8 21

8_20

9_1

Visit 8 21's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 8 21's page at Knotilus! Visit 8 21's page at the original Knot Atlas! 8 21 Quick Notes

8 21 Further Notes and Views

Knot presentations

Planar diagram presentation	X1425 X3849 X12,6,13,5 X13,16,14,1 X9,14,10,15 X15,10,16,11 X6,12,7,11 X7283
Gauss code	-1, 8, -2, 1, 3, -7, -8, 2, -5, 6, 7, -3, -4, 5, -6, 4
Dowker-Thistlethwaite code	4 8 -12 2 14 -6 16 10
Conway Notation	[21,21,2-]

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 2 Bridge index 3 Super bridge index 4 Nakanishi index 1 Maximal Thurston-Bennequin number [-9][1] Hyperbolic Volume 6.78371 A-Polynomial See Data:8 21/A-polynomial

[edit Notes for 8 21'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 21's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{2}+4t-5+4t^{-1}-t^{-2}$
Conway polynomial	$1-z^{4}$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 15, -2 }
Jones polynomial	$2q^{-1}-2q^{-2}+3q^{-3}-3q^{-4}+2q^{-5}-2q^{-6}+q^{-7}$
HOMFLY-PT polynomial (db, data sources)	$z^{2}a^{6}+a^{6}-z^{4}a^{4}-3z^{2}a^{4}-3a^{4}+2z^{2}a^{2}+3a^{2}$
Kauffman polynomial (db, data sources)	$z^{4}a^{8}-2z^{2}a^{8}+2z^{5}a^{7}-5z^{3}a^{7}+2za^{7}+z^{6}a^{6}-z^{4}a^{6}-a^{6}+3z^{5}a^{5}-6z^{3}a^{5}+4za^{5}+z^{6}a^{4}-2z^{4}a^{4}+5z^{2}a^{4}-3a^{4}+z^{5}a^{3}-z^{3}a^{3}+2za^{3}+3z^{2}a^{2}-3a^{2}$
The A2 invariant	$q^{22}-2q^{14}-q^{12}-q^{10}+q^{8}+2q^{6}+q^{4}+2q^{2}$
The G2 invariant	$q^{114}-q^{112}+2q^{110}-3q^{108}+q^{104}-4q^{102}+5q^{100}-4q^{98}+3q^{96}+q^{94}-4q^{92}+5q^{90}-2q^{88}+2q^{86}+2q^{84}-4q^{82}+4q^{80}+2q^{78}-2q^{76}+4q^{74}-4q^{72}+3q^{70}-4q^{66}+3q^{64}-7q^{62}+5q^{60}-4q^{58}-3q^{56}+2q^{54}-6q^{52}+4q^{50}-5q^{48}-q^{46}+2q^{44}-3q^{42}+2q^{40}-3q^{36}+7q^{34}-2q^{32}+3q^{28}-3q^{26}+7q^{24}-2q^{22}+2q^{20}+q^{18}-q^{16}+4q^{14}-q^{12}+2q^{10}+q^{8}$

Further Quantum Invariants

Further quantum knot invariants for 8_21.

A1 Invariants.

Weight	Invariant
1	$q^{15}-q^{13}-q^{9}+q^{5}+2q$
2	$q^{42}-q^{40}-2q^{38}+2q^{36}+q^{34}-2q^{32}+q^{30}+2q^{28}-2q^{26}+q^{22}-q^{20}-q^{18}+2q^{14}-2q^{12}-q^{10}+3q^{8}-q^{6}-q^{4}+3q^{2}+1$
3	$q^{81}-q^{79}-2q^{77}+3q^{73}+3q^{71}-3q^{69}-4q^{67}+q^{65}+5q^{63}+q^{61}-6q^{59}-4q^{57}+5q^{55}+4q^{53}-3q^{51}-4q^{49}+4q^{47}+5q^{45}-2q^{43}-3q^{41}+2q^{37}-q^{35}-2q^{33}-3q^{31}+2q^{29}+4q^{27}-6q^{23}-q^{21}+7q^{19}+3q^{17}-6q^{15}-3q^{13}+4q^{11}+4q^{9}-2q^{7}-4q^{5}+2q^{3}+2q+2q^{-1}$
4	$q^{132}-q^{130}-2q^{128}+q^{124}+5q^{122}+q^{120}-3q^{118}-5q^{116}-6q^{114}+6q^{112}+8q^{110}+4q^{108}-4q^{106}-14q^{104}-4q^{102}+7q^{100}+15q^{98}+8q^{96}-13q^{94}-14q^{92}-3q^{90}+15q^{88}+15q^{86}-6q^{84}-16q^{82}-11q^{80}+9q^{78}+14q^{76}-q^{74}-10q^{72}-8q^{70}+6q^{68}+11q^{66}+2q^{64}-5q^{62}-5q^{60}+2q^{58}+5q^{56}+4q^{54}-2q^{52}-5q^{50}-6q^{48}+11q^{44}+4q^{42}-6q^{40}-15q^{38}-7q^{36}+15q^{34}+14q^{32}+q^{30}-18q^{28}-15q^{26}+11q^{24}+16q^{22}+10q^{20}-9q^{18}-15q^{16}+7q^{12}+10q^{10}-7q^{6}-3q^{4}-q^{2}+3+3q^{-2}+q^{-4}$
5	$q^{195}-q^{193}-2q^{191}+q^{187}+3q^{185}+3q^{183}+q^{181}-5q^{179}-7q^{177}-4q^{175}+q^{173}+8q^{171}+12q^{169}+7q^{167}-8q^{165}-16q^{163}-15q^{161}-3q^{159}+16q^{157}+26q^{155}+18q^{153}-8q^{151}-29q^{149}-32q^{147}-11q^{145}+25q^{143}+42q^{141}+26q^{139}-14q^{137}-45q^{135}-41q^{133}+41q^{129}+49q^{127}+17q^{125}-33q^{123}-49q^{121}-23q^{119}+22q^{117}+46q^{115}+27q^{113}-14q^{111}-40q^{109}-26q^{107}+9q^{105}+30q^{103}+21q^{101}-7q^{99}-24q^{97}-18q^{95}+5q^{93}+18q^{91}+11q^{89}-q^{87}-11q^{85}-12q^{83}+9q^{79}+10q^{77}+9q^{75}-2q^{73}-15q^{71}-15q^{69}-q^{67}+17q^{65}+27q^{63}+12q^{61}-19q^{59}-37q^{57}-25q^{55}+15q^{53}+45q^{51}+36q^{49}-9q^{47}-50q^{45}-48q^{43}-7q^{41}+46q^{39}+54q^{37}+16q^{35}-32q^{33}-52q^{31}-27q^{29}+16q^{27}+43q^{25}+33q^{23}-q^{21}-24q^{19}-25q^{17}-9q^{15}+10q^{13}+21q^{11}+10q^{9}-2q^{7}-6q^{5}-10q^{3}-4q+2q^{-1}+4q^{-3}+2q^{-5}+2q^{-7}$
6	$q^{270}-q^{268}-2q^{266}+q^{262}+3q^{260}+q^{258}+3q^{256}-q^{254}-7q^{252}-6q^{250}-4q^{248}+2q^{246}+5q^{244}+15q^{242}+11q^{240}+q^{238}-10q^{236}-19q^{234}-19q^{232}-16q^{230}+14q^{228}+31q^{226}+34q^{224}+21q^{222}-5q^{220}-35q^{218}-61q^{216}-40q^{214}-q^{212}+46q^{210}+71q^{208}+68q^{206}+18q^{204}-60q^{202}-94q^{200}-86q^{198}-21q^{196}+59q^{194}+123q^{192}+109q^{190}+19q^{188}-76q^{186}-137q^{184}-114q^{182}-24q^{180}+99q^{178}+150q^{176}+105q^{174}-113q^{170}-148q^{168}-94q^{166}+35q^{164}+126q^{162}+130q^{160}+56q^{158}-58q^{156}-121q^{154}-104q^{152}-6q^{150}+78q^{148}+101q^{146}+57q^{144}-24q^{142}-77q^{140}-75q^{138}-11q^{136}+42q^{134}+58q^{132}+33q^{130}-12q^{128}-40q^{126}-40q^{124}-6q^{122}+22q^{120}+32q^{118}+19q^{116}-3q^{114}-20q^{112}-26q^{110}-15q^{108}+3q^{106}+25q^{104}+27q^{102}+19q^{100}-6q^{98}-32q^{96}-47q^{94}-28q^{92}+23q^{90}+55q^{88}+68q^{86}+28q^{84}-36q^{82}-95q^{80}-90q^{78}-11q^{76}+72q^{74}+129q^{72}+96q^{70}-3q^{68}-122q^{66}-156q^{64}-80q^{62}+39q^{60}+148q^{58}+156q^{56}+68q^{54}-81q^{52}-163q^{50}-134q^{48}-39q^{46}+86q^{44}+147q^{42}+118q^{40}+7q^{38}-86q^{36}-112q^{34}-83q^{32}-6q^{30}+64q^{28}+91q^{26}+51q^{24}+3q^{22}-35q^{20}-53q^{18}-35q^{16}-4q^{14}+26q^{12}+27q^{10}+19q^{8}+8q^{6}-8q^{4}-15q^{2}-11-3q^{-2}+q^{-4}+3q^{-6}+3q^{-8}+3q^{-10}+q^{-12}$

A2 Invariants.

Weight	Invariant
1,0	$q^{22}-2q^{14}-q^{12}-q^{10}+q^{8}+2q^{6}+q^{4}+2q^{2}$
1,1	$q^{60}-2q^{58}+4q^{56}-8q^{54}+9q^{52}-10q^{50}+12q^{48}-8q^{46}+4q^{44}-8q^{40}+10q^{38}-15q^{36}+18q^{34}-16q^{32}+20q^{30}-12q^{28}+12q^{26}-6q^{24}-2q^{22}+q^{20}-14q^{18}+8q^{16}-12q^{14}+10q^{12}-4q^{10}+8q^{8}+2q^{6}+4q^{4}+2q^{2}$
2,0	$q^{56}-q^{52}-q^{50}+q^{46}-q^{44}+q^{40}-q^{36}+2q^{32}+2q^{30}+q^{28}+q^{26}-q^{24}-3q^{22}-2q^{20}-2q^{18}-3q^{16}-q^{14}+2q^{12}+2q^{10}+2q^{8}+2q^{6}+4q^{4}+q^{2}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{48}-q^{46}-3q^{40}+q^{38}+q^{36}+3q^{32}+4q^{30}+q^{28}-2q^{24}-4q^{22}-5q^{20}-5q^{18}+2q^{12}+6q^{10}+3q^{8}+q^{6}+3q^{4}$
1,0,0	$q^{29}+q^{25}-2q^{19}-2q^{17}-2q^{15}-q^{13}+q^{11}+2q^{9}+3q^{7}+q^{5}+2q^{3}$
1,0,1	$q^{78}-2q^{76}+3q^{74}-3q^{72}-q^{70}+5q^{68}-8q^{66}+10q^{64}-6q^{62}+2q^{60}+5q^{58}-11q^{56}+12q^{54}-12q^{52}+2q^{50}-q^{48}-6q^{46}+q^{44}+q^{42}+6q^{40}+3q^{38}+18q^{36}-3q^{34}+13q^{32}-5q^{30}-11q^{28}-2q^{26}-25q^{24}-2q^{22}-11q^{20}-4q^{18}+9q^{16}+3q^{14}+12q^{12}+10q^{10}+5q^{8}+5q^{6}+2q^{4}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{62}+q^{56}-q^{54}-4q^{52}-3q^{50}-q^{48}-q^{46}+q^{44}+6q^{42}+9q^{40}+6q^{38}+5q^{36}+3q^{34}-4q^{32}-9q^{30}-8q^{28}-10q^{26}-9q^{24}-4q^{22}+q^{20}+3q^{18}+4q^{16}+8q^{14}+7q^{12}+4q^{10}+2q^{8}+3q^{6}$
1,0,0,0	$q^{36}+q^{32}+q^{30}-2q^{24}-2q^{22}-3q^{20}-2q^{18}-q^{16}+q^{14}+2q^{12}+3q^{10}+3q^{8}+q^{6}+2q^{4}$

B2 Invariants.

Weight	Invariant
0,1	$q^{48}-q^{46}+2q^{44}-2q^{42}+q^{40}-q^{38}+q^{36}-q^{32}+2q^{30}-3q^{28}+2q^{26}-4q^{24}+2q^{22}-3q^{20}+q^{18}+2q^{12}+3q^{8}-q^{6}+3q^{4}$
1,0	$q^{78}-q^{74}-q^{72}+q^{70}+q^{68}-2q^{66}-2q^{64}+2q^{60}+q^{58}-q^{56}+2q^{52}+2q^{50}+q^{48}+q^{44}+q^{42}-3q^{38}-2q^{36}-q^{34}-q^{32}-3q^{30}-3q^{28}+q^{24}-q^{20}+2q^{18}+3q^{16}+3q^{14}+q^{8}+3q^{6}$

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{66}-q^{64}+q^{62}-2q^{60}+q^{58}-2q^{56}-q^{52}+q^{50}+q^{48}+q^{46}+4q^{44}+2q^{42}+4q^{40}-q^{38}+2q^{36}-4q^{34}-q^{32}-7q^{30}-4q^{28}-6q^{26}-2q^{24}-q^{22}+3q^{18}+3q^{16}+6q^{14}+3q^{12}+4q^{10}+3q^{6}$

G2 Invariants.

Weight	Invariant
1,0	$q^{114}-q^{112}+2q^{110}-3q^{108}+q^{104}-4q^{102}+5q^{100}-4q^{98}+3q^{96}+q^{94}-4q^{92}+5q^{90}-2q^{88}+2q^{86}+2q^{84}-4q^{82}+4q^{80}+2q^{78}-2q^{76}+4q^{74}-4q^{72}+3q^{70}-4q^{66}+3q^{64}-7q^{62}+5q^{60}-4q^{58}-3q^{56}+2q^{54}-6q^{52}+4q^{50}-5q^{48}-q^{46}+2q^{44}-3q^{42}+2q^{40}-3q^{36}+7q^{34}-2q^{32}+3q^{28}-3q^{26}+7q^{24}-2q^{22}+2q^{20}+q^{18}-q^{16}+4q^{14}-q^{12}+2q^{10}+q^{8}$

.

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 21"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t^{2}+4t-5+4t^{-1}-t^{-2}$

In[5]:=

Conway[K][z]

Out[5]=

$1-z^{4}$

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]=

{ 15, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

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

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 $0$ $8$ $0$ $-16$ $8$ $0$ $-{\frac {16}{3}}$ $-{\frac {64}{3}}$ $-24$ $0$ $32$ $0$ $0$ $136$ $-{\frac {8}{3}}$ ${\frac {424}{3}}$ ${\frac {40}{3}}$ $8$

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

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

Integral Khovanov Homology (db, data source)

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

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, 21]]
Out[2]=	8
In[3]:=	PD[Knot[8, 21]]
Out[3]=	PD[X[1, 4, 2, 5], X[3, 8, 4, 9], X[12, 6, 13, 5], X[13, 16, 14, 1], X[9, 14, 10, 15], X[15, 10, 16, 11], X[6, 12, 7, 11], X[7, 2, 8, 3]]
In[4]:=	GaussCode[Knot[8, 21]]
Out[4]=	GaussCode[-1, 8, -2, 1, 3, -7, -8, 2, -5, 6, 7, -3, -4, 5, -6, 4]
In[5]:=	BR[Knot[8, 21]]
Out[5]=	BR[3, {-1, -1, -1, -2, 1, 1, -2, -2}]
In[6]:=	alex = Alexander[Knot[8, 21]][t]
Out[6]=	-2 4 2 -5 - t + - + 4 t - t t
In[7]:=	Conway[Knot[8, 21]][z]
Out[7]=	4 1 - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[8, 21], Knot[10, 136]}
In[9]:=	{KnotDet[Knot[8, 21]], KnotSignature[Knot[8, 21]]}
Out[9]=	{15, -2}
In[10]:=	J=Jones[Knot[8, 21]][q]
Out[10]=	-7 2 2 3 3 2 2 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, 21]}
In[12]:=	A2Invariant[Knot[8, 21]][q]
Out[12]=	-22 2 -12 -10 -8 2 -4 2 q - --- - q - q + q + -- + q + -- 14 6 2 q q q
In[13]:=	Kauffman[Knot[8, 21]][a, z]
Out[13]=	2 4 6 3 5 7 2 2 4 2 -3 a - 3 a - a + 2 a z + 4 a z + 2 a z + 3 a z + 5 a z - 8 2 3 3 5 3 7 3 4 4 6 4 8 4 2 a z - a z - 6 a z - 5 a z - 2 a z - a z + a z + 3 5 5 5 7 5 4 6 6 6 a z + 3 a z + 2 a z + a z + a z
In[14]:=	{Vassiliev[2][Knot[8, 21]], Vassiliev[3][Knot[8, 21]]}
Out[14]=	{0, 1}
In[15]:=	Kh[Knot[8, 21]][q, t]
Out[15]=	-3 2 1 1 1 1 1 2 1 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 1 2 1 1 ----- + ----- + ---- + ---- 7 2 5 2 5 3 q t q t q t q t