8 11

8_10

8_12

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8 11 Quick Notes

8 11 Further Notes and Views

Knot presentations

Planar diagram presentation	X1425 X5,12,6,13 X3,11,4,10 X11,3,12,2 X9,16,10,1 X15,6,16,7 X7,14,8,15 X13,8,14,9
Gauss code	-1, 4, -3, 1, -2, 6, -7, 8, -5, 3, -4, 2, -8, 7, -6, 5
Dowker-Thistlethwaite code	4 10 12 14 16 2 8 6
Conway Notation	[3212]

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 2 Bridge index 2 Super bridge index $\{4,5\}$ Nakanishi index 1 Maximal Thurston-Bennequin number [-9][-1] Hyperbolic Volume 8.28632 A-Polynomial See Data:8 11/A-polynomial

[edit Notes for 8 11's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for 8 11's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-2t^{2}+7t-9+7t^{-1}-2t^{-2}$
Conway polynomial	$-2z^{4}-z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 27, -2 }
Jones polynomial	$q-2+4q^{-1}-4q^{-2}+5q^{-3}-5q^{-4}+3q^{-5}-2q^{-6}+q^{-7}$
HOMFLY-PT polynomial (db, data sources)	$z^{2}a^{6}+a^{6}-z^{4}a^{4}-2z^{2}a^{4}-2a^{4}-z^{4}a^{2}-z^{2}a^{2}+a^{2}+z^{2}+1$
Kauffman polynomial (db, data sources)	$z^{4}a^{8}-2z^{2}a^{8}+2z^{5}a^{7}-4z^{3}a^{7}+2za^{7}+2z^{6}a^{6}-3z^{4}a^{6}+2z^{2}a^{6}-a^{6}+z^{7}a^{5}+z^{5}a^{5}-3z^{3}a^{5}+3za^{5}+4z^{6}a^{4}-7z^{4}a^{4}+6z^{2}a^{4}-2a^{4}+z^{7}a^{3}+z^{5}a^{3}-2z^{3}a^{3}+za^{3}+2z^{6}a^{2}-2z^{4}a^{2}-a^{2}+2z^{5}a-3z^{3}a+z^{4}-2z^{2}+1$
The A2 invariant	$q^{22}+q^{16}-2q^{14}-q^{12}-q^{10}+2q^{6}+2q^{2}+q^{-4}$
The G2 invariant	$q^{114}-q^{112}+2q^{110}-3q^{108}+q^{106}-q^{104}-3q^{102}+7q^{100}-8q^{98}+8q^{96}-6q^{94}+2q^{92}+7q^{90}-13q^{88}+16q^{86}-13q^{84}+7q^{82}+3q^{80}-10q^{78}+13q^{76}-9q^{74}+8q^{72}+4q^{70}-10q^{68}+7q^{66}-2q^{64}-8q^{62}+14q^{60}-18q^{58}+10q^{56}-3q^{54}-9q^{52}+18q^{50}-25q^{48}+19q^{46}-14q^{44}-q^{42}+11q^{40}-17q^{38}+16q^{36}-9q^{34}+4q^{32}+6q^{30}-9q^{28}+7q^{26}-7q^{22}+14q^{20}-11q^{18}+5q^{16}+6q^{14}-11q^{12}+17q^{10}-14q^{8}+9q^{6}-2q^{4}-7q^{2}+10-9q^{-2}+8q^{-4}-3q^{-6}+q^{-8}+2q^{-10}-3q^{-12}+3q^{-14}-q^{-16}+q^{-18}$

Further Quantum Invariants

Further quantum knot invariants for 8_11.

A1 Invariants.

Weight	Invariant
1	$q^{15}-q^{13}+q^{11}-2q^{9}+q^{5}+2q-q^{-1}+q^{-3}$
2	$q^{42}-q^{40}-q^{38}+3q^{36}-2q^{34}-4q^{32}+5q^{30}-5q^{26}+6q^{24}+2q^{22}-4q^{20}+q^{16}-4q^{12}+2q^{10}+4q^{8}-5q^{6}+q^{4}+6q^{2}-4-q^{-2}+4q^{-4}-q^{-6}-q^{-8}+q^{-10}$
3	$q^{81}-q^{79}-q^{77}+q^{75}+2q^{73}-2q^{71}-5q^{69}+2q^{67}+8q^{65}-11q^{61}-2q^{59}+14q^{57}+8q^{55}-15q^{53}-11q^{51}+12q^{49}+14q^{47}-13q^{45}-15q^{43}+8q^{41}+13q^{39}-q^{37}-11q^{35}+6q^{31}+6q^{29}-4q^{27}-9q^{25}-q^{23}+14q^{21}+3q^{19}-16q^{17}-6q^{15}+16q^{13}+10q^{11}-14q^{9}-12q^{7}+11q^{5}+14q^{3}-7q-11q^{-1}+2q^{-3}+10q^{-5}+q^{-7}-6q^{-9}-q^{-11}+3q^{-13}+q^{-15}-q^{-17}-q^{-19}+q^{-21}$
4	$q^{132}-q^{130}-q^{128}+q^{126}+2q^{122}-4q^{120}-3q^{118}+4q^{116}+3q^{114}+8q^{112}-8q^{110}-13q^{108}+q^{106}+9q^{104}+25q^{102}-5q^{100}-28q^{98}-20q^{96}+3q^{94}+49q^{92}+19q^{90}-28q^{88}-47q^{86}-23q^{84}+56q^{82}+47q^{80}-7q^{78}-54q^{76}-47q^{74}+43q^{72}+53q^{70}+12q^{68}-39q^{66}-49q^{64}+15q^{62}+36q^{60}+22q^{58}-16q^{56}-33q^{54}-7q^{52}+17q^{50}+24q^{48}+6q^{46}-15q^{44}-31q^{42}-5q^{40}+30q^{38}+29q^{36}-2q^{34}-52q^{32}-25q^{30}+32q^{28}+51q^{26}+18q^{24}-60q^{22}-43q^{20}+16q^{18}+54q^{16}+40q^{14}-43q^{12}-48q^{10}-11q^{8}+35q^{6}+49q^{4}-13q^{2}-29-23q^{-2}+6q^{-4}+32q^{-6}+5q^{-8}-6q^{-10}-16q^{-12}-6q^{-14}+12q^{-16}+3q^{-18}+2q^{-20}-4q^{-22}-3q^{-24}+3q^{-26}+q^{-30}-q^{-32}-q^{-34}+q^{-36}$
5	$q^{195}-q^{193}-q^{191}+q^{189}-2q^{181}-2q^{179}+4q^{177}+6q^{175}+q^{173}-4q^{171}-9q^{169}-8q^{167}+4q^{165}+18q^{163}+18q^{161}-q^{159}-24q^{157}-35q^{155}-16q^{153}+25q^{151}+58q^{149}+43q^{147}-15q^{145}-74q^{143}-84q^{141}-17q^{139}+82q^{137}+126q^{135}+65q^{133}-65q^{131}-161q^{129}-120q^{127}+33q^{125}+172q^{123}+175q^{121}+18q^{119}-170q^{117}-209q^{115}-64q^{113}+138q^{111}+220q^{109}+109q^{107}-107q^{105}-212q^{103}-124q^{101}+62q^{99}+182q^{97}+130q^{95}-23q^{93}-146q^{91}-124q^{89}+3q^{87}+103q^{85}+101q^{83}+25q^{81}-67q^{79}-93q^{77}-33q^{75}+37q^{73}+71q^{71}+61q^{69}-5q^{67}-69q^{65}-76q^{63}-24q^{61}+61q^{59}+107q^{57}+56q^{55}-63q^{53}-136q^{51}-92q^{49}+49q^{47}+164q^{45}+130q^{43}-34q^{41}-179q^{39}-170q^{37}+5q^{35}+181q^{33}+199q^{31}+37q^{29}-161q^{27}-211q^{25}-79q^{23}+118q^{21}+208q^{19}+115q^{17}-67q^{15}-176q^{13}-131q^{11}+9q^{9}+131q^{7}+134q^{5}+28q^{3}-77q-106q^{-1}-52q^{-3}+28q^{-5}+76q^{-7}+55q^{-9}-q^{-11}-40q^{-13}-43q^{-15}-13q^{-17}+16q^{-19}+27q^{-21}+16q^{-23}-5q^{-25}-13q^{-27}-9q^{-29}+3q^{-33}+5q^{-35}+2q^{-37}-3q^{-39}-q^{-41}+q^{-43}+q^{-49}-q^{-51}-q^{-53}+q^{-55}$

A2 Invariants.

Weight	Invariant
1,0	$q^{22}+q^{16}-2q^{14}-q^{12}-q^{10}+2q^{6}+2q^{2}+q^{-4}$
1,1	$q^{60}-2q^{58}+4q^{56}-8q^{54}+15q^{52}-20q^{50}+26q^{48}-38q^{46}+43q^{44}-44q^{42}+40q^{40}-34q^{38}+20q^{36}+4q^{34}-26q^{32}+50q^{30}-66q^{28}+84q^{26}-88q^{24}+86q^{22}-79q^{20}+56q^{18}-38q^{16}+14q^{14}+6q^{12}-26q^{10}+42q^{8}-44q^{6}+44q^{4}-40q^{2}+36-22q^{-2}+17q^{-4}-10q^{-6}+6q^{-8}-2q^{-10}+q^{-12}$
2,0	$q^{56}+q^{48}-q^{46}-4q^{44}-q^{42}+2q^{40}-3q^{36}+2q^{34}+5q^{32}+3q^{30}-q^{28}+2q^{26}-4q^{22}-q^{20}-q^{18}-2q^{16}-q^{14}+3q^{12}-2q^{8}+q^{6}+4q^{4}-q^{2}-2+3q^{-2}+3q^{-4}-q^{-8}+q^{-12}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{48}-q^{46}+q^{42}-4q^{40}+4q^{36}-3q^{34}+2q^{32}+7q^{30}-2q^{28}+q^{24}-3q^{22}-4q^{20}-3q^{18}+q^{16}-q^{14}-2q^{12}+5q^{10}+2q^{8}-4q^{6}+5q^{4}+2q^{2}-2+3q^{-2}+q^{-4}-q^{-6}+q^{-8}$
1,0,0	$q^{29}+q^{25}+q^{21}-2q^{19}-q^{17}-2q^{15}-q^{13}+q^{9}+2q^{7}+2q^{3}+q^{-1}+q^{-5}$

A4 Invariants.

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

B2 Invariants.

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

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{66}-q^{64}+q^{62}-2q^{60}+2q^{58}-4q^{56}+2q^{54}-3q^{52}+4q^{50}-2q^{48}+3q^{46}+q^{44}+4q^{42}+3q^{40}-2q^{38}+3q^{36}-5q^{34}+4q^{32}-9q^{30}+2q^{28}-9q^{26}+4q^{24}-5q^{22}+3q^{20}-3q^{18}+3q^{16}+2q^{14}+q^{12}+2q^{10}-2q^{8}+5q^{6}-2q^{4}+4q^{2}-2+4q^{-2}-q^{-4}+2q^{-6}-q^{-8}+q^{-10}$

G2 Invariants.

Weight

Invariant

1,0

$q^{114}-q^{112}+2q^{110}-3q^{108}+q^{106}-q^{104}-3q^{102}+7q^{100}-8q^{98}+8q^{96}-6q^{94}+2q^{92}+7q^{90}-13q^{88}+16q^{86}-13q^{84}+7q^{82}+3q^{80}-10q^{78}+13q^{76}-9q^{74}+8q^{72}+4q^{70}-10q^{68}+7q^{66}-2q^{64}-8q^{62}+14q^{60}-18q^{58}+10q^{56}-3q^{54}-9q^{52}+18q^{50}-25q^{48}+19q^{46}-14q^{44}-q^{42}+11q^{40}-17q^{38}+16q^{36}-9q^{34}+4q^{32}+6q^{30}-9q^{28}+7q^{26}-7q^{22}+14q^{20}-11q^{18}+5q^{16}+6q^{14}-11q^{12}+17q^{10}-14q^{8}+9q^{6}-2q^{4}-7q^{2}+10-9q^{-2}+8q^{-4}-3q^{-6}+q^{-8}+2q^{-10}-3q^{-12}+3q^{-14}-q^{-16}+q^{-18}$

.

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["8 11"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-2t^{2}+7t-9+7t^{-1}-2t^{-2}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 27, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

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

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 $-4$ $16$ $8$ $-{\frac {14}{3}}$ ${\frac {62}{3}}$ $-64$ $-{\frac {416}{3}}$ $-{\frac {128}{3}}$ $-80$ $-{\frac {32}{3}}$ $128$ ${\frac {56}{3}}$ $-{\frac {248}{3}}$ ${\frac {14849}{30}}$ $-{\frac {578}{15}}$ ${\frac {11698}{45}}$ ${\frac {1567}{18}}$ $-{\frac {31}{30}}$

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 8 11. 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								1		-1
-1							3	1		2
-3						2	2			0
-5					3	2				1
-7				2	2					0
-9			1	3						-2
-11		1	2							1
-13		1								-1
-15	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[8, 11]]
Out[2]=	8
In[3]:=	PD[Knot[8, 11]]
Out[3]=	PD[X[1, 4, 2, 5], X[5, 12, 6, 13], X[3, 11, 4, 10], X[11, 3, 12, 2], X[9, 16, 10, 1], X[15, 6, 16, 7], X[7, 14, 8, 15], X[13, 8, 14, 9]]
In[4]:=	GaussCode[Knot[8, 11]]
Out[4]=	GaussCode[-1, 4, -3, 1, -2, 6, -7, 8, -5, 3, -4, 2, -8, 7, -6, 5]
In[5]:=	BR[Knot[8, 11]]
Out[5]=	BR[4, {-1, -1, -2, 1, -2, -2, 3, -2, 3}]
In[6]:=	alex = Alexander[Knot[8, 11]][t]
Out[6]=	2 7 2 -9 - -- + - + 7 t - 2 t 2 t t
In[7]:=	Conway[Knot[8, 11]][z]
Out[7]=	2 4 1 - z - 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[8, 11], Knot[10, 147], Knot[11, NonAlternating, 122]}
In[9]:=	{KnotDet[Knot[8, 11]], KnotSignature[Knot[8, 11]]}
Out[9]=	{27, -2}
In[10]:=	J=Jones[Knot[8, 11]][q]
Out[10]=	-7 2 3 5 5 4 4 -2 + 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, 11]}
In[12]:=	A2Invariant[Knot[8, 11]][q]
Out[12]=	-22 -16 2 -12 -10 2 2 4 q + q - --- - q - q + -- + -- + q 14 6 2 q q q
In[13]:=	Kauffman[Knot[8, 11]][a, z]
Out[13]=	2 4 6 3 5 7 2 4 2 1 - a - 2 a - a + a z + 3 a z + 2 a z - 2 z + 6 a z + 6 2 8 2 3 3 3 5 3 7 3 4 2 a z - 2 a z - 3 a z - 2 a z - 3 a z - 4 a z + z - 2 4 4 4 6 4 8 4 5 3 5 5 5 2 a z - 7 a z - 3 a z + a z + 2 a z + a z + a z + 7 5 2 6 4 6 6 6 3 7 5 7 2 a z + 2 a z + 4 a z + 2 a z + a z + a z
In[14]:=	{Vassiliev[2][Knot[8, 11]], Vassiliev[3][Knot[8, 11]]}
Out[14]=	{0, 2}
In[15]:=	Kh[Knot[8, 11]][q, t]
Out[15]=	2 3 1 1 1 2 1 3 2 -- + - + ------ + ------ + ------ + ------ + ----- + ----- + ----- + 3 q 15 6 13 5 11 5 11 4 9 4 9 3 7 3 q q t q t q t q t q t q t q t 2 3 2 2 t 3 2 ----- + ----- + ---- + ---- + - + q t + q t 7 2 5 2 5 3 q q t q t q t q t