8 20

8_19

8_21

Visit 8 20's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 8 20's page at Knotilus! Visit 8 20's page at the original Knot Atlas!

8_20 is also known as the pretzel knot P(3,-3,2). Its complement contains no complete totally geodesic immersed surfaces.[citation needed]

The Oysterman's stopper[1]

Knot presentations

Planar diagram presentation	X4251 X8493 X5,12,6,13 X13,16,14,1 X9,14,10,15 X15,10,16,11 X11,6,12,7 X2837
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	[3,21,2-]

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 [-6][-2] Hyperbolic Volume 4.1249 A-Polynomial See Data:8 20/A-polynomial

[edit Notes for 8 20's three dimensional invariants] 8_20 ribbon diagram from A. Kawauchi's text. Ribbon diagram for 8_20

Four dimensional invariants

Smooth 4 genus $0$ Topological 4 genus $0$ Concordance genus $0$ Rasmussen s-Invariant 0

[edit Notes for 8 20's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{2}-2t+3-2t^{-1}+t^{-2}$
Conway polynomial	$z^{4}+2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 9, 0 }
Jones polynomial	$-q+2-q^{-1}+2q^{-2}-q^{-3}+q^{-4}-q^{-5}$
HOMFLY-PT polynomial (db, data sources)	$-z^{2}a^{4}-2a^{4}+z^{4}a^{2}+4z^{2}a^{2}+4a^{2}-z^{2}-1$
Kauffman polynomial (db, data sources)	$a^{4}z^{6}+a^{2}z^{6}+a^{5}z^{5}+2a^{3}z^{5}+az^{5}-4a^{4}z^{4}-4a^{2}z^{4}-4a^{5}z^{3}-7a^{3}z^{3}-3az^{3}+4a^{4}z^{2}+6a^{2}z^{2}+2z^{2}+3a^{5}z+5a^{3}z+3az+za^{-1}-2a^{4}-4a^{2}-1$
The A2 invariant	$-q^{16}-q^{14}-q^{12}+2q^{8}+2q^{6}+2q^{4}+q^{2}-q^{-4}$
The G2 invariant	$q^{80}+q^{76}-q^{74}-2q^{68}+q^{66}-q^{64}-q^{62}-q^{60}-2q^{58}-3q^{52}-q^{50}-q^{48}+q^{44}-2q^{42}+q^{40}+q^{38}+2q^{36}+q^{34}+2q^{30}+2q^{28}+3q^{26}+2q^{22}+3q^{20}+q^{18}+q^{16}+q^{14}+3q^{10}-2q^{6}+q^{4}+1-q^{-2}-2q^{-4}-q^{-12}-q^{-14}-q^{-20}+q^{-24}$

Further Quantum Invariants

Further quantum knot invariants for 8_20.

A1 Invariants.

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

A2 Invariants.

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

A3 Invariants.

Weight	Invariant
0,1,0	$q^{34}+q^{30}-q^{26}-2q^{24}-4q^{22}-4q^{20}-3q^{18}+2q^{14}+6q^{12}+6q^{10}+6q^{8}+3q^{6}+q^{4}-q^{2}-2-2q^{-2}-q^{-4}-q^{-6}+q^{-10}$
1,0,0	$-q^{21}-q^{19}-2q^{17}-q^{15}+2q^{11}+3q^{9}+3q^{7}+2q^{5}+q^{3}-q^{-1}-q^{-5}$
1,0,1	$q^{56}+2q^{52}+q^{48}-q^{44}+q^{42}-2q^{40}+q^{38}-4q^{36}-2q^{34}-6q^{32}-5q^{30}-5q^{28}-6q^{26}-2q^{22}+6q^{20}+5q^{18}+10q^{16}+9q^{14}+9q^{12}+8q^{10}+2q^{8}+3q^{6}-2q^{4}-3-q^{-2}-q^{-4}-2q^{-6}-q^{-8}-q^{-10}+q^{-16}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{44}+q^{42}+2q^{40}+2q^{38}+q^{36}-q^{34}-3q^{32}-6q^{30}-9q^{28}-9q^{26}-7q^{24}-3q^{22}+q^{20}+8q^{18}+12q^{16}+13q^{14}+12q^{12}+9q^{10}+3q^{8}-q^{6}-4q^{4}-5q^{2}-5-3q^{-2}-q^{-4}+q^{-10}+q^{-12}$
1,0,0,0	$-q^{26}-q^{24}-2q^{22}-2q^{20}-q^{18}+2q^{14}+3q^{12}+4q^{10}+3q^{8}+2q^{6}+q^{4}-1-q^{-2}-q^{-6}$

B2 Invariants.

Weight	Invariant
0,1	$-q^{34}-q^{30}-q^{26}+q^{18}+2q^{14}+2q^{10}+q^{6}+q^{4}+q^{2}+q^{-4}-q^{-6}-q^{-10}$
1,0	$q^{56}+q^{48}-q^{44}-q^{42}-q^{38}-2q^{36}-2q^{34}-2q^{32}-q^{30}-q^{28}+2q^{22}+2q^{20}+3q^{18}+2q^{16}+3q^{14}+3q^{12}+2q^{10}+q^{6}-1-q^{-2}-q^{-4}-q^{-8}-q^{-10}+q^{-16}$

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{46}+q^{42}+q^{38}-q^{36}-2q^{34}-3q^{32}-4q^{30}-4q^{28}-4q^{26}-2q^{24}-q^{22}+3q^{20}+4q^{18}+7q^{16}+6q^{14}+7q^{12}+4q^{10}+3q^{8}+q^{6}-q^{4}-2q^{2}-2-2q^{-2}-2q^{-4}-q^{-8}+q^{-14}$

G2 Invariants.

Weight	Invariant
1,0	$q^{80}+q^{76}-q^{74}-2q^{68}+q^{66}-q^{64}-q^{62}-q^{60}-2q^{58}-3q^{52}-q^{50}-q^{48}+q^{44}-2q^{42}+q^{40}+q^{38}+2q^{36}+q^{34}+2q^{30}+2q^{28}+3q^{26}+2q^{22}+3q^{20}+q^{18}+q^{16}+q^{14}+3q^{10}-2q^{6}+q^{4}+1-q^{-2}-2q^{-4}-q^{-12}-q^{-14}-q^{-20}+q^{-24}$

.

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 20"];

In[4]:=

Alexander[K][t]

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

Out[4]=

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

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 9, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

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

Vassiliev invariants

V2 and V3:

(2, -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 $8$ $-16$ $32$ ${\frac {172}{3}}$ ${\frac {20}{3}}$ $-128$ $-{\frac {640}{3}}$ $-{\frac {64}{3}}$ $-48$ ${\frac {256}{3}}$ $128$ ${\frac {1376}{3}}$ ${\frac {160}{3}}$ ${\frac {11911}{15}}$ $-{\frac {524}{15}}$ ${\frac {15484}{45}}$ ${\frac {233}{9}}$ ${\frac {631}{15}}$

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=$0 is the signature of 8 20. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \	-5	-4	-3	-2	-1	0	1	χ
3							1	-1
1						1		1
-1					1	2		1
-3				1				1
-5				1				1
-7		1	1					0
-9								0
-11	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, 20]]
Out[2]=	8
In[3]:=	PD[Knot[8, 20]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 4, 9, 3], X[5, 12, 6, 13], X[13, 16, 14, 1], X[9, 14, 10, 15], X[15, 10, 16, 11], X[11, 6, 12, 7], X[2, 8, 3, 7]]
In[4]:=	GaussCode[Knot[8, 20]]
Out[4]=	GaussCode[1, -8, 2, -1, -3, 7, 8, -2, -5, 6, -7, 3, -4, 5, -6, 4]
In[5]:=	BR[Knot[8, 20]]
Out[5]=	BR[3, {1, 1, 1, -2, -1, -1, -1, -2}]
In[6]:=	alex = Alexander[Knot[8, 20]][t]
Out[6]=	-2 2 2 3 + t - - - 2 t + t t
In[7]:=	Conway[Knot[8, 20]][z]
Out[7]=	2 4 1 + 2 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[8, 20], Knot[10, 140], Knot[11, NonAlternating, 73], Knot[11, NonAlternating, 74]}
In[9]:=	{KnotDet[Knot[8, 20]], KnotSignature[Knot[8, 20]]}
Out[9]=	{9, 0}
In[10]:=	J=Jones[Knot[8, 20]][q]
Out[10]=	-5 -4 -3 2 1 2 - q + q - q + -- - - - q 2 q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[8, 20]}
In[12]:=	A2Invariant[Knot[8, 20]][q]
Out[12]=	-16 -14 -12 2 2 2 -2 4 -q - q - q + -- + -- + -- + q - q 8 6 4 q q q
In[13]:=	Kauffman[Knot[8, 20]][a, z]
Out[13]=	2 4 z 3 5 2 2 2 -1 - 4 a - 2 a + - + 3 a z + 5 a z + 3 a z + 2 z + 6 a z + a 4 2 3 3 3 5 3 2 4 4 4 5 4 a z - 3 a z - 7 a z - 4 a z - 4 a z - 4 a z + a z + 3 5 5 5 2 6 4 6 2 a z + a z + a z + a z
In[14]:=	{Vassiliev[2][Knot[8, 20]], Vassiliev[3][Knot[8, 20]]}
Out[14]=	{0, -2}
In[15]:=	Kh[Knot[8, 20]][q, t]
Out[15]=	2 1 1 1 1 1 1 3 - + q + ------ + ----- + ----- + ----- + ----- + --- + q t q 11 5 7 4 7 3 5 2 3 2 q t q t q t q t q t q t