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] This appears to be the Ashley/oysterman stopper knot of practical knot tying.

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-]

Minimum Braid Representative: Length is 8, width is 3. Braid index is 3.

A Morse Link Presentation:

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]=

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

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_140, K11n73, K11n74, ...}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {...}

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

Integral Khovanov Homology (db, data source)

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

The Coloured Jones Polynomials

The Coloured Jones Polynomials (in the $n+1$-dimensional representation of $sl(2)$)

$n$	$J_{n}$
2	$-q^{2}+q+1-2q^{-1}+2q^{-2}+q^{-3}-2q^{-4}+q^{-5}+2q^{-6}-2q^{-7}+2q^{-9}-2q^{-10}-q^{-11}+2q^{-12}-q^{-13}-q^{-14}+q^{-15}$
3	$q^{7}-q^{6}-q^{5}-q^{4}+2q^{3}+2q^{2}-3q-1+2q^{-1}+4q^{-2}-3q^{-3}-2q^{-4}+q^{-5}+4q^{-6}-3q^{-7}-q^{-8}+q^{-9}+2q^{-10}-2q^{-11}+q^{-14}-q^{-17}-q^{-18}+q^{-19}+q^{-20}-q^{-21}-2q^{-22}+q^{-23}+2q^{-24}-2q^{-26}+q^{-28}+q^{-29}-q^{-30}$
4	$-q^{12}+q^{11}+2q^{10}-q^{8}-5q^{7}+5q^{5}+3q^{4}-10q^{2}-2q+8+6q^{-1}+2q^{-2}-11q^{-3}-4q^{-4}+7q^{-5}+7q^{-6}+2q^{-7}-11q^{-8}-4q^{-9}+7q^{-10}+5q^{-11}+2q^{-12}-10q^{-13}-4q^{-14}+7q^{-15}+4q^{-16}+2q^{-17}-8q^{-18}-4q^{-19}+6q^{-20}+2q^{-21}+3q^{-22}-5q^{-23}-4q^{-24}+4q^{-25}+3q^{-27}-2q^{-28}-3q^{-29}+2q^{-30}-2q^{-31}+2q^{-32}-q^{-34}+3q^{-35}-3q^{-36}+4q^{-40}-2q^{-41}-q^{-42}-q^{-43}-q^{-44}+3q^{-45}-q^{-48}-q^{-49}+q^{-50}$
5	$-q^{16}+2q^{14}+2q^{13}-2q^{11}-6q^{10}-2q^{9}+5q^{8}+8q^{7}+4q^{6}-6q^{5}-11q^{4}-7q^{3}+5q^{2}+14q+10-6q^{-1}-13q^{-2}-10q^{-3}+3q^{-4}+15q^{-5}+13q^{-6}-5q^{-7}-13q^{-8}-11q^{-9}+2q^{-10}+14q^{-11}+13q^{-12}-5q^{-13}-13q^{-14}-10q^{-15}+2q^{-16}+13q^{-17}+11q^{-18}-5q^{-19}-11q^{-20}-9q^{-21}+q^{-22}+11q^{-23}+10q^{-24}-2q^{-25}-9q^{-26}-9q^{-27}-q^{-28}+8q^{-29}+10q^{-30}+q^{-31}-6q^{-32}-8q^{-33}-4q^{-34}+5q^{-35}+8q^{-36}+4q^{-37}-3q^{-38}-6q^{-39}-5q^{-40}+5q^{-42}+5q^{-43}-2q^{-45}-4q^{-46}-2q^{-47}+q^{-48}+3q^{-49}+q^{-50}+q^{-51}-q^{-52}-q^{-53}-q^{-56}+q^{-58}+q^{-59}+q^{-60}-2q^{-62}-2q^{-63}+q^{-65}+q^{-66}+2q^{-67}-2q^{-69}-q^{-70}+q^{-73}+q^{-74}-q^{-75}$
6	$q^{26}-q^{25}-q^{24}-q^{20}+5q^{19}+q^{18}-4q^{15}-7q^{14}-6q^{13}+9q^{12}+7q^{11}+8q^{10}+5q^{9}-6q^{8}-19q^{7}-17q^{6}+10q^{5}+11q^{4}+17q^{3}+13q^{2}-4q-24-25q^{-1}+7q^{-2}+10q^{-3}+20q^{-4}+18q^{-5}-24q^{-7}-27q^{-8}+4q^{-9}+8q^{-10}+19q^{-11}+19q^{-12}+q^{-13}-23q^{-14}-26q^{-15}+4q^{-16}+8q^{-17}+18q^{-18}+17q^{-19}-22q^{-21}-25q^{-22}+5q^{-23}+8q^{-24}+17q^{-25}+15q^{-26}-q^{-27}-19q^{-28}-23q^{-29}+5q^{-30}+5q^{-31}+14q^{-32}+14q^{-33}+q^{-34}-13q^{-35}-20q^{-36}+2q^{-37}+q^{-38}+10q^{-39}+13q^{-40}+5q^{-41}-6q^{-42}-17q^{-43}-2q^{-44}-4q^{-45}+5q^{-46}+12q^{-47}+9q^{-48}+q^{-49}-12q^{-50}-4q^{-51}-9q^{-52}-q^{-53}+8q^{-54}+9q^{-55}+7q^{-56}-5q^{-57}-2q^{-58}-10q^{-59}-6q^{-60}+2q^{-61}+5q^{-62}+9q^{-63}+q^{-64}+3q^{-65}-6q^{-66}-6q^{-67}-3q^{-68}-q^{-69}+6q^{-70}+q^{-71}+5q^{-72}-2q^{-74}-3q^{-75}-3q^{-76}+3q^{-77}-3q^{-78}+2q^{-79}+q^{-80}+q^{-81}-q^{-83}+4q^{-84}-4q^{-85}-q^{-86}-q^{-87}+5q^{-91}-q^{-92}-q^{-94}-q^{-95}-2q^{-96}-q^{-97}+3q^{-98}+q^{-100}-q^{-103}-q^{-104}+q^{-105}$
7	$-q^{35}+q^{34}+q^{33}+q^{32}-2q^{30}-q^{29}-2q^{28}-3q^{27}+4q^{25}+6q^{24}+7q^{23}-q^{21}-7q^{20}-14q^{19}-10q^{18}+11q^{16}+21q^{15}+16q^{14}+6q^{13}-9q^{12}-28q^{11}-26q^{10}-16q^{9}+6q^{8}+34q^{7}+35q^{6}+21q^{5}-3q^{4}-33q^{3}-38q^{2}-30q-3+34q^{-1}+44q^{-2}+31q^{-3}+6q^{-4}-31q^{-5}-39q^{-6}-35q^{-7}-10q^{-8}+30q^{-9}+42q^{-10}+32q^{-11}+10q^{-12}-28q^{-13}-38q^{-14}-34q^{-15}-11q^{-16}+29q^{-17}+40q^{-18}+31q^{-19}+9q^{-20}-28q^{-21}-38q^{-22}-33q^{-23}-9q^{-24}+30q^{-25}+40q^{-26}+30q^{-27}+6q^{-28}-28q^{-29}-37q^{-30}-31q^{-31}-7q^{-32}+28q^{-33}+38q^{-34}+27q^{-35}+5q^{-36}-24q^{-37}-34q^{-38}-28q^{-39}-7q^{-40}+23q^{-41}+33q^{-42}+24q^{-43}+6q^{-44}-17q^{-45}-28q^{-46}-25q^{-47}-9q^{-48}+14q^{-49}+26q^{-50}+22q^{-51}+10q^{-52}-8q^{-53}-20q^{-54}-21q^{-55}-13q^{-56}+3q^{-57}+16q^{-58}+19q^{-59}+13q^{-60}+3q^{-61}-10q^{-62}-15q^{-63}-14q^{-64}-8q^{-65}+4q^{-66}+11q^{-67}+14q^{-68}+11q^{-69}-5q^{-71}-10q^{-72}-13q^{-73}-7q^{-74}+7q^{-76}+13q^{-77}+6q^{-78}+6q^{-79}-10q^{-81}-9q^{-82}-8q^{-83}-3q^{-84}+5q^{-85}+5q^{-86}+10q^{-87}+9q^{-88}-2q^{-89}-3q^{-90}-6q^{-91}-9q^{-92}-3q^{-93}-3q^{-94}+5q^{-95}+9q^{-96}+2q^{-97}+4q^{-98}+q^{-99}-5q^{-100}-3q^{-101}-6q^{-102}-2q^{-103}+4q^{-104}-q^{-105}+3q^{-106}+3q^{-107}+2q^{-109}-2q^{-110}-2q^{-111}+2q^{-112}-3q^{-113}-q^{-114}-2q^{-116}+3q^{-117}+q^{-118}+3q^{-120}-q^{-123}-4q^{-124}-q^{-127}+2q^{-128}+q^{-129}+2q^{-130}+q^{-131}-2q^{-132}-q^{-133}-q^{-135}+q^{-138}+q^{-139}-q^{-140}$

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 29, 2005, 15:27:48)...
In[2]:=	PD[Knot[8, 20]]
Out[2]=	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[3]:=	GaussCode[Knot[8, 20]]
Out[3]=	GaussCode[1, -8, 2, -1, -3, 7, 8, -2, -5, 6, -7, 3, -4, 5, -6, 4]
In[4]:=	DTCode[Knot[8, 20]]
Out[4]=	DTCode[4, 8, -12, 2, -14, -6, -16, -10]
In[5]:=	br = BR[Knot[8, 20]]
Out[5]=	BR[3, {1, 1, 1, -2, -1, -1, -1, -2}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{3, 8}
In[7]:=	BraidIndex[Knot[8, 20]]
Out[7]=	3
In[8]:=	Show[DrawMorseLink[Knot[8, 20]]]
Out[8]=	-Graphics-
In[9]:=	(#[Knot[8, 20]]&) /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 1, 2, 3, 4, 1}
In[10]:=	alex = Alexander[Knot[8, 20]][t]
Out[10]=	-2 2 2 3 + t - - - 2 t + t t
In[11]:=	Conway[Knot[8, 20]][z]
Out[11]=	2 4 1 + 2 z + z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[8, 20], Knot[10, 140], Knot[11, NonAlternating, 73], Knot[11, NonAlternating, 74]}
In[13]:=	{KnotDet[Knot[8, 20]], KnotSignature[Knot[8, 20]]}
Out[13]=	{9, 0}
In[14]:=	Jones[Knot[8, 20]][q]
Out[14]=	-5 -4 -3 2 1 2 - q + q - q + -- - - - q 2 q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[8, 20]}
In[16]:=	A2Invariant[Knot[8, 20]][q]
Out[16]=	-16 -14 -12 2 2 2 -2 4 -q - q - q + -- + -- + -- + q - q 8 6 4 q q q
In[17]:=	HOMFLYPT[Knot[8, 20]][a, z]
Out[17]=	2 4 2 2 2 4 2 2 4 -1 + 4 a - 2 a - z + 4 a z - a z + a z
In[18]:=	Kauffman[Knot[8, 20]][a, z]
Out[18]=	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[19]:=	{Vassiliev[2][Knot[8, 20]], Vassiliev[3][Knot[8, 20]]}
Out[19]=	{2, -2}
In[20]:=	Kh[Knot[8, 20]][q, t]
Out[20]=	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
In[21]:=	ColouredJones[Knot[8, 20], 2][q]
Out[21]=	-15 -14 -13 2 -11 2 2 2 2 -5 2 1 + q - q - q + --- - q - --- + -- - -- + -- + q - -- + 12 10 9 7 6 4 q q q q q q -3 2 2 2 q + -- - - + q - q 2 q q

See/edit the Rolfsen_Splice_Template.