8 2

8_1

8_3

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8 2 Quick Notes

8 2 Further Notes and Views

Knot presentations

Planar diagram presentation	X1425 X5,12,6,13 X3,11,4,10 X11,3,12,2 X7,14,8,15 X9,16,10,1 X13,6,14,7 X15,8,16,9
Gauss code	-1, 4, -3, 1, -2, 7, -5, 8, -6, 3, -4, 2, -7, 5, -8, 6
Dowker-Thistlethwaite code	4 10 12 14 16 2 6 8
Conway Notation	[512]

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 3 Bridge index 2 Super bridge index $\{4,5\}$ Nakanishi index 1 Maximal Thurston-Bennequin number [-11][1] Hyperbolic Volume 4.93524 A-Polynomial See Data:8 2/A-polynomial

[edit Notes for 8 2's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus $2$ Topological 4 genus $2$ Concordance genus $3$ Rasmussen s-Invariant -4

[edit Notes for 8 2's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{3}+3t^{2}-3t+3-3t^{-1}+3t^{-2}-t^{-3}$
Conway polynomial	$-z^{6}-3z^{4}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 17, -4 }
Jones polynomial	$1-q^{-1}+2q^{-2}-2q^{-3}+3q^{-4}-3q^{-5}+2q^{-6}-2q^{-7}+q^{-8}$
HOMFLY-PT polynomial (db, data sources)	$z^{4}a^{6}+3z^{2}a^{6}+a^{6}-z^{6}a^{4}-5z^{4}a^{4}-7z^{2}a^{4}-3a^{4}+z^{4}a^{2}+4z^{2}a^{2}+3a^{2}$
Kauffman polynomial (db, data sources)	$z^{2}a^{10}+2z^{3}a^{9}-za^{9}+2z^{4}a^{8}-z^{2}a^{8}+2z^{5}a^{7}-2z^{3}a^{7}-za^{7}+2z^{6}a^{6}-5z^{4}a^{6}+3z^{2}a^{6}-a^{6}+z^{7}a^{5}-2z^{5}a^{5}-z^{3}a^{5}+za^{5}+3z^{6}a^{4}-12z^{4}a^{4}+12z^{2}a^{4}-3a^{4}+z^{7}a^{3}-4z^{5}a^{3}+3z^{3}a^{3}+za^{3}+z^{6}a^{2}-5z^{4}a^{2}+7z^{2}a^{2}-3a^{2}$
The A2 invariant	$q^{24}-q^{18}-q^{16}-q^{12}+q^{10}+q^{6}+q^{4}+q^{2}+1$
The G2 invariant	$q^{134}-q^{132}+q^{130}-q^{128}-q^{126}-q^{122}+2q^{120}-2q^{118}+q^{116}+q^{110}+q^{106}-q^{104}+q^{102}+q^{96}+2q^{92}-q^{84}+q^{82}-2q^{78}+q^{76}-q^{74}+q^{70}-4q^{68}+2q^{66}-3q^{64}-3q^{58}+3q^{56}-2q^{54}+q^{52}-q^{50}-q^{48}+q^{46}-2q^{44}+q^{42}-q^{38}+q^{36}+q^{32}+2q^{30}-2q^{28}+3q^{26}-q^{24}+q^{22}+3q^{20}-3q^{18}+4q^{16}+q^{12}+q^{10}-q^{8}+2q^{6}+q^{2}$

Further Quantum Invariants

Further quantum knot invariants for 8_2.

A1 Invariants.

Weight	Invariant
1	$q^{17}-q^{15}-q^{11}+q^{7}+q^{3}+q^{-1}$
2	$q^{46}-q^{44}-q^{42}+q^{40}-q^{38}+q^{36}+q^{34}-q^{32}+q^{28}-q^{24}-q^{18}-q^{16}+q^{14}+2q^{8}+q^{6}-q^{4}+q^{2}+1-q^{-2}+q^{-6}$
3	$q^{87}-q^{85}-q^{83}+q^{79}+q^{77}-q^{75}-q^{71}+q^{69}+q^{67}-2q^{63}+2q^{59}-q^{55}-q^{53}+q^{49}+2q^{47}-q^{45}-q^{43}+2q^{39}-q^{37}-2q^{35}+q^{31}-q^{29}-q^{27}+q^{23}+q^{21}-q^{19}-q^{17}+q^{15}+2q^{13}+q^{11}-q^{9}-q^{7}+2q^{5}+2q^{3}-2q^{-1}+2q^{-5}+q^{-7}-q^{-9}-q^{-11}+q^{-15}$
4	$q^{140}-q^{138}-q^{136}+3q^{130}-q^{128}-q^{126}-2q^{124}-q^{122}+5q^{120}+q^{118}-q^{116}-4q^{114}-2q^{112}+5q^{110}+4q^{108}-q^{106}-6q^{104}-4q^{102}+4q^{100}+5q^{98}+2q^{96}-4q^{94}-6q^{92}+3q^{88}+4q^{86}-2q^{82}-q^{80}-2q^{78}+q^{76}+4q^{74}+2q^{72}-q^{70}-3q^{68}-q^{66}+3q^{64}+2q^{62}-2q^{60}-3q^{58}-q^{56}+3q^{54}+2q^{52}-3q^{50}-3q^{48}+4q^{44}+3q^{42}-3q^{40}-4q^{38}-2q^{36}+2q^{34}+3q^{32}-q^{30}-q^{28}-2q^{26}+3q^{22}+2q^{20}+2q^{18}-q^{16}-2q^{14}+4q^{8}+2q^{6}-q^{4}-2q^{2}-3+2q^{-2}+3q^{-4}+2q^{-6}-4q^{-10}-q^{-12}+q^{-14}+2q^{-16}+2q^{-18}-q^{-20}-q^{-22}-q^{-24}+q^{-28}$
5	$q^{205}-q^{203}-q^{201}+2q^{195}+q^{193}-q^{191}-3q^{189}-q^{187}+q^{185}+4q^{183}+3q^{181}-q^{179}-5q^{177}-5q^{175}+3q^{173}+6q^{171}+4q^{169}-2q^{167}-9q^{165}-7q^{163}+5q^{161}+11q^{159}+5q^{157}-5q^{155}-11q^{153}-9q^{151}+5q^{149}+14q^{147}+10q^{145}-2q^{143}-12q^{141}-10q^{139}-q^{137}+9q^{135}+12q^{133}+5q^{131}-5q^{129}-9q^{127}-8q^{125}-2q^{123}+4q^{121}+5q^{119}+4q^{117}-5q^{113}-5q^{111}-3q^{109}+2q^{107}+6q^{105}+6q^{103}-q^{101}-5q^{99}-3q^{97}+q^{95}+5q^{93}+4q^{91}-3q^{89}-5q^{87}-q^{85}+3q^{83}+6q^{81}+3q^{79}-4q^{77}-8q^{75}-3q^{73}+4q^{71}+8q^{69}+5q^{67}-3q^{65}-10q^{63}-7q^{61}+q^{59}+8q^{57}+7q^{55}-8q^{51}-9q^{49}-3q^{47}+7q^{45}+9q^{43}+4q^{41}-3q^{39}-8q^{37}-5q^{35}+5q^{31}+6q^{29}+2q^{27}-q^{25}-3q^{23}-3q^{21}-q^{19}+2q^{17}+4q^{15}+3q^{13}+3q^{11}-3q^{7}-4q^{5}-2q^{3}+q+4q^{-1}+5q^{-3}+2q^{-5}-2q^{-7}-5q^{-9}-4q^{-11}+3q^{-15}+5q^{-17}+3q^{-19}-q^{-21}-4q^{-23}-3q^{-25}-q^{-27}+q^{-29}+3q^{-31}+2q^{-33}-q^{-37}-q^{-39}-q^{-41}+q^{-45}$
6	$q^{282}-q^{280}-q^{278}+2q^{272}+q^{268}-3q^{266}-2q^{264}+q^{262}+q^{260}+4q^{258}+2q^{256}+q^{254}-7q^{252}-4q^{250}+2q^{246}+6q^{244}+3q^{242}-q^{240}-9q^{238}-3q^{236}+2q^{234}+4q^{232}+6q^{230}-6q^{226}-10q^{224}+2q^{222}+11q^{220}+10q^{218}+6q^{216}-8q^{214}-17q^{212}-14q^{210}+4q^{208}+19q^{206}+19q^{204}+8q^{202}-10q^{200}-24q^{198}-24q^{196}-5q^{194}+14q^{192}+24q^{190}+20q^{188}+4q^{186}-14q^{184}-24q^{182}-15q^{180}-q^{178}+11q^{176}+18q^{174}+15q^{172}+5q^{170}-7q^{168}-11q^{166}-11q^{164}-7q^{162}+q^{160}+8q^{158}+12q^{156}+8q^{154}-8q^{150}-12q^{148}-8q^{146}+9q^{142}+9q^{140}+q^{138}-6q^{136}-8q^{134}-4q^{132}+3q^{130}+10q^{128}+5q^{126}-5q^{124}-9q^{122}-7q^{120}+q^{118}+9q^{116}+14q^{114}+8q^{112}-8q^{110}-14q^{108}-10q^{106}+11q^{102}+16q^{100}+11q^{98}-5q^{96}-15q^{94}-14q^{92}-7q^{90}+6q^{88}+16q^{86}+16q^{84}+2q^{82}-12q^{80}-17q^{78}-15q^{76}-2q^{74}+13q^{72}+21q^{70}+13q^{68}-2q^{66}-14q^{64}-21q^{62}-14q^{60}+2q^{58}+17q^{56}+19q^{54}+10q^{52}-3q^{50}-17q^{48}-19q^{46}-10q^{44}+6q^{42}+14q^{40}+15q^{38}+10q^{36}-3q^{34}-10q^{32}-12q^{30}-5q^{28}+q^{26}+6q^{24}+10q^{22}+6q^{20}+3q^{18}-3q^{16}-3q^{14}-5q^{12}-5q^{10}-q^{8}+q^{6}+5q^{4}+4q^{2}+7+3q^{-2}-3q^{-4}-5q^{-6}-7q^{-8}-4q^{-10}-2q^{-12}+6q^{-14}+8q^{-16}+6q^{-18}+2q^{-20}-3q^{-22}-6q^{-24}-9q^{-26}-2q^{-28}+2q^{-30}+5q^{-32}+6q^{-34}+4q^{-36}+q^{-38}-5q^{-40}-4q^{-42}-3q^{-44}-q^{-46}+q^{-48}+3q^{-50}+3q^{-52}-q^{-58}-q^{-60}-q^{-62}+q^{-66}$

A2 Invariants.

Weight	Invariant
1,0	$q^{24}-q^{18}-q^{16}-q^{12}+q^{10}+q^{6}+q^{4}+q^{2}+1$
1,1	$q^{68}-2q^{66}+2q^{64}-2q^{62}+3q^{60}-4q^{58}+2q^{56}+2q^{52}-2q^{50}-2q^{46}+q^{44}-2q^{42}+2q^{40}+2q^{38}+6q^{34}-6q^{32}+6q^{30}-11q^{28}+8q^{26}-10q^{24}+6q^{22}-6q^{20}+4q^{18}-2q^{14}+5q^{12}-4q^{10}+8q^{8}-6q^{6}+6q^{4}-2q^{2}+4+q^{-4}$
2,0	$q^{60}-q^{54}-q^{52}-q^{44}+q^{42}+q^{40}+q^{38}+2q^{34}+q^{32}-q^{30}-q^{28}-2q^{26}-2q^{24}-2q^{22}+q^{16}+2q^{14}+2q^{12}+q^{6}+q^{4}+1+q^{-2}+q^{-4}$
3,0	$q^{108}-q^{102}-q^{100}+q^{94}-2q^{92}-q^{90}+3q^{86}+q^{84}-q^{82}+2q^{78}+4q^{76}-3q^{72}-3q^{70}+q^{66}-3q^{62}-q^{60}+2q^{56}+q^{50}+q^{48}-q^{44}+q^{42}-q^{38}-2q^{36}+2q^{32}+q^{30}-2q^{28}-3q^{26}-q^{24}+2q^{22}+q^{20}-q^{16}+2q^{14}+4q^{12}+3q^{10}-q^{6}+2q^{2}+1-q^{-4}+q^{-8}+q^{-10}+q^{-12}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{56}-q^{54}-q^{52}+q^{50}-q^{48}-q^{46}+q^{44}+q^{42}+2q^{38}+q^{36}-2q^{26}-q^{24}-q^{22}-3q^{20}-2q^{18}+2q^{12}+3q^{10}+2q^{8}+2q^{6}+2q^{4}+1$
1,0,0	$q^{31}+q^{27}-q^{25}-q^{21}-q^{19}-q^{17}-q^{15}+2q^{9}+q^{7}+2q^{5}+q^{3}+q$
1,0,1	$q^{90}-2q^{88}+q^{86}+q^{84}-2q^{82}+3q^{80}-2q^{78}-q^{76}+q^{74}+q^{70}-q^{68}+q^{62}-2q^{60}+2q^{58}-3q^{56}-2q^{54}+3q^{52}-4q^{50}+5q^{48}-q^{46}+4q^{44}+q^{40}+3q^{38}-2q^{36}+4q^{34}-5q^{32}+2q^{30}-5q^{28}-q^{26}-5q^{24}-4q^{22}-q^{20}-3q^{18}+4q^{16}-2q^{14}+6q^{12}+2q^{10}+5q^{8}+6q^{6}+2q^{4}+4q^{2}+q^{-2}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{70}-q^{66}-2q^{60}-2q^{58}+q^{52}+2q^{50}+4q^{48}+2q^{46}+2q^{44}+2q^{42}-q^{38}-q^{34}-4q^{32}-3q^{30}-4q^{28}-4q^{26}-4q^{24}-2q^{22}+2q^{18}+3q^{16}+4q^{14}+4q^{12}+4q^{10}+3q^{8}+2q^{6}+q^{4}+q^{2}$
1,0,0,0	$q^{38}+q^{34}-q^{26}-q^{24}-2q^{22}-q^{20}-2q^{18}+2q^{12}+2q^{10}+2q^{8}+2q^{6}+q^{4}+q^{2}$

B2 Invariants.

Weight	Invariant
0,1	$q^{56}-q^{54}+q^{52}-q^{50}+q^{48}-q^{46}+q^{44}-q^{42}-q^{36}+2q^{34}-2q^{32}+2q^{30}-2q^{28}+2q^{26}-3q^{24}+q^{22}-q^{20}+2q^{12}-q^{10}+2q^{8}+2q^{4}+1$
1,0	$q^{90}-q^{86}-q^{84}+q^{80}-q^{76}-q^{74}+q^{70}+q^{68}+q^{62}+2q^{60}-q^{56}+q^{52}-q^{48}-q^{46}-q^{40}-q^{38}-q^{32}-2q^{30}-q^{28}+q^{26}+q^{24}-q^{20}+q^{18}+2q^{16}+2q^{14}+q^{8}+2q^{6}+q^{-2}$

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{78}-q^{76}-q^{72}+q^{70}-q^{68}-q^{64}+q^{62}+q^{58}+q^{54}+q^{52}+2q^{48}-q^{46}+2q^{44}-q^{42}+2q^{40}-2q^{38}+q^{36}-3q^{34}-q^{32}-4q^{30}-2q^{28}-3q^{26}-2q^{24}-q^{22}+3q^{18}+2q^{16}+4q^{14}+2q^{12}+4q^{10}+q^{8}+2q^{6}+q^{2}$

G2 Invariants.

Weight	Invariant
1,0	$q^{134}-q^{132}+q^{130}-q^{128}-q^{126}-q^{122}+2q^{120}-2q^{118}+q^{116}+q^{110}+q^{106}-q^{104}+q^{102}+q^{96}+2q^{92}-q^{84}+q^{82}-2q^{78}+q^{76}-q^{74}+q^{70}-4q^{68}+2q^{66}-3q^{64}-3q^{58}+3q^{56}-2q^{54}+q^{52}-q^{50}-q^{48}+q^{46}-2q^{44}+q^{42}-q^{38}+q^{36}+q^{32}+2q^{30}-2q^{28}+3q^{26}-q^{24}+q^{22}+3q^{20}-3q^{18}+4q^{16}+q^{12}+q^{10}-q^{8}+2q^{6}+q^{2}$

.

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 2"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t^{3}+3t^{2}-3t+3-3t^{-1}+3t^{-2}-t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

$-z^{6}-3z^{4}+1$

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]=

{ 17, -4 }

In[8]:=

Jones[K][q]

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

Out[8]=

$1-q^{-1}+2q^{-2}-2q^{-3}+3q^{-4}-3q^{-5}+2q^{-6}-2q^{-7}+q^{-8}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{2}a^{10}+2z^{3}a^{9}-za^{9}+2z^{4}a^{8}-z^{2}a^{8}+2z^{5}a^{7}-2z^{3}a^{7}-za^{7}+2z^{6}a^{6}-5z^{4}a^{6}+3z^{2}a^{6}-a^{6}+z^{7}a^{5}-2z^{5}a^{5}-z^{3}a^{5}+za^{5}+3z^{6}a^{4}-12z^{4}a^{4}+12z^{2}a^{4}-3a^{4}+z^{7}a^{3}-4z^{5}a^{3}+3z^{3}a^{3}+za^{3}+z^{6}a^{2}-5z^{4}a^{2}+7z^{2}a^{2}-3a^{2}$

Vassiliev invariants

V2 and V3:

(0, 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 $0$ $8$ $0$ $0$ $24$ $0$ $-{\frac {304}{3}}$ $-{\frac {160}{3}}$ $-88$ $0$ $32$ $0$ $0$ $432$ $-104$ $472$ $80$ $32$

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 $s=$-4 is the signature of 8 2. 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
1									1	1
-1										0
-3							2	1		1
-5						1	1			0
-7					2	1				1
-9				1	1					0
-11			1	2						-1
-13		1	1							0
-15		1								-1
-17	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, 2]]
Out[2]=	8
In[3]:=	PD[Knot[8, 2]]
Out[3]=	PD[X[1, 4, 2, 5], X[5, 12, 6, 13], X[3, 11, 4, 10], X[11, 3, 12, 2], X[7, 14, 8, 15], X[9, 16, 10, 1], X[13, 6, 14, 7], X[15, 8, 16, 9]]
In[4]:=	GaussCode[Knot[8, 2]]
Out[4]=	GaussCode[-1, 4, -3, 1, -2, 7, -5, 8, -6, 3, -4, 2, -7, 5, -8, 6]
In[5]:=	BR[Knot[8, 2]]
Out[5]=	BR[3, {-1, -1, -1, -1, -1, 2, -1, 2}]
In[6]:=	alex = Alexander[Knot[8, 2]][t]
Out[6]=	-3 3 3 2 3 3 - t + -- - - - 3 t + 3 t - t 2 t t
In[7]:=	Conway[Knot[8, 2]][z]
Out[7]=	4 6 1 - 3 z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[8, 2], Knot[11, NonAlternating, 6]}
In[9]:=	{KnotDet[Knot[8, 2]], KnotSignature[Knot[8, 2]]}
Out[9]=	{17, -4}
In[10]:=	J=Jones[Knot[8, 2]][q]
Out[10]=	-8 2 2 3 3 2 2 1 1 + q - -- + -- - -- + -- - -- + -- - - 7 6 5 4 3 2 q q q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[8, 2]}
In[12]:=	A2Invariant[Knot[8, 2]][q]
Out[12]=	-24 -18 -16 -12 -10 -6 -4 -2 1 + q - q - q - q + q + q + q + q
In[13]:=	Kauffman[Knot[8, 2]][a, z]
Out[13]=	2 4 6 3 5 7 9 2 2 4 2 -3 a - 3 a - a + a z + a z - a z - a z + 7 a z + 12 a z + 6 2 8 2 10 2 3 3 5 3 7 3 9 3 3 a z - a z + a z + 3 a z - a z - 2 a z + 2 a z - 2 4 4 4 6 4 8 4 3 5 5 5 5 a z - 12 a z - 5 a z + 2 a z - 4 a z - 2 a z + 7 5 2 6 4 6 6 6 3 7 5 7 2 a z + a z + 3 a z + 2 a z + a z + a z
In[14]:=	{Vassiliev[2][Knot[8, 2]], Vassiliev[3][Knot[8, 2]]}
Out[14]=	{0, 1}
In[15]:=	Kh[Knot[8, 2]][q, t]
Out[15]=	-5 2 1 1 1 1 1 2 q + -- + ------ + ------ + ------ + ------ + ------ + ------ + 3 17 6 15 5 13 5 13 4 11 4 11 3 q q t q t q t q t q t q t 1 1 2 1 1 t 2 ----- + ----- + ----- + ---- + ---- + -- + q t 9 3 9 2 7 2 7 5 3 q t q t q t q t q t q