10 20

10_19

10_21

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10 20 Further Notes and Views

Knot presentations

Planar diagram presentation	X1425 X11,14,12,15 X3,13,4,12 X13,3,14,2 X5,18,6,19 X7,20,8,1 X19,6,20,7 X9,16,10,17 X15,10,16,11 X17,8,18,9
Gauss code	-1, 4, -3, 1, -5, 7, -6, 10, -8, 9, -2, 3, -4, 2, -9, 8, -10, 5, -7, 6
Dowker-Thistlethwaite code	4 12 18 20 16 14 2 10 8 6
Conway Notation	[352]

Minimum Braid Representative: Length is 12, width is 5. Braid index is 5.

A Morse Link Presentation:

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 2 Bridge index 2 Super bridge index Missing Nakanishi index 1 Maximal Thurston-Bennequin number [-11][-1] Hyperbolic Volume 8.31738 A-Polynomial See Data:10 20/A-polynomial

[edit Notes for 10 20'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 10 20's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-3t^{2}+9t-11+9t^{-1}-3t^{-2}$
Conway polynomial	$-3z^{4}-3z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 35, -2 }
Jones polynomial	$q-1+3q^{-1}-4q^{-2}+5q^{-3}-6q^{-4}+5q^{-5}-4q^{-6}+3q^{-7}-2q^{-8}+q^{-9}$
HOMFLY-PT polynomial (db, data sources)	$z^{2}a^{8}+a^{8}-z^{4}a^{6}-2z^{2}a^{6}-a^{6}-z^{4}a^{4}-z^{2}a^{4}-z^{4}a^{2}-2z^{2}a^{2}-a^{2}+z^{2}+2$
Kauffman polynomial (db, data sources)	$z^{6}a^{10}-4z^{4}a^{10}+3z^{2}a^{10}+2z^{7}a^{9}-8z^{5}a^{9}+7z^{3}a^{9}-za^{9}+2z^{8}a^{8}-8z^{6}a^{8}+9z^{4}a^{8}-5z^{2}a^{8}+a^{8}+z^{9}a^{7}-3z^{7}a^{7}+3z^{5}a^{7}-4z^{3}a^{7}+2za^{7}+3z^{8}a^{6}-12z^{6}a^{6}+17z^{4}a^{6}-9z^{2}a^{6}+a^{6}+z^{9}a^{5}-4z^{7}a^{5}+9z^{5}a^{5}-8z^{3}a^{5}+3za^{5}+z^{8}a^{4}-2z^{6}a^{4}+3z^{4}a^{4}+z^{7}a^{3}-z^{5}a^{3}+2z^{3}a^{3}-za^{3}+z^{6}a^{2}-2z^{2}a^{2}+a^{2}+z^{5}a-z^{3}a-za+z^{4}-3z^{2}+2$
The A2 invariant	$q^{28}+q^{22}-q^{20}-q^{14}-q^{10}-q^{4}+2q^{2}+1+q^{-2}+q^{-4}$
The G2 invariant	$q^{142}-q^{140}+2q^{138}-3q^{136}+2q^{134}-2q^{132}-q^{130}+6q^{128}-9q^{126}+10q^{124}-9q^{122}+3q^{120}+4q^{118}-11q^{116}+16q^{114}-13q^{112}+9q^{110}+q^{108}-8q^{106}+12q^{104}-9q^{102}+3q^{100}+3q^{98}-7q^{96}+7q^{94}-2q^{92}-4q^{90}+11q^{88}-12q^{86}+9q^{84}-4q^{82}-5q^{80}+10q^{78}-15q^{76}+15q^{74}-9q^{72}+3q^{70}+5q^{68}-12q^{66}+13q^{64}-10q^{62}+4q^{60}+q^{58}-7q^{56}+7q^{54}-3q^{52}-2q^{50}+5q^{48}-6q^{46}+q^{44}+2q^{42}-7q^{40}+7q^{38}-5q^{36}+3q^{34}-q^{32}-3q^{30}+4q^{28}-5q^{26}+5q^{24}-5q^{22}+4q^{20}-3q^{18}-q^{16}+4q^{14}-6q^{12}+8q^{10}-4q^{8}+3q^{6}+q^{4}-q^{2}+4-3q^{-2}+4q^{-4}-q^{-6}+q^{-8}+q^{-10}-q^{-12}+2q^{-14}+q^{-18}$

Further Quantum Invariants

Further quantum knot invariants for 10_20.

A1 Invariants.

Weight	Invariant
1	$q^{19}-q^{17}+q^{15}-q^{13}+q^{11}-q^{9}-q^{7}+q^{5}-q^{3}+2q+q^{-3}$
2	$q^{54}-q^{52}-q^{50}+3q^{48}-q^{46}-4q^{44}+3q^{42}+2q^{40}-4q^{38}+q^{36}+3q^{34}-3q^{32}-q^{30}+3q^{28}-q^{26}-q^{24}+2q^{22}+2q^{20}-2q^{18}-2q^{16}+4q^{14}-q^{12}-3q^{10}+3q^{8}-q^{6}-2q^{4}+2q^{2}-1+2q^{-4}+q^{-10}$
3	$q^{105}-q^{103}-q^{101}+q^{99}+2q^{97}-q^{95}-5q^{93}+7q^{89}+3q^{87}-6q^{85}-7q^{83}+4q^{81}+9q^{79}-q^{77}-8q^{75}-4q^{73}+6q^{71}+7q^{69}-2q^{67}-8q^{65}-2q^{63}+8q^{61}+6q^{59}-7q^{57}-6q^{55}+6q^{53}+7q^{51}-6q^{49}-7q^{47}+2q^{45}+6q^{43}-2q^{41}-6q^{39}+5q^{35}+6q^{33}-4q^{31}-8q^{29}+10q^{25}+3q^{23}-7q^{21}-7q^{19}+6q^{17}+6q^{15}+q^{13}-5q^{11}-2q^{9}+3q^{7}+3q^{5}-4q-q^{-1}+2q^{-3}+2q^{-5}-2q^{-7}-q^{-9}+2q^{-13}+q^{-21}$
4	$q^{172}-q^{170}-q^{168}+q^{166}+2q^{162}-3q^{160}-3q^{158}+2q^{156}+2q^{154}+9q^{152}-3q^{150}-11q^{148}-5q^{146}-q^{144}+18q^{142}+9q^{140}-6q^{138}-13q^{136}-18q^{134}+10q^{132}+16q^{130}+10q^{128}-20q^{124}-8q^{122}-2q^{120}+11q^{118}+21q^{116}+2q^{114}-8q^{112}-25q^{110}-14q^{108}+20q^{106}+27q^{104}+13q^{102}-28q^{100}-36q^{98}+4q^{96}+33q^{94}+30q^{92}-15q^{90}-40q^{88}-10q^{86}+24q^{84}+29q^{82}-5q^{80}-30q^{78}-11q^{76}+16q^{74}+22q^{72}+q^{70}-19q^{68}-11q^{66}+10q^{64}+14q^{62}+8q^{60}-8q^{58}-20q^{56}-8q^{54}+8q^{52}+29q^{50}+14q^{48}-26q^{46}-34q^{44}-13q^{42}+37q^{40}+45q^{38}-7q^{36}-43q^{34}-41q^{32}+18q^{30}+51q^{28}+17q^{26}-20q^{24}-43q^{22}-6q^{20}+30q^{18}+22q^{16}+3q^{14}-26q^{12}-13q^{10}+10q^{8}+13q^{6}+11q^{4}-11q^{2}-10+q^{-2}+5q^{-4}+9q^{-6}-4q^{-8}-5q^{-10}-2q^{-12}+5q^{-16}-q^{-18}-q^{-20}-q^{-22}-q^{-24}+2q^{-26}+q^{-36}$
5	$q^{255}-q^{253}-q^{251}+q^{249}-q^{241}-2q^{239}+2q^{237}+5q^{235}+2q^{233}-q^{231}-6q^{229}-9q^{227}-5q^{225}+9q^{223}+17q^{221}+11q^{219}-q^{217}-19q^{215}-25q^{213}-13q^{211}+14q^{209}+31q^{207}+26q^{205}+5q^{203}-24q^{201}-36q^{199}-23q^{197}+8q^{195}+29q^{193}+29q^{191}+16q^{189}-6q^{187}-22q^{185}-28q^{183}-21q^{181}-6q^{179}+18q^{177}+43q^{175}+44q^{173}+10q^{171}-45q^{169}-75q^{167}-54q^{165}+19q^{163}+95q^{161}+99q^{159}+19q^{157}-88q^{155}-130q^{153}-66q^{151}+60q^{149}+144q^{147}+108q^{145}-23q^{143}-139q^{141}-133q^{139}-15q^{137}+115q^{135}+146q^{133}+48q^{131}-91q^{129}-141q^{127}-62q^{125}+62q^{123}+123q^{121}+72q^{119}-42q^{117}-103q^{115}-63q^{113}+28q^{111}+79q^{109}+53q^{107}-20q^{105}-65q^{103}-42q^{101}+16q^{99}+48q^{97}+35q^{95}-6q^{93}-42q^{91}-42q^{89}-8q^{87}+38q^{85}+54q^{83}+36q^{81}-20q^{79}-78q^{77}-75q^{75}-3q^{73}+89q^{71}+121q^{69}+48q^{67}-83q^{65}-158q^{63}-99q^{61}+55q^{59}+181q^{57}+146q^{55}-18q^{53}-164q^{51}-174q^{49}-33q^{47}+134q^{45}+174q^{43}+58q^{41}-87q^{39}-144q^{37}-81q^{35}+46q^{33}+110q^{31}+71q^{29}-17q^{27}-72q^{25}-55q^{23}+45q^{19}+41q^{17}+4q^{15}-28q^{13}-25q^{11}-3q^{9}+18q^{7}+20q^{5}+3q^{3}-13q-13q^{-1}-4q^{-3}+10q^{-5}+13q^{-7}+3q^{-9}-6q^{-11}-7q^{-13}-6q^{-15}+2q^{-17}+9q^{-19}+3q^{-21}-q^{-23}-2q^{-25}-4q^{-27}-2q^{-29}+3q^{-31}+q^{-33}-q^{-39}-q^{-41}+q^{-43}+q^{-55}$

A2 Invariants.

Weight	Invariant
1,0	$q^{28}+q^{22}-q^{20}-q^{14}-q^{10}-q^{4}+2q^{2}+1+q^{-2}+q^{-4}$
1,1	$q^{76}-2q^{74}+4q^{72}-8q^{70}+15q^{68}-22q^{66}+30q^{64}-38q^{62}+43q^{60}-44q^{58}+38q^{56}-28q^{54}+11q^{52}+8q^{50}-26q^{48}+44q^{46}-59q^{44}+64q^{42}-68q^{40}+64q^{38}-55q^{36}+46q^{34}-28q^{32}+18q^{30}-3q^{28}-4q^{26}+12q^{24}-12q^{22}+13q^{20}-12q^{18}+10q^{16}-14q^{14}+11q^{12}-18q^{10}+14q^{8}-18q^{6}+16q^{4}-12q^{2}+12-6q^{-2}+8q^{-4}-2q^{-6}+4q^{-8}+q^{-12}$
2,0	$q^{72}+q^{64}-3q^{60}-q^{58}+q^{56}+q^{54}-q^{52}-q^{50}+2q^{48}+q^{46}-2q^{44}-2q^{42}+q^{40}+q^{38}-q^{36}+2q^{32}+2q^{30}+3q^{26}+q^{24}-q^{22}+2q^{20}+q^{18}-3q^{16}-3q^{14}+q^{12}-5q^{8}-3q^{6}+2q^{4}+2q^{-2}+3q^{-4}+q^{-6}+q^{-8}+q^{-10}+q^{-12}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{60}-q^{58}+q^{54}-2q^{52}+q^{50}+q^{48}-2q^{46}+2q^{44}-3q^{40}+2q^{38}+2q^{36}-2q^{34}+2q^{32}+3q^{30}-q^{28}+q^{22}-3q^{20}-q^{18}+2q^{16}-4q^{14}-3q^{12}+2q^{10}-2q^{8}-3q^{6}+4q^{4}+2q^{2}+1+3q^{-2}+2q^{-4}+q^{-8}$
1,0,0	$q^{37}+q^{33}+q^{29}-q^{27}-q^{23}-q^{19}-q^{13}-q^{9}-q^{5}+2q^{3}+q+2q^{-1}+q^{-3}+q^{-5}$

B2 Invariants.

Weight	Invariant
0,1	$q^{60}-q^{58}+2q^{56}-3q^{54}+4q^{52}-5q^{50}+5q^{48}-4q^{46}+4q^{44}-2q^{42}+q^{40}+2q^{38}-4q^{36}+6q^{34}-8q^{32}+7q^{30}-9q^{28}+8q^{26}-8q^{24}+5q^{22}-3q^{20}+q^{18}-2q^{14}+3q^{12}-4q^{10}+4q^{8}-3q^{6}+4q^{4}-2q^{2}+3-q^{-2}+2q^{-4}+q^{-8}$
1,0	$q^{98}-q^{94}-q^{92}+q^{90}+2q^{88}-q^{86}-3q^{84}+4q^{80}+2q^{78}-4q^{76}-4q^{74}+2q^{72}+5q^{70}-5q^{66}-3q^{64}+3q^{62}+4q^{60}-3q^{56}+3q^{52}+q^{50}-q^{48}-q^{46}+3q^{44}+2q^{42}-2q^{40}-3q^{38}+2q^{36}+3q^{34}-q^{32}-4q^{30}-q^{28}+3q^{26}+q^{24}-4q^{22}-4q^{20}+3q^{16}+q^{14}-3q^{12}-3q^{10}+4q^{6}+2q^{4}-1+q^{-2}+2q^{-4}+2q^{-6}+q^{-14}$

G2 Invariants.

Weight

Invariant

1,0

$q^{142}-q^{140}+2q^{138}-3q^{136}+2q^{134}-2q^{132}-q^{130}+6q^{128}-9q^{126}+10q^{124}-9q^{122}+3q^{120}+4q^{118}-11q^{116}+16q^{114}-13q^{112}+9q^{110}+q^{108}-8q^{106}+12q^{104}-9q^{102}+3q^{100}+3q^{98}-7q^{96}+7q^{94}-2q^{92}-4q^{90}+11q^{88}-12q^{86}+9q^{84}-4q^{82}-5q^{80}+10q^{78}-15q^{76}+15q^{74}-9q^{72}+3q^{70}+5q^{68}-12q^{66}+13q^{64}-10q^{62}+4q^{60}+q^{58}-7q^{56}+7q^{54}-3q^{52}-2q^{50}+5q^{48}-6q^{46}+q^{44}+2q^{42}-7q^{40}+7q^{38}-5q^{36}+3q^{34}-q^{32}-3q^{30}+4q^{28}-5q^{26}+5q^{24}-5q^{22}+4q^{20}-3q^{18}-q^{16}+4q^{14}-6q^{12}+8q^{10}-4q^{8}+3q^{6}+q^{4}-q^{2}+4-3q^{-2}+4q^{-4}-q^{-6}+q^{-8}+q^{-10}-q^{-12}+2q^{-14}+q^{-18}$

.

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["10 20"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-3t^{2}+9t-11+9t^{-1}-3t^{-2}$

In[5]:=

Conway[K][z]

Out[5]=

$-3z^{4}-3z^{2}+1$

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]=

{ 35, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q-1+3q^{-1}-4q^{-2}+5q^{-3}-6q^{-4}+5q^{-5}-4q^{-6}+3q^{-7}-2q^{-8}+q^{-9}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^{2}a^{8}+a^{8}-z^{4}a^{6}-2z^{2}a^{6}-a^{6}-z^{4}a^{4}-z^{2}a^{4}-z^{4}a^{2}-2z^{2}a^{2}-a^{2}+z^{2}+2$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{6}a^{10}-4z^{4}a^{10}+3z^{2}a^{10}+2z^{7}a^{9}-8z^{5}a^{9}+7z^{3}a^{9}-za^{9}+2z^{8}a^{8}-8z^{6}a^{8}+9z^{4}a^{8}-5z^{2}a^{8}+a^{8}+z^{9}a^{7}-3z^{7}a^{7}+3z^{5}a^{7}-4z^{3}a^{7}+2za^{7}+3z^{8}a^{6}-12z^{6}a^{6}+17z^{4}a^{6}-9z^{2}a^{6}+a^{6}+z^{9}a^{5}-4z^{7}a^{5}+9z^{5}a^{5}-8z^{3}a^{5}+3za^{5}+z^{8}a^{4}-2z^{6}a^{4}+3z^{4}a^{4}+z^{7}a^{3}-z^{5}a^{3}+2z^{3}a^{3}-za^{3}+z^{6}a^{2}-2z^{2}a^{2}+a^{2}+z^{5}a-z^{3}a-za+z^{4}-3z^{2}+2$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_162, K11n117, ...}

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

Vassiliev invariants

V2 and V3:

(-3, 6)

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 $-12$ $48$ $72$ $18$ $62$ $-576$ $-896$ $-256$ $-272$ $-288$ $1152$ $-216$ $-744$ ${\frac {30689}{10}}$ ${\frac {6826}{15}}$ ${\frac {7858}{15}}$ ${\frac {2527}{6}}$ $-{\frac {1951}{10}}$

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

\   r    \    j   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 3                     1 1 1                       0 -1                 3 1   2 -3               2 1     -1 -5             3 2       1 -7           3 2         -1 -9         2 3           -1 -11       2 3             1 -13     1 2               -1 -15   1 2                 1 -17   1                   -1 -19 1                     1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-3$ $i=-1$ $r=-8$ ${\mathbb {Z} }$ $r=-7$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-3$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-2$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-1$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=0$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{3}$ $r=1$ ${\mathbb {Z} }$ $r=2$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

The Coloured Jones Polynomials

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

$n$	$J_{n}$
2	$q^{4}-q^{3}+3q-3-q^{-1}+6q^{-2}-7q^{-3}+10q^{-5}-13q^{-6}+2q^{-7}+15q^{-8}-19q^{-9}+2q^{-10}+19q^{-11}-19q^{-12}-q^{-13}+19q^{-14}-15q^{-15}-5q^{-16}+17q^{-17}-9q^{-18}-7q^{-19}+12q^{-20}-3q^{-21}-6q^{-22}+5q^{-23}-2q^{-25}+q^{-26}$
3	$q^{9}-q^{8}+3q^{5}-3q^{4}-q^{3}-q^{2}+7q-3-4q^{-1}-4q^{-2}+11q^{-3}-4q^{-5}-9q^{-6}+8q^{-7}+6q^{-8}+q^{-9}-9q^{-10}-5q^{-11}+6q^{-12}+11q^{-13}-2q^{-14}-15q^{-15}-2q^{-16}+15q^{-17}+8q^{-18}-16q^{-19}-7q^{-20}+9q^{-21}+12q^{-22}-8q^{-23}-11q^{-24}+13q^{-26}+5q^{-27}-12q^{-28}-12q^{-29}+12q^{-30}+18q^{-31}-10q^{-32}-22q^{-33}+6q^{-34}+24q^{-35}-q^{-36}-23q^{-37}-4q^{-38}+20q^{-39}+6q^{-40}-13q^{-41}-9q^{-42}+9q^{-43}+7q^{-44}-4q^{-45}-5q^{-46}+2q^{-47}+2q^{-48}-2q^{-50}+q^{-51}$
4	$q^{16}-q^{15}+3q^{11}-4q^{10}+9q^{6}-9q^{5}-2q^{4}-3q^{3}+q^{2}+22q-13-6q^{-1}-14q^{-2}+44q^{-4}-11q^{-5}-9q^{-6}-37q^{-7}-13q^{-8}+73q^{-9}+8q^{-10}-q^{-11}-73q^{-12}-50q^{-13}+96q^{-14}+45q^{-15}+33q^{-16}-106q^{-17}-109q^{-18}+94q^{-19}+81q^{-20}+85q^{-21}-114q^{-22}-159q^{-23}+73q^{-24}+89q^{-25}+125q^{-26}-99q^{-27}-180q^{-28}+57q^{-29}+77q^{-30}+137q^{-31}-83q^{-32}-174q^{-33}+53q^{-34}+56q^{-35}+129q^{-36}-63q^{-37}-153q^{-38}+47q^{-39}+29q^{-40}+110q^{-41}-38q^{-42}-119q^{-43}+42q^{-44}-5q^{-45}+80q^{-46}-13q^{-47}-74q^{-48}+45q^{-49}-34q^{-50}+40q^{-51}-5q^{-52}-33q^{-53}+59q^{-54}-41q^{-55}+6q^{-56}-16q^{-57}-16q^{-58}+69q^{-59}-22q^{-60}-4q^{-61}-29q^{-62}-22q^{-63}+57q^{-64}-2q^{-65}+6q^{-66}-23q^{-67}-28q^{-68}+29q^{-69}+3q^{-70}+13q^{-71}-8q^{-72}-19q^{-73}+10q^{-74}-q^{-75}+7q^{-76}-7q^{-78}+3q^{-79}-q^{-80}+2q^{-81}-2q^{-83}+q^{-84}$
5	Not Available
6	Not Available
7	Not Available

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[10, 20]]
Out[2]=	PD[X[1, 4, 2, 5], X[11, 14, 12, 15], X[3, 13, 4, 12], X[13, 3, 14, 2], X[5, 18, 6, 19], X[7, 20, 8, 1], X[19, 6, 20, 7], X[9, 16, 10, 17], X[15, 10, 16, 11], X[17, 8, 18, 9]]
In[3]:=	GaussCode[Knot[10, 20]]
Out[3]=	GaussCode[-1, 4, -3, 1, -5, 7, -6, 10, -8, 9, -2, 3, -4, 2, -9, 8, -10, 5, -7, 6]
In[4]:=	DTCode[Knot[10, 20]]
Out[4]=	DTCode[4, 12, 18, 20, 16, 14, 2, 10, 8, 6]
In[5]:=	br = BR[Knot[10, 20]]
Out[5]=	BR[5, {-1, -1, -1, -1, -2, 1, -2, -3, 2, 4, -3, 4}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{5, 12}
In[7]:=	BraidIndex[Knot[10, 20]]
Out[7]=	5
In[8]:=	Show[DrawMorseLink[Knot[10, 20]]]
Out[8]=	-Graphics-
In[9]:=	(#[Knot[10, 20]]&) /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 2, 2, 2, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 20]][t]
Out[10]=	3 9 2 -11 - -- + - + 9 t - 3 t 2 t t
In[11]:=	Conway[Knot[10, 20]][z]
Out[11]=	2 4 1 - 3 z - 3 z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 20], Knot[10, 162], Knot[11, NonAlternating, 117]}
In[13]:=	{KnotDet[Knot[10, 20]], KnotSignature[Knot[10, 20]]}
Out[13]=	{35, -2}
In[14]:=	Jones[Knot[10, 20]][q]
Out[14]=	-9 2 3 4 5 6 5 4 3 -1 + q - -- + -- - -- + -- - -- + -- - -- + - + q 8 7 6 5 4 3 2 q q q q q q q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[10, 20]}
In[16]:=	A2Invariant[Knot[10, 20]][q]
Out[16]=	-28 -22 -20 -14 -10 -4 2 2 4 1 + q + q - q - q - q - q + -- + q + q 2 q
In[17]:=	HOMFLYPT[Knot[10, 20]][a, z]
Out[17]=	2 6 8 2 2 2 4 2 6 2 8 2 2 4 2 - a - a + a + z - 2 a z - a z - 2 a z + a z - a z - 4 4 6 4 a z - a z
In[18]:=	Kauffman[Knot[10, 20]][a, z]
Out[18]=	2 6 8 3 5 7 9 2 2 + a + a + a - a z - a z + 3 a z + 2 a z - a z - 3 z - 2 2 6 2 8 2 10 2 3 3 3 5 3 2 a z - 9 a z - 5 a z + 3 a z - a z + 2 a z - 8 a z - 7 3 9 3 4 4 4 6 4 8 4 10 4 4 a z + 7 a z + z + 3 a z + 17 a z + 9 a z - 4 a z + 5 3 5 5 5 7 5 9 5 2 6 4 6 a z - a z + 9 a z + 3 a z - 8 a z + a z - 2 a z - 6 6 8 6 10 6 3 7 5 7 7 7 9 7 12 a z - 8 a z + a z + a z - 4 a z - 3 a z + 2 a z + 4 8 6 8 8 8 5 9 7 9 a z + 3 a z + 2 a z + a z + a z
In[19]:=	{Vassiliev[2][Knot[10, 20]], Vassiliev[3][Knot[10, 20]]}
Out[19]=	{-3, 6}
In[20]:=	Kh[Knot[10, 20]][q, t]
Out[20]=	-3 3 1 1 1 2 1 2 q + - + ------ + ------ + ------ + ------ + ------ + ------ + q 19 8 17 7 15 7 15 6 13 6 13 5 q t q t q t q t q t q t 2 3 2 3 3 2 3 2 ------ + ------ + ----- + ----- + ----- + ----- + ----- + ---- + 11 5 11 4 9 4 9 3 7 3 7 2 5 2 5 q t q t q t q t q t q t q t q t 2 t 3 2 ---- + - + q t 3 q q t
In[21]:=	ColouredJones[Knot[10, 20], 2][q]
Out[21]=	-26 2 5 6 3 12 7 9 17 5 15 -3 + q - --- + --- - --- - --- + --- - --- - --- + --- - --- - --- + 25 23 22 21 20 19 18 17 16 15 q q q q q q q q q q 19 -13 19 19 2 19 15 2 13 10 7 6 1 --- - q - --- + --- + --- - -- + -- + -- - -- + -- - -- + -- - - + 14 12 11 10 9 8 7 6 5 3 2 q q q q q q q q q q q q 3 4 3 q - q + q

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