}-26q^{46}+25q^{44}-24q^{42}+16q^{40}-8q^{38}-5q^{36}+18q^{34}-30q^{32}+40q^{30}-46q^{28}+50q^{26}-44q^{24}+42q^{22}-28q^{20}+18q^{18}-4q^{16}-8q^{14}+15q^{12}-24q^{10}+22q^{8}-24q^{6}+18q^{4}-16q^{2}+12-6q^{-2}+8q^{-4}-2q^{-6}+4q^{-8}+q^{-12}$
2,0	$q^{56}+q^{48}-3q^{44}-q^{42}+q^{40}-2q^{36}+2q^{32}-q^{28}+2q^{26}+2q^{24}+2q^{20}+2q^{18}-q^{16}+q^{12}-q^{10}-4q^{8}-2q^{6}+q^{4}-2q^{2}-1+2q^{-2}+3q^{-4}+q^{-6}+q^{-8}+q^{-10}+q^{-12}$
3,0	$q^{102}+q^{92}-2q^{90}-2q^{88}-3q^{86}+q^{84}+5q^{82}+2q^{80}-2q^{78}-7q^{76}-q^{74}+6q^{72}+5q^{70}-2q^{68}-8q^{66}+10q^{62}+8q^{60}-q^{58}-7q^{56}+6q^{52}+q^{50}-7q^{48}-7q^{46}+q^{42}-3q^{40}-4q^{38}+q^{36}+3q^{34}-2q^{32}-2q^{30}+2q^{28}+7q^{26}+3q^{24}-6q^{22}-q^{20}+7q^{18}+12q^{16}+q^{14}-8q^{12}-2q^{10}+6q^{8}+7q^{6}-5q^{4}-10q^{2}-5+2q^{-2}+3q^{-4}-q^{-6}-3q^{-8}+2q^{-12}+3q^{-14}+q^{-16}+q^{-18}+q^{-20}+q^{-22}+q^{-24}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{48}-q^{46}+q^{42}-3q^{40}+q^{38}+3q^{36}-3q^{34}+q^{32}+3q^{30}-2q^{28}+2q^{24}+q^{22}+q^{16}-2q^{14}-5q^{12}+q^{10}-2q^{8}-4q^{6}+4q^{4}+2q^{2}+1+3q^{-2}+2q^{-4}+q^{-8}$
1,0,0	$q^{29}+q^{25}+q^{21}-q^{19}-q^{15}-q^{13}-q^{11}-q^{9}-q^{5}+2q^{3}+q+2q^{-1}+q^{-3}+q^{-5}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{62}-q^{58}+q^{56}+q^{54}-2q^{52}-q^{50}+2q^{48}+q^{46}-3q^{44}-q^{42}+2q^{40}-q^{38}-2q^{36}+3q^{34}+3q^{32}+4q^{28}+4q^{26}+q^{24}-q^{22}+q^{20}-2q^{18}-7q^{16}-5q^{14}-2q^{12}-4q^{10}-4q^{8}+2q^{6}+3q^{4}+3q^{2}+3+4q^{-2}+3q^{-4}+2q^{-6}+q^{-8}+q^{-10}$
1,0,0,0	$q^{36}+q^{32}+q^{30}+q^{26}-q^{24}-q^{20}-q^{18}-q^{16}-q^{14}-q^{12}-q^{10}-q^{6}+2q^{4}+q^{2}+2+2q^{-2}+q^{-4}+q^{-6}$

B2 Invariants.

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

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{66}-q^{64}+q^{62}-2q^{60}+2q^{58}-3q^{56}+2q^{54}-2q^{52}+3q^{50}-q^{48}+q^{46}+q^{44}+q^{42}+2q^{40}-3q^{38}+3q^{36}-3q^{34}+5q^{32}-4q^{30}+4q^{28}-4q^{26}+4q^{24}-3q^{22}-4q^{18}-2q^{16}-q^{14}-3q^{12}-3q^{8}+4q^{6}+4q^{2}+1+4q^{-2}+q^{-4}+2q^{-6}+q^{-10}$

G2 Invariants.

Weight

Invariant

1,0

$q^{114}-q^{112}+2q^{110}-3q^{108}+q^{106}-3q^{102}+6q^{100}-7q^{98}+7q^{96}-4q^{94}-2q^{92}+7q^{90}-9q^{88}+11q^{86}-6q^{84}+q^{82}+5q^{80}-7q^{78}+7q^{76}-2q^{74}-3q^{72}+6q^{70}-5q^{68}+2q^{66}+4q^{64}-9q^{62}+12q^{60}-10q^{58}+5q^{56}+q^{54}-10q^{52}+13q^{50}-13q^{48}+10q^{46}-5q^{44}-3q^{42}+7q^{40}-10q^{38}+6q^{36}-3q^{34}-4q^{32}+5q^{30}-5q^{28}-q^{26}+6q^{24}-8q^{22}+8q^{20}-6q^{18}-q^{16}+6q^{14}-8q^{12}+10q^{10}-5q^{8}+3q^{6}+2q^{4}-2q^{2}+4-3q^{-2}+4q^{-4}-q^{-6}+q^{-8}+q^{-10}-q^{-12}+2q^{-14}+q^{-18}$

.

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["8 6"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-2t^{2}+6t-7+6t^{-1}-2t^{-2}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 23, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q-1+3q^{-1}-4q^{-2}+4q^{-3}-4q^{-4}+3q^{-5}-2q^{-6}+q^{-7}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

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

Vassiliev invariants

V2 and V3:

(-2, 3)

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 $-8$ $24$ $32$ ${\frac {68}{3}}$ ${\frac {100}{3}}$ $-192$ $-336$ $-96$ $-104$ $-{\frac {256}{3}}$ $288$ $-{\frac {544}{3}}$ $-{\frac {800}{3}}$ ${\frac {10529}{15}}$ ${\frac {2804}{15}}$ $-{\frac {964}{45}}$ ${\frac {1231}{9}}$ $-{\frac {1471}{15}}$

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 8 6. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -6 -5 -4 -3 -2 -1 0 1 2 χ 3                 1 1 1                   0 -1             3 1   2 -3           2 1     -1 -5         2 2       0 -7       2 2         0 -9     1 2           -1 -11   1 2             1 -13   1               -1 -15 1                 1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-3$ $i=-1$ $r=-6$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-2$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-1$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=0$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{3}$ $r=1$ ${\mathbb {Z} }$ $r=2$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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[8, 6]]
Out[2]=	8
In[3]:=	PD[Knot[8, 6]]
Out[3]=	PD[X[1, 4, 2, 5], X[9, 12, 10, 13], X[3, 11, 4, 10], X[11, 3, 12, 2], X[5, 14, 6, 15], X[7, 16, 8, 1], X[15, 6, 16, 7], X[13, 8, 14, 9]]
In[4]:=	GaussCode[Knot[8, 6]]
Out[4]=	GaussCode[-1, 4, -3, 1, -5, 7, -6, 8, -2, 3, -4, 2, -8, 5, -7, 6]
In[5]:=	BR[Knot[8, 6]]
Out[5]=	BR[4, {-1, -1, -1, -1, -2, 1, 3, -2, 3}]
In[6]:=	alex = Alexander[Knot[8, 6]][t]
Out[6]=	2 6 2 -7 - -- + - + 6 t - 2 t 2 t t
In[7]:=	Conway[Knot[8, 6]][z]
Out[7]=	2 4 1 - 2 z - 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[8, 6], Knot[11, NonAlternating, 20], Knot[11, NonAlternating, 151], Knot[11, NonAlternating, 152]}
In[9]:=	{KnotDet[Knot[8, 6]], KnotSignature[Knot[8, 6]]}
Out[9]=	{23, -2}
In[10]:=	J=Jones[Knot[8, 6]][q]
Out[10]=	-7 2 3 4 4 4 3 -1 + q - -- + -- - -- + -- - -- + - + q 6 5 4 3 2 q q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[8, 6]}
In[12]:=	A2Invariant[Knot[8, 6]][q]
Out[12]=	-22 -16 -14 -10 -8 -4 2 2 4 1 + q + q - q - q - q - q + -- + q + q 2 q
In[13]:=	Kauffman[Knot[8, 6]][a, z]
Out[13]=	2 4 6 3 5 7 2 2 2 2 + a - a - a - a z - 3 a z - a z + a z - 3 z - 2 a z + 4 2 6 2 8 2 3 3 3 5 3 7 3 6 a z + 3 a z - 2 a z - a z + 5 a z + 2 a z - 4 a z + 4 4 4 6 4 8 4 5 3 5 5 5 7 5 z - 6 a z - 4 a z + a z + a z - 2 a z - a z + 2 a z + 2 6 4 6 6 6 3 7 5 7 a z + 3 a z + 2 a z + a z + a z
In[14]:=	{Vassiliev[2][Knot[8, 6]], Vassiliev[3][Knot[8, 6]]}
Out[14]=	{0, 3}
In[15]:=	Kh[Knot[8, 6]][q, t]
Out[15]=	-3 3 1 1 1 2 1 2 2 q + - + ------ + ------ + ------ + ------ + ----- + ----- + ----- + q 15 6 13 5 11 5 11 4 9 4 9 3 7 3 q t q t q t q t q t q t q t 2 2 2 2 t 3 2 ----- + ----- + ---- + ---- + - + q t 7 2 5 2 5 3 q q t q t q t q t