8 16

8_15

8_17

Visit 8 16's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 8 16's page at Knotilus! Visit 8 16's page at the original Knot Atlas! 8 16 Quick Notes

Square depiction.

Knot presentations

Planar diagram presentation	X6271 X14,6,15,5 X16,11,1,12 X12,7,13,8 X8394 X4,9,5,10 X10,15,11,16 X2,14,3,13
Gauss code	1, -8, 5, -6, 2, -1, 4, -5, 6, -7, 3, -4, 8, -2, 7, -3
Dowker-Thistlethwaite code	6 8 14 12 4 16 2 10
Conway Notation	[.2.20]

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 2 3-genus 3 Bridge index 3 Super bridge index 4 Nakanishi index 1 Maximal Thurston-Bennequin number [-8][-2] Hyperbolic Volume 10.579 A-Polynomial See Data:8 16/A-polynomial

[edit Notes for 8 16's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus $1$ Topological 4 genus $1$ Concordance genus $3$ Rasmussen s-Invariant 2

[edit Notes for 8 16's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{3}-4t^{2}+8t-9+8t^{-1}-4t^{-2}+t^{-3}$
Conway polynomial	$z^{6}+2z^{4}+z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 35, -2 }
Jones polynomial	$-q^{2}+3q-4+6q^{-1}-6q^{-2}+6q^{-3}-5q^{-4}+3q^{-5}-q^{-6}$
HOMFLY-PT polynomial (db, data sources)	$a^{2}z^{6}-a^{4}z^{4}+4a^{2}z^{4}-z^{4}-2a^{4}z^{2}+5a^{2}z^{2}-2z^{2}-a^{4}+2a^{2}$
Kauffman polynomial (db, data sources)	$z^{3}a^{7}+3z^{4}a^{6}-z^{2}a^{6}+5z^{5}a^{5}-5z^{3}a^{5}+2za^{5}+5z^{6}a^{4}-7z^{4}a^{4}+4z^{2}a^{4}-a^{4}+2z^{7}a^{3}+3z^{5}a^{3}-10z^{3}a^{3}+4za^{3}+8z^{6}a^{2}-18z^{4}a^{2}+10z^{2}a^{2}-2a^{2}+2z^{7}a-z^{5}a-6z^{3}a+3za+3z^{6}-8z^{4}+5z^{2}+z^{5}a^{-1}-2z^{3}a^{-1}+za^{-1}$
The A2 invariant	$-q^{18}+q^{16}-q^{14}+q^{10}-q^{8}+2q^{6}-q^{4}+2q^{2}+1+q^{-4}-q^{-6}$
The G2 invariant	$q^{100}-2q^{98}+3q^{96}-4q^{94}+2q^{92}-q^{90}-2q^{88}+9q^{86}-12q^{84}+15q^{82}-14q^{80}+7q^{78}+2q^{76}-16q^{74}+28q^{72}-31q^{70}+24q^{68}-10q^{66}-11q^{64}+26q^{62}-30q^{60}+21q^{58}-5q^{56}-15q^{54}+23q^{52}-19q^{50}+2q^{48}+22q^{46}-36q^{44}+36q^{42}-20q^{40}-4q^{38}+30q^{36}-45q^{34}+46q^{32}-33q^{30}+12q^{28}+14q^{26}-32q^{24}+39q^{22}-30q^{20}+14q^{18}+5q^{16}-20q^{14}+24q^{12}-14q^{10}-q^{8}+21q^{6}-29q^{4}+26q^{2}-6-17q^{-2}+35q^{-4}-37q^{-6}+28q^{-8}-10q^{-10}-12q^{-12}+24q^{-14}-25q^{-16}+20q^{-18}-9q^{-20}-2q^{-22}+6q^{-24}-8q^{-26}+5q^{-28}-2q^{-30}+q^{-32}$

Further Quantum Invariants

Further quantum knot invariants for 8_16.

A1 Invariants.

Weight	Invariant
1	$-q^{13}+2q^{11}-2q^{9}+q^{7}+2q-q^{-1}+2q^{-3}-q^{-5}$
2	$q^{36}-2q^{34}+5q^{30}-7q^{28}-2q^{26}+11q^{24}-6q^{22}-5q^{20}+8q^{18}-q^{16}-5q^{14}+q^{12}+5q^{10}-2q^{8}-5q^{6}+7q^{4}+2q^{2}-9+6q^{-2}+6q^{-4}-8q^{-6}+q^{-8}+6q^{-10}-3q^{-12}-2q^{-14}+q^{-16}$
3	$-q^{69}+2q^{67}-3q^{63}+q^{61}+6q^{59}-16q^{55}+26q^{51}+6q^{49}-33q^{47}-19q^{45}+35q^{43}+28q^{41}-30q^{39}-32q^{37}+19q^{35}+34q^{33}-5q^{31}-28q^{29}-8q^{27}+20q^{25}+15q^{23}-12q^{21}-24q^{19}+5q^{17}+29q^{15}+2q^{13}-32q^{11}-6q^{9}+34q^{7}+16q^{5}-32q^{3}-25q+28q^{-1}+31q^{-3}-16q^{-5}-35q^{-7}+5q^{-9}+33q^{-11}+7q^{-13}-24q^{-15}-13q^{-17}+12q^{-19}+15q^{-21}-4q^{-23}-9q^{-25}-2q^{-27}+3q^{-29}+2q^{-31}-q^{-33}$
4	$q^{112}-2q^{110}+3q^{106}-3q^{104}-4q^{100}+8q^{98}+13q^{96}-16q^{94}-17q^{92}-17q^{90}+36q^{88}+65q^{86}-17q^{84}-75q^{82}-91q^{80}+44q^{78}+160q^{76}+60q^{74}-91q^{72}-198q^{70}-38q^{68}+185q^{66}+163q^{64}-9q^{62}-209q^{60}-134q^{58}+86q^{56}+165q^{54}+89q^{52}-103q^{50}-139q^{48}-29q^{46}+81q^{44}+115q^{42}+12q^{40}-83q^{38}-95q^{36}+3q^{34}+104q^{32}+79q^{30}-45q^{28}-130q^{26}-38q^{24}+99q^{22}+130q^{20}-20q^{18}-163q^{16}-79q^{14}+87q^{12}+180q^{10}+32q^{8}-165q^{6}-139q^{4}+17q^{2}+193+119q^{-2}-89q^{-4}-164q^{-6}-94q^{-8}+114q^{-10}+158q^{-12}+35q^{-14}-91q^{-16}-144q^{-18}-9q^{-20}+90q^{-22}+89q^{-24}+18q^{-26}-80q^{-28}-56q^{-30}-4q^{-32}+41q^{-34}+45q^{-36}-6q^{-38}-20q^{-40}-20q^{-42}-3q^{-44}+12q^{-46}+5q^{-48}+2q^{-50}-3q^{-52}-2q^{-54}+q^{-56}$
5	$-q^{165}+2q^{163}-3q^{159}+3q^{157}+2q^{155}-2q^{153}-4q^{151}-5q^{149}+16q^{145}+23q^{143}-5q^{141}-47q^{139}-56q^{137}+q^{135}+88q^{133}+133q^{131}+54q^{129}-148q^{127}-270q^{125}-158q^{123}+161q^{121}+436q^{119}+366q^{117}-87q^{115}-593q^{113}-639q^{111}-92q^{109}+642q^{107}+901q^{105}+381q^{103}-545q^{101}-1081q^{99}-686q^{97}+319q^{95}+1079q^{93}+926q^{91}-9q^{89}-909q^{87}-1027q^{85}-283q^{83}+632q^{81}+953q^{79}+492q^{77}-306q^{75}-772q^{73}-588q^{71}+26q^{69}+532q^{67}+573q^{65}+188q^{63}-301q^{61}-515q^{59}-314q^{57}+121q^{55}+446q^{53}+394q^{51}-6q^{49}-406q^{47}-447q^{45}-64q^{43}+409q^{41}+512q^{39}+102q^{37}-444q^{35}-591q^{33}-159q^{31}+485q^{29}+704q^{27}+242q^{25}-502q^{23}-819q^{21}-375q^{19}+450q^{17}+918q^{15}+557q^{13}-325q^{11}-945q^{9}-746q^{7}+108q^{5}+868q^{3}+905q+178q^{-1}-685q^{-3}-956q^{-5}-454q^{-7}+383q^{-9}+878q^{-11}+668q^{-13}-46q^{-15}-663q^{-17}-733q^{-19}-257q^{-21}+361q^{-23}+642q^{-25}+439q^{-27}-53q^{-29}-440q^{-31}-465q^{-33}-162q^{-35}+194q^{-37}+354q^{-39}+259q^{-41}+4q^{-43}-201q^{-45}-224q^{-47}-99q^{-49}+52q^{-51}+134q^{-53}+115q^{-55}+21q^{-57}-52q^{-59}-68q^{-61}-39q^{-63}+26q^{-67}+28q^{-69}+8q^{-71}-5q^{-73}-8q^{-75}-5q^{-77}-2q^{-79}+3q^{-81}+2q^{-83}-q^{-85}$
6	$q^{228}-2q^{226}+3q^{222}-3q^{220}-2q^{218}+10q^{214}+q^{212}-8q^{210}-19q^{206}-10q^{204}+21q^{202}+57q^{200}+33q^{198}-31q^{196}-73q^{194}-135q^{192}-75q^{190}+106q^{188}+304q^{186}+281q^{184}+25q^{182}-308q^{180}-659q^{178}-562q^{176}+42q^{174}+883q^{172}+1266q^{170}+813q^{168}-284q^{166}-1689q^{164}-2162q^{162}-1163q^{160}+975q^{158}+2814q^{156}+3056q^{154}+1343q^{152}-1837q^{150}-4170q^{148}-3998q^{146}-992q^{144}+2915q^{142}+5291q^{140}+4527q^{138}+469q^{136}-4039q^{134}-6182q^{132}-4287q^{130}+289q^{128}+4762q^{126}+6374q^{124}+3726q^{122}-1108q^{120}-5157q^{118}-5719q^{116}-2863q^{114}+1612q^{112}+4916q^{110}+4828q^{108}+1928q^{106}-1918q^{104}-4121q^{102}-3794q^{100}-1232q^{98}+1821q^{96}+3366q^{94}+2838q^{92}+684q^{90}-1514q^{88}-2714q^{86}-2150q^{84}-380q^{82}+1434q^{80}+2265q^{78}+1615q^{76}+57q^{74}-1565q^{72}-2025q^{70}-1146q^{68}+573q^{66}+1878q^{64}+1816q^{62}+420q^{60}-1428q^{58}-2213q^{56}-1420q^{54}+671q^{52}+2378q^{50}+2383q^{48}+523q^{46}-1971q^{44}-3147q^{42}-2159q^{40}+730q^{38}+3282q^{36}+3646q^{34}+1333q^{32}-2119q^{30}-4283q^{28}-3695q^{26}-273q^{24}+3444q^{22}+5009q^{20}+3166q^{18}-818q^{16}-4378q^{14}-5287q^{12}-2546q^{10}+1786q^{8}+5046q^{6}+5012q^{4}+1936q^{2}-2346-5245q^{-2}-4675q^{-4}-1385q^{-6}+2685q^{-8}+4881q^{-10}+4252q^{-12}+1110q^{-14}-2616q^{-16}-4443q^{-18}-3709q^{-20}-876q^{-22}+2111q^{-24}+3844q^{-26}+3229q^{-28}+835q^{-30}-1658q^{-32}-3060q^{-34}-2633q^{-36}-978q^{-38}+1160q^{-40}+2332q^{-42}+2106q^{-44}+900q^{-46}-623q^{-48}-1572q^{-50}-1682q^{-52}-795q^{-54}+263q^{-56}+992q^{-58}+1131q^{-60}+696q^{-62}+16q^{-64}-602q^{-66}-702q^{-68}-501q^{-70}-106q^{-72}+244q^{-74}+412q^{-76}+347q^{-78}+92q^{-80}-78q^{-82}-190q^{-84}-179q^{-86}-97q^{-88}+17q^{-90}+83q^{-92}+70q^{-94}+51q^{-96}+7q^{-98}-20q^{-100}-34q^{-102}-16q^{-104}+q^{-108}+8q^{-110}+5q^{-112}+2q^{-114}-3q^{-116}-2q^{-118}+q^{-120}$

A2 Invariants.

Weight	Invariant
1,0	$-q^{18}+q^{16}-q^{14}+q^{10}-q^{8}+2q^{6}-q^{4}+2q^{2}+1+q^{-4}-q^{-6}$
1,1	$q^{52}-4q^{50}+8q^{48}-12q^{46}+22q^{44}-36q^{42}+46q^{40}-62q^{38}+82q^{36}-92q^{34}+96q^{32}-90q^{30}+67q^{28}-34q^{26}-18q^{24}+70q^{22}-120q^{20}+166q^{18}-188q^{16}+200q^{14}-190q^{12}+164q^{10}-124q^{8}+70q^{6}-20q^{4}-28q^{2}+76-100q^{-2}+115q^{-4}-106q^{-6}+94q^{-8}-70q^{-10}+44q^{-12}-28q^{-14}+12q^{-16}-4q^{-18}+q^{-20}$
2,0	$q^{46}-q^{44}+2q^{40}-2q^{38}-2q^{36}+3q^{32}-2q^{30}-4q^{28}+4q^{26}+2q^{24}-3q^{22}+4q^{18}-q^{16}-q^{14}+2q^{12}-3q^{8}-q^{6}+4q^{4}-2q^{2}+6q^{-2}+3q^{-4}-2q^{-6}+2q^{-10}-3q^{-14}-q^{-16}+q^{-18}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{42}-2q^{40}-q^{38}+5q^{36}-3q^{34}-3q^{32}+8q^{30}-4q^{28}-5q^{26}+6q^{24}-4q^{22}-4q^{20}+2q^{18}+q^{16}-q^{12}+5q^{10}+4q^{8}-5q^{6}+4q^{4}+6q^{2}-6+2q^{-2}+4q^{-4}-6q^{-6}+2q^{-8}+q^{-10}-2q^{-12}+q^{-14}$
1,0,0	$-q^{23}+q^{21}-2q^{19}+q^{17}-q^{15}+q^{13}+q^{9}+q^{7}+2q^{3}+2q^{-1}-q^{-3}+q^{-5}-q^{-7}$
1,0,1	$q^{68}-4q^{66}+6q^{64}-2q^{62}-7q^{60}+19q^{58}-24q^{56}+9q^{54}+19q^{52}-46q^{50}+54q^{48}-25q^{46}-26q^{44}+76q^{42}-97q^{40}+71q^{38}-10q^{36}-66q^{34}+109q^{32}-114q^{30}+64q^{28}-2q^{26}-41q^{24}+53q^{22}-21q^{20}+3q^{18}+3q^{16}+30q^{14}-75q^{12}+94q^{10}-80q^{8}+18q^{6}+56q^{4}-104q^{2}+124-82q^{-2}+36q^{-4}+28q^{-6}-59q^{-8}+60q^{-10}-43q^{-12}+10q^{-14}+7q^{-16}-14q^{-18}+10q^{-20}-4q^{-22}+q^{-24}$

A4 Invariants.

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

B2 Invariants.

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

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{58}-2q^{56}+q^{54}-2q^{52}+5q^{50}-5q^{48}+4q^{46}-5q^{44}+8q^{42}-5q^{40}+3q^{38}-4q^{36}+q^{32}-5q^{30}+2q^{28}-9q^{26}+9q^{24}-9q^{22}+11q^{20}-9q^{18}+12q^{16}-5q^{14}+10q^{12}-4q^{10}+4q^{8}+q^{6}+2q^{2}-4+6q^{-2}-6q^{-4}+5q^{-6}-6q^{-8}+5q^{-10}-4q^{-12}+3q^{-14}-2q^{-16}+q^{-18}$

G2 Invariants.

Weight

Invariant

1,0

$q^{100}-2q^{98}+3q^{96}-4q^{94}+2q^{92}-q^{90}-2q^{88}+9q^{86}-12q^{84}+15q^{82}-14q^{80}+7q^{78}+2q^{76}-16q^{74}+28q^{72}-31q^{70}+24q^{68}-10q^{66}-11q^{64}+26q^{62}-30q^{60}+21q^{58}-5q^{56}-15q^{54}+23q^{52}-19q^{50}+2q^{48}+22q^{46}-36q^{44}+36q^{42}-20q^{40}-4q^{38}+30q^{36}-45q^{34}+46q^{32}-33q^{30}+12q^{28}+14q^{26}-32q^{24}+39q^{22}-30q^{20}+14q^{18}+5q^{16}-20q^{14}+24q^{12}-14q^{10}-q^{8}+21q^{6}-29q^{4}+26q^{2}-6-17q^{-2}+35q^{-4}-37q^{-6}+28q^{-8}-10q^{-10}-12q^{-12}+24q^{-14}-25q^{-16}+20q^{-18}-9q^{-20}-2q^{-22}+6q^{-24}-8q^{-26}+5q^{-28}-2q^{-30}+q^{-32}$

.

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 16"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^{3}-4t^{2}+8t-9+8t^{-1}-4t^{-2}+t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

$z^{6}+2z^{4}+z^{2}+1$

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]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{3}a^{7}+3z^{4}a^{6}-z^{2}a^{6}+5z^{5}a^{5}-5z^{3}a^{5}+2za^{5}+5z^{6}a^{4}-7z^{4}a^{4}+4z^{2}a^{4}-a^{4}+2z^{7}a^{3}+3z^{5}a^{3}-10z^{3}a^{3}+4za^{3}+8z^{6}a^{2}-18z^{4}a^{2}+10z^{2}a^{2}-2a^{2}+2z^{7}a-z^{5}a-6z^{3}a+3za+3z^{6}-8z^{4}+5z^{2}+z^{5}a^{-1}-2z^{3}a^{-1}+za^{-1}$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_156, K11n15, K11n56, K11n58, ...}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {10_156, ...}

Vassiliev invariants

V2 and V3:

(1, -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 $4$ $-8$ $8$ ${\frac {14}{3}}$ $-{\frac {38}{3}}$ $-32$ $-{\frac {80}{3}}$ ${\frac {64}{3}}$ $-8$ ${\frac {32}{3}}$ $32$ ${\frac {56}{3}}$ $-{\frac {152}{3}}$ ${\frac {1951}{30}}$ $-{\frac {114}{5}}$ $-{\frac {298}{45}}$ $-{\frac {31}{18}}$ $-{\frac {449}{30}}$

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

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

Integral Khovanov Homology (db, data source)

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

The Coloured Jones Polynomials

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

$n$	$J_{n}$
2	$q^{7}-3q^{6}-q^{5}+10q^{4}-8q^{3}-10q^{2}+24q-8-25q^{-1}+35q^{-2}-3q^{-3}-37q^{-4}+38q^{-5}+4q^{-6}-41q^{-7}+32q^{-8}+8q^{-9}-32q^{-10}+19q^{-11}+7q^{-12}-15q^{-13}+6q^{-14}+2q^{-15}-3q^{-16}+q^{-17}$
3	$-q^{15}+3q^{14}+q^{13}-5q^{12}-8q^{11}+8q^{10}+20q^{9}-8q^{8}-33q^{7}-3q^{6}+51q^{5}+18q^{4}-61q^{3}-43q^{2}+70q+65-64q^{-1}-96q^{-2}+63q^{-3}+113q^{-4}-46q^{-5}-136q^{-6}+37q^{-7}+147q^{-8}-19q^{-9}-160q^{-10}+8q^{-11}+159q^{-12}+8q^{-13}-155q^{-14}-20q^{-15}+139q^{-16}+31q^{-17}-116q^{-18}-35q^{-19}+88q^{-20}+33q^{-21}-58q^{-22}-28q^{-23}+34q^{-24}+19q^{-25}-19q^{-26}-8q^{-27}+8q^{-28}+3q^{-29}-3q^{-30}-2q^{-31}+3q^{-32}-q^{-33}$
4	$q^{26}-3q^{25}-q^{24}+5q^{23}+3q^{22}+8q^{21}-18q^{20}-18q^{19}+5q^{18}+17q^{17}+59q^{16}-22q^{15}-63q^{14}-47q^{13}-7q^{12}+157q^{11}+49q^{10}-62q^{9}-146q^{8}-142q^{7}+210q^{6}+175q^{5}+61q^{4}-190q^{3}-350q^{2}+140q+250+269q^{-1}-116q^{-2}-526q^{-3}-16q^{-4}+224q^{-5}+466q^{-6}+32q^{-7}-619q^{-8}-182q^{-9}+140q^{-10}+609q^{-11}+182q^{-12}-650q^{-13}-319q^{-14}+48q^{-15}+694q^{-16}+306q^{-17}-625q^{-18}-420q^{-19}-50q^{-20}+706q^{-21}+401q^{-22}-522q^{-23}-454q^{-24}-160q^{-25}+596q^{-26}+437q^{-27}-330q^{-28}-378q^{-29}-239q^{-30}+376q^{-31}+362q^{-32}-130q^{-33}-206q^{-34}-217q^{-35}+153q^{-36}+202q^{-37}-23q^{-38}-55q^{-39}-117q^{-40}+37q^{-41}+67q^{-42}-7q^{-43}+3q^{-44}-35q^{-45}+8q^{-46}+14q^{-47}-7q^{-48}+4q^{-49}-6q^{-50}+3q^{-51}+2q^{-52}-3q^{-53}+q^{-54}$
5	$-q^{40}+3q^{39}+q^{38}-5q^{37}-3q^{36}-3q^{35}+2q^{34}+16q^{33}+21q^{32}-7q^{31}-29q^{30}-42q^{29}-27q^{28}+32q^{27}+94q^{26}+87q^{25}-10q^{24}-124q^{23}-178q^{22}-93q^{21}+117q^{20}+292q^{19}+245q^{18}-29q^{17}-338q^{16}-449q^{15}-186q^{14}+317q^{13}+632q^{12}+463q^{11}-135q^{10}-730q^{9}-804q^{8}-159q^{7}+702q^{6}+1080q^{5}+579q^{4}-520q^{3}-1299q^{2}-996q+200+1351q^{-1}+1442q^{-2}+207q^{-3}-1336q^{-4}-1756q^{-5}-654q^{-6}+1152q^{-7}+2062q^{-8}+1089q^{-9}-975q^{-10}-2224q^{-11}-1479q^{-12}+708q^{-13}+2379q^{-14}+1833q^{-15}-513q^{-16}-2443q^{-17}-2123q^{-18}+276q^{-19}+2526q^{-20}+2379q^{-21}-103q^{-22}-2546q^{-23}-2596q^{-24}-107q^{-25}+2567q^{-26}+2779q^{-27}+297q^{-28}-2494q^{-29}-2921q^{-30}-542q^{-31}+2366q^{-32}+2993q^{-33}+786q^{-34}-2109q^{-35}-2962q^{-36}-1048q^{-37}+1752q^{-38}+2809q^{-39}+1252q^{-40}-1311q^{-41}-2501q^{-42}-1369q^{-43}+837q^{-44}+2065q^{-45}+1370q^{-46}-411q^{-47}-1566q^{-48}-1216q^{-49}+77q^{-50}+1060q^{-51}+975q^{-52}+125q^{-53}-640q^{-54}-696q^{-55}-182q^{-56}+326q^{-57}+428q^{-58}+171q^{-59}-134q^{-60}-243q^{-61}-112q^{-62}+51q^{-63}+109q^{-64}+59q^{-65}-12q^{-66}-41q^{-67}-33q^{-68}+6q^{-69}+22q^{-70}+2q^{-71}-3q^{-72}+q^{-73}-5q^{-74}-q^{-75}+6q^{-76}-3q^{-77}-2q^{-78}+3q^{-79}-q^{-80}$
6	$q^{57}-3q^{56}-q^{55}+5q^{54}+3q^{53}+3q^{52}-7q^{51}-19q^{49}-19q^{48}+19q^{47}+30q^{46}+47q^{45}+12q^{44}+13q^{43}-85q^{42}-133q^{41}-63q^{40}+19q^{39}+159q^{38}+182q^{37}+268q^{36}-20q^{35}-301q^{34}-413q^{33}-376q^{32}-42q^{31}+282q^{30}+886q^{29}+660q^{28}+134q^{27}-552q^{26}-1105q^{25}-1100q^{24}-605q^{23}+996q^{22}+1609q^{21}+1657q^{20}+654q^{19}-879q^{18}-2272q^{17}-2743q^{16}-659q^{15}+1182q^{14}+3059q^{13}+3147q^{12}+1515q^{11}-1657q^{10}-4476q^{9}-3646q^{8}-1651q^{7}+2325q^{6}+4974q^{5}+5241q^{4}+1485q^{3}-3847q^{2}-5842q-5721-965q^{-1}+4404q^{-2}+8140q^{-3}+5767q^{-4}-771q^{-5}-5808q^{-6}-8981q^{-7}-5297q^{-8}+1663q^{-9}+9049q^{-10}+9327q^{-11}+3213q^{-12}-3965q^{-13}-10546q^{-14}-9014q^{-15}-1759q^{-16}+8461q^{-17}+11491q^{-18}+6665q^{-19}-1652q^{-20}-10910q^{-21}-11566q^{-22}-4648q^{-23}+7473q^{-24}+12667q^{-25}+9159q^{-26}+208q^{-27}-10915q^{-28}-13273q^{-29}-6739q^{-30}+6680q^{-31}+13452q^{-32}+11007q^{-33}+1604q^{-34}-10853q^{-35}-14578q^{-36}-8458q^{-37}+5801q^{-38}+13912q^{-39}+12629q^{-40}+3162q^{-41}-10203q^{-42}-15409q^{-43}-10272q^{-44}+4031q^{-45}+13348q^{-46}+13829q^{-47}+5360q^{-48}-8049q^{-49}-14881q^{-50}-11817q^{-51}+978q^{-52}+10786q^{-53}+13502q^{-54}+7563q^{-55}-4203q^{-56}-11981q^{-57}-11729q^{-58}-2326q^{-59}+6311q^{-60}+10646q^{-61}+8125q^{-62}-154q^{-63}-7147q^{-64}-9081q^{-65}-3938q^{-66}+1838q^{-67}+6070q^{-68}+6230q^{-69}+1989q^{-70}-2639q^{-71}-5023q^{-72}-3174q^{-73}-538q^{-74}+2163q^{-75}+3224q^{-76}+1817q^{-77}-306q^{-78}-1843q^{-79}-1461q^{-80}-780q^{-81}+324q^{-82}+1086q^{-83}+818q^{-84}+167q^{-85}-438q^{-86}-364q^{-87}-327q^{-88}-59q^{-89}+245q^{-90}+214q^{-91}+70q^{-92}-87q^{-93}-31q^{-94}-71q^{-95}-36q^{-96}+47q^{-97}+33q^{-98}+10q^{-99}-25q^{-100}+11q^{-101}-7q^{-102}-12q^{-103}+11q^{-104}+2q^{-105}+q^{-106}-6q^{-107}+3q^{-108}+2q^{-109}-3q^{-110}+q^{-111}$
7	$-q^{77}+3q^{76}+q^{75}-5q^{74}-3q^{73}-3q^{72}+7q^{71}+5q^{70}+3q^{69}+17q^{68}+7q^{67}-20q^{66}-35q^{65}-50q^{64}-13q^{63}+24q^{62}+39q^{61}+119q^{60}+117q^{59}+50q^{58}-63q^{57}-236q^{56}-265q^{55}-206q^{54}-102q^{53}+231q^{52}+496q^{51}+605q^{50}+507q^{49}-59q^{48}-577q^{47}-990q^{46}-1207q^{45}-684q^{44}+181q^{43}+1236q^{42}+2115q^{41}+1858q^{40}+879q^{39}-694q^{38}-2595q^{37}-3326q^{36}-2899q^{35}-954q^{34}+2197q^{33}+4399q^{32}+5232q^{31}+3816q^{30}-72q^{29}-4116q^{28}-7274q^{27}-7578q^{26}-3711q^{25}+1847q^{24}+7752q^{23}+10998q^{22}+8840q^{21}+2904q^{20}-5758q^{19}-13028q^{18}-14181q^{17}-9460q^{16}+851q^{15}+12286q^{14}+18243q^{13}+16969q^{12}+6719q^{11}-8340q^{10}-19818q^{9}-23784q^{8}-15878q^{7}+1107q^{6}+18085q^{5}+28705q^{4}+25303q^{3}+8356q^{2}-12909q-30600-33687q^{-1}-19117q^{-2}+4988q^{-3}+29521q^{-4}+39865q^{-5}+29561q^{-6}+4823q^{-7}-25363q^{-8}-43600q^{-9}-39112q^{-10}-15197q^{-11}+19379q^{-12}+44807q^{-13}+46617q^{-14}+25329q^{-15}-11993q^{-16}-44047q^{-17}-52503q^{-18}-34421q^{-19}+4638q^{-20}+41971q^{-21}+56398q^{-22}+42097q^{-23}+2550q^{-24}-39239q^{-25}-59199q^{-26}-48383q^{-27}-8627q^{-28}+36527q^{-29}+60871q^{-30}+53343q^{-31}+13846q^{-32}-34098q^{-33}-62288q^{-34}-57325q^{-35}-17878q^{-36}+32218q^{-37}+63392q^{-38}+60622q^{-39}+21301q^{-40}-30840q^{-41}-64672q^{-42}-63577q^{-43}-24173q^{-44}+29684q^{-45}+65879q^{-46}+66514q^{-47}+27181q^{-48}-28413q^{-49}-67044q^{-50}-69481q^{-51}-30530q^{-52}+26397q^{-53}+67545q^{-54}+72462q^{-55}+34715q^{-56}-23142q^{-57}-66976q^{-58}-75019q^{-59}-39541q^{-60}+18181q^{-61}+64426q^{-62}+76462q^{-63}+44850q^{-64}-11370q^{-65}-59471q^{-66}-75993q^{-67}-49674q^{-68}+3119q^{-69}+51715q^{-70}+72699q^{-71}+53060q^{-72}+5813q^{-73}-41513q^{-74}-66212q^{-75}-53989q^{-76}-14044q^{-77}+29826q^{-78}+56667q^{-79}+51610q^{-80}+20314q^{-81}-17896q^{-82}-44973q^{-83}-46094q^{-84}-23700q^{-85}+7529q^{-86}+32625q^{-87}+37962q^{-88}+23751q^{-89}+298q^{-90}-21075q^{-91}-28709q^{-92}-21093q^{-93}-4955q^{-94}+11823q^{-95}+19723q^{-96}+16633q^{-97}+6638q^{-98}-5311q^{-99}-12093q^{-100}-11767q^{-101}-6321q^{-102}+1507q^{-103}+6674q^{-104}+7423q^{-105}+4806q^{-106}+201q^{-107}-3153q^{-108}-4130q^{-109}-3212q^{-110}-698q^{-111}+1305q^{-112}+2093q^{-113}+1843q^{-114}+559q^{-115}-456q^{-116}-903q^{-117}-921q^{-118}-363q^{-119}+107q^{-120}+370q^{-121}+465q^{-122}+154q^{-123}-63q^{-124}-123q^{-125}-162q^{-126}-53q^{-127}-15q^{-128}+35q^{-129}+99q^{-130}+17q^{-131}-25q^{-132}-15q^{-133}-16q^{-134}+9q^{-135}-8q^{-136}-6q^{-137}+22q^{-138}-8q^{-140}-2q^{-141}-q^{-142}+6q^{-143}-3q^{-144}-2q^{-145}+3q^{-146}-q^{-147}$

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, 16]]
Out[2]=	PD[X[6, 2, 7, 1], X[14, 6, 15, 5], X[16, 11, 1, 12], X[12, 7, 13, 8], X[8, 3, 9, 4], X[4, 9, 5, 10], X[10, 15, 11, 16], X[2, 14, 3, 13]]
In[3]:=	GaussCode[Knot[8, 16]]
Out[3]=	GaussCode[1, -8, 5, -6, 2, -1, 4, -5, 6, -7, 3, -4, 8, -2, 7, -3]
In[4]:=	DTCode[Knot[8, 16]]
Out[4]=	DTCode[6, 8, 14, 12, 4, 16, 2, 10]
In[5]:=	br = BR[Knot[8, 16]]
Out[5]=	BR[3, {-1, -1, 2, -1, -1, 2, -1, 2}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{3, 8}
In[7]:=	BraidIndex[Knot[8, 16]]
Out[7]=	3
In[8]:=	Show[DrawMorseLink[Knot[8, 16]]]
Out[8]=	-Graphics-
In[9]:=	(#[Knot[8, 16]]&) /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 2, 3, 3, 4, 1}
In[10]:=	alex = Alexander[Knot[8, 16]][t]
Out[10]=	-3 4 8 2 3 -9 + t - -- + - + 8 t - 4 t + t 2 t t
In[11]:=	Conway[Knot[8, 16]][z]
Out[11]=	2 4 6 1 + z + 2 z + z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[8, 16], Knot[10, 156], Knot[11, NonAlternating, 15], Knot[11, NonAlternating, 56], Knot[11, NonAlternating, 58]}
In[13]:=	{KnotDet[Knot[8, 16]], KnotSignature[Knot[8, 16]]}
Out[13]=	{35, -2}
In[14]:=	Jones[Knot[8, 16]][q]
Out[14]=	-6 3 5 6 6 6 2 -4 - q + -- - -- + -- - -- + - + 3 q - q 5 4 3 2 q q q q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[8, 16], Knot[10, 156]}
In[16]:=	A2Invariant[Knot[8, 16]][q]
Out[16]=	-18 -16 -14 -10 -8 2 -4 2 4 6 1 - q + q - q + q - q + -- - q + -- + q - q 6 2 q q
In[17]:=	HOMFLYPT[Knot[8, 16]][a, z]
Out[17]=	2 4 2 2 2 4 2 4 2 4 4 4 2 6 2 a - a - 2 z + 5 a z - 2 a z - z + 4 a z - a z + a z
In[18]:=	Kauffman[Knot[8, 16]][a, z]
Out[18]=	2 4 z 3 5 2 2 2 4 2 -2 a - a + - + 3 a z + 4 a z + 2 a z + 5 z + 10 a z + 4 a z - a 3 6 2 2 z 3 3 3 5 3 7 3 4 a z - ---- - 6 a z - 10 a z - 5 a z + a z - 8 z - a 5 2 4 4 4 6 4 z 5 3 5 5 5 6 18 a z - 7 a z + 3 a z + -- - a z + 3 a z + 5 a z + 3 z + a 2 6 4 6 7 3 7 8 a z + 5 a z + 2 a z + 2 a z
In[19]:=	{Vassiliev[2][Knot[8, 16]], Vassiliev[3][Knot[8, 16]]}
Out[19]=	{1, -1}
In[20]:=	Kh[Knot[8, 16]][q, t]
Out[20]=	3 4 1 2 1 3 2 3 3 -- + - + ------ + ------ + ----- + ----- + ----- + ----- + ----- + 3 q 13 5 11 4 9 4 9 3 7 3 7 2 5 2 q q t q t q t q t q t q t q t 3 3 2 t 2 3 2 5 3 ---- + ---- + --- + 2 q t + q t + 2 q t + q t 5 3 q q t q t
In[21]:=	ColouredJones[Knot[8, 16], 2][q]
Out[21]=	-17 3 2 6 15 7 19 32 8 32 41 -8 + q - --- + --- + --- - --- + --- + --- - --- + -- + -- - -- + 16 15 14 13 12 11 10 9 8 7 q q q q q q q q q q 4 38 37 3 35 25 2 3 4 5 -- + -- - -- - -- + -- - -- + 24 q - 10 q - 8 q + 10 q - q - 6 5 4 3 2 q q q q q q 6 7 3 q + q

See/edit the Rolfsen_Splice_Template.