8 9

8_8

8_10

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8 9 Further Notes and Views

Knot presentations

Planar diagram presentation	X6271 X14,8,15,7 X10,3,11,4 X2,13,3,14 X12,5,13,6 X4,11,5,12 X16,10,1,9 X8,16,9,15
Gauss code	1, -4, 3, -6, 5, -1, 2, -8, 7, -3, 6, -5, 4, -2, 8, -7
Dowker-Thistlethwaite code	6 10 12 14 16 4 2 8
Conway Notation	[3113]

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

A Morse Link Presentation:

Three dimensional invariants

Symmetry type Fully amphicheiral Unknotting number 1 3-genus 3 Bridge index 2 Super bridge index $\{3,6\}$ Nakanishi index 1 Maximal Thurston-Bennequin number [-5][-5] Hyperbolic Volume 7.58818 A-Polynomial See Data:8 9/A-polynomial

[edit Notes for 8 9's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for 8 9's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{3}+3t^{2}-5t+7-5t^{-1}+3t^{-2}-t^{-3}$
Conway polynomial	$-z^{6}-3z^{4}-2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 25, 0 }
Jones polynomial	$q^{4}-2q^{3}+3q^{2}-4q+5-4q^{-1}+3q^{-2}-2q^{-3}+q^{-4}$
HOMFLY-PT polynomial (db, data sources)	$-z^{6}+a^{2}z^{4}+z^{4}a^{-2}-5z^{4}+3a^{2}z^{2}+3z^{2}a^{-2}-8z^{2}+2a^{2}+2a^{-2}-3$
Kauffman polynomial (db, data sources)	$az^{7}+z^{7}a^{-1}+2a^{2}z^{6}+2z^{6}a^{-2}+4z^{6}+2a^{3}z^{5}+2z^{5}a^{-3}+a^{4}z^{4}-4a^{2}z^{4}-4z^{4}a^{-2}+z^{4}a^{-4}-10z^{4}-4a^{3}z^{3}-az^{3}-z^{3}a^{-1}-4z^{3}a^{-3}-2a^{4}z^{2}+4a^{2}z^{2}+4z^{2}a^{-2}-2z^{2}a^{-4}+12z^{2}+a^{3}z+az+za^{-1}+za^{-3}-2a^{2}-2a^{-2}-3$
The A2 invariant	$q^{12}+q^{8}-q^{4}+q^{2}-1+q^{-2}-q^{-4}+q^{-8}+q^{-12}$
The G2 invariant	$q^{66}-q^{64}+2q^{62}-3q^{60}+q^{58}-3q^{54}+6q^{52}-6q^{50}+7q^{48}-5q^{46}+6q^{42}-10q^{40}+12q^{38}-8q^{36}+4q^{34}+3q^{32}-6q^{30}+11q^{28}-6q^{26}+2q^{24}+4q^{22}-7q^{20}+5q^{18}-8q^{14}+11q^{12}-10q^{10}+9q^{8}-3q^{6}-10q^{4}+14q^{2}-17+14q^{-2}-10q^{-4}-3q^{-6}+9q^{-8}-10q^{-10}+11q^{-12}-8q^{-14}+5q^{-18}-7q^{-20}+4q^{-22}+2q^{-24}-6q^{-26}+11q^{-28}-6q^{-30}+3q^{-32}+4q^{-34}-8q^{-36}+12q^{-38}-10q^{-40}+6q^{-42}-5q^{-46}+7q^{-48}-6q^{-50}+6q^{-52}-3q^{-54}+q^{-58}-3q^{-60}+2q^{-62}-q^{-64}+q^{-66}$

Further Quantum Invariants

Further quantum knot invariants for 8_9.

A1 Invariants.

Weight	Invariant
1	$q^{9}-q^{7}+q^{5}-q^{3}+q+q^{-1}-q^{-3}+q^{-5}-q^{-7}+q^{-9}$
2	$q^{26}-q^{24}-q^{22}+3q^{20}-q^{18}-3q^{16}+4q^{14}-5q^{10}+3q^{8}+q^{6}-3q^{4}+2q^{2}+3+2q^{-2}-3q^{-4}+q^{-6}+3q^{-8}-5q^{-10}+4q^{-14}-3q^{-16}-q^{-18}+3q^{-20}-q^{-22}-q^{-24}+q^{-26}$
3	$q^{51}-q^{49}-q^{47}+q^{45}+2q^{43}-q^{41}-4q^{39}+q^{37}+6q^{35}-8q^{31}-3q^{29}+9q^{27}+7q^{25}-9q^{23}-9q^{21}+8q^{19}+11q^{17}-6q^{15}-11q^{13}+4q^{11}+9q^{9}-q^{7}-8q^{5}+5q+5q^{-1}-8q^{-5}-q^{-7}+9q^{-9}+4q^{-11}-11q^{-13}-6q^{-15}+11q^{-17}+8q^{-19}-9q^{-21}-9q^{-23}+7q^{-25}+9q^{-27}-3q^{-29}-8q^{-31}+6q^{-35}+q^{-37}-4q^{-39}-q^{-41}+2q^{-43}+q^{-45}-q^{-47}-q^{-49}+q^{-51}$
4	$q^{84}-q^{82}-q^{80}+q^{78}+2q^{74}-3q^{72}-2q^{70}+3q^{68}+q^{66}+6q^{64}-6q^{62}-9q^{60}+q^{58}+5q^{56}+17q^{54}-3q^{52}-17q^{50}-13q^{48}-2q^{46}+31q^{44}+16q^{42}-12q^{40}-29q^{38}-21q^{36}+31q^{34}+32q^{32}+4q^{30}-31q^{28}-36q^{26}+18q^{24}+31q^{22}+14q^{20}-19q^{18}-33q^{16}+5q^{14}+20q^{12}+16q^{10}-6q^{8}-20q^{6}-4q^{4}+8q^{2}+15+8q^{-2}-4q^{-4}-20q^{-6}-6q^{-8}+16q^{-10}+20q^{-12}+5q^{-14}-33q^{-16}-19q^{-18}+14q^{-20}+31q^{-22}+18q^{-24}-36q^{-26}-31q^{-28}+4q^{-30}+32q^{-32}+31q^{-34}-21q^{-36}-29q^{-38}-12q^{-40}+16q^{-42}+31q^{-44}-2q^{-46}-13q^{-48}-17q^{-50}-3q^{-52}+17q^{-54}+5q^{-56}+q^{-58}-9q^{-60}-6q^{-62}+6q^{-64}+q^{-66}+3q^{-68}-2q^{-70}-3q^{-72}+2q^{-74}+q^{-78}-q^{-80}-q^{-82}+q^{-84}$
5	$q^{125}-q^{123}-q^{121}+q^{119}-q^{111}-q^{109}+3q^{107}+4q^{105}-q^{103}-4q^{101}-6q^{99}-4q^{97}+4q^{95}+13q^{93}+11q^{91}-3q^{89}-17q^{87}-22q^{85}-8q^{83}+17q^{81}+36q^{79}+28q^{77}-8q^{75}-42q^{73}-51q^{71}-18q^{69}+40q^{67}+73q^{65}+50q^{63}-23q^{61}-87q^{59}-83q^{57}-6q^{55}+85q^{53}+111q^{51}+39q^{49}-72q^{47}-124q^{45}-67q^{43}+49q^{41}+122q^{39}+87q^{37}-25q^{35}-108q^{33}-91q^{31}+5q^{29}+85q^{27}+84q^{25}+11q^{23}-63q^{21}-73q^{19}-15q^{17}+42q^{15}+54q^{13}+23q^{11}-22q^{9}-45q^{7}-22q^{5}+8q^{3}+31q+31q^{-1}+8q^{-3}-22q^{-5}-45q^{-7}-22q^{-9}+23q^{-11}+54q^{-13}+42q^{-15}-15q^{-17}-73q^{-19}-63q^{-21}+11q^{-23}+84q^{-25}+85q^{-27}+5q^{-29}-91q^{-31}-108q^{-33}-25q^{-35}+87q^{-37}+122q^{-39}+49q^{-41}-67q^{-43}-124q^{-45}-72q^{-47}+39q^{-49}+111q^{-51}+85q^{-53}-6q^{-55}-83q^{-57}-87q^{-59}-23q^{-61}+50q^{-63}+73q^{-65}+40q^{-67}-18q^{-69}-51q^{-71}-42q^{-73}-8q^{-75}+28q^{-77}+36q^{-79}+17q^{-81}-8q^{-83}-22q^{-85}-17q^{-87}-3q^{-89}+11q^{-91}+13q^{-93}+4q^{-95}-4q^{-97}-6q^{-99}-4q^{-101}-q^{-103}+4q^{-105}+3q^{-107}-q^{-109}-q^{-111}+q^{-119}-q^{-121}-q^{-123}+q^{-125}$
6	$q^{174}-q^{172}-q^{170}+q^{168}-2q^{162}+2q^{160}-q^{156}+5q^{154}+2q^{152}-2q^{150}-9q^{148}-2q^{146}-3q^{144}+15q^{140}+14q^{138}+5q^{136}-17q^{134}-14q^{132}-22q^{130}-16q^{128}+21q^{126}+41q^{124}+41q^{122}+4q^{120}-14q^{118}-58q^{116}-75q^{114}-28q^{112}+36q^{110}+91q^{108}+86q^{106}+70q^{104}-36q^{102}-135q^{100}-149q^{98}-82q^{96}+48q^{94}+153q^{92}+232q^{90}+123q^{88}-71q^{86}-232q^{84}-271q^{82}-145q^{80}+72q^{78}+322q^{76}+329q^{74}+140q^{72}-149q^{70}-359q^{68}-355q^{66}-134q^{64}+243q^{62}+408q^{60}+333q^{58}+39q^{56}-274q^{54}-412q^{52}-292q^{50}+73q^{48}+316q^{46}+360q^{44}+164q^{42}-116q^{40}-311q^{38}-296q^{36}-43q^{34}+162q^{32}+254q^{30}+165q^{28}-6q^{26}-163q^{24}-198q^{22}-67q^{20}+51q^{18}+132q^{16}+109q^{14}+35q^{12}-60q^{10}-105q^{8}-64q^{6}-5q^{4}+61q^{2}+79+61q^{-2}-5q^{-4}-64q^{-6}-105q^{-8}-60q^{-10}+35q^{-12}+109q^{-14}+132q^{-16}+51q^{-18}-67q^{-20}-198q^{-22}-163q^{-24}-6q^{-26}+165q^{-28}+254q^{-30}+162q^{-32}-43q^{-34}-296q^{-36}-311q^{-38}-116q^{-40}+164q^{-42}+360q^{-44}+316q^{-46}+73q^{-48}-292q^{-50}-412q^{-52}-274q^{-54}+39q^{-56}+333q^{-58}+408q^{-60}+243q^{-62}-134q^{-64}-355q^{-66}-359q^{-68}-149q^{-70}+140q^{-72}+329q^{-74}+322q^{-76}+72q^{-78}-145q^{-80}-271q^{-82}-232q^{-84}-71q^{-86}+123q^{-88}+232q^{-90}+153q^{-92}+48q^{-94}-82q^{-96}-149q^{-98}-135q^{-100}-36q^{-102}+70q^{-104}+86q^{-106}+91q^{-108}+36q^{-110}-28q^{-112}-75q^{-114}-58q^{-116}-14q^{-118}+4q^{-120}+41q^{-122}+41q^{-124}+21q^{-126}-16q^{-128}-22q^{-130}-14q^{-132}-17q^{-134}+5q^{-136}+14q^{-138}+15q^{-140}-3q^{-144}-2q^{-146}-9q^{-148}-2q^{-150}+2q^{-152}+5q^{-154}-q^{-156}+2q^{-160}-2q^{-162}+q^{-168}-q^{-170}-q^{-172}+q^{-174}$

A2 Invariants.

Weight	Invariant
1,0	$q^{12}+q^{8}-q^{4}+q^{2}-1+q^{-2}-q^{-4}+q^{-8}+q^{-12}$
1,1	$q^{36}-2q^{34}+4q^{32}-8q^{30}+13q^{28}-14q^{26}+22q^{24}-28q^{22}+27q^{20}-28q^{18}+24q^{16}-20q^{14}+7q^{12}+10q^{10}-20q^{8}+36q^{6}-47q^{4}+56q^{2}-58+56q^{-2}-47q^{-4}+36q^{-6}-20q^{-8}+10q^{-10}+7q^{-12}-20q^{-14}+24q^{-16}-28q^{-18}+27q^{-20}-28q^{-22}+22q^{-24}-14q^{-26}+13q^{-28}-8q^{-30}+4q^{-32}-2q^{-34}+q^{-36}$
2,0	$q^{32}+q^{26}+q^{24}-q^{22}-q^{20}+q^{18}-3q^{14}-q^{12}+q^{10}-2q^{8}-q^{6}+3q^{4}+3q^{2}+2+3q^{-2}+3q^{-4}-q^{-6}-2q^{-8}+q^{-10}-q^{-12}-3q^{-14}+q^{-18}-q^{-20}-q^{-22}+q^{-24}+q^{-26}+q^{-32}$
3,0	$q^{60}+q^{52}+q^{50}-2q^{48}-2q^{46}+4q^{42}+q^{40}-5q^{38}-6q^{36}+q^{34}+7q^{32}+2q^{30}-7q^{28}-5q^{26}+5q^{24}+11q^{22}+3q^{20}-6q^{18}-q^{16}+5q^{14}+5q^{12}-5q^{10}-7q^{8}+q^{6}+4q^{4}+q^{2}-4+q^{-2}+4q^{-4}+q^{-6}-7q^{-8}-5q^{-10}+5q^{-12}+5q^{-14}-q^{-16}-6q^{-18}+3q^{-20}+11q^{-22}+5q^{-24}-5q^{-26}-7q^{-28}+2q^{-30}+7q^{-32}+q^{-34}-6q^{-36}-5q^{-38}+q^{-40}+4q^{-42}-2q^{-46}-2q^{-48}+q^{-50}+q^{-52}+q^{-60}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{28}-q^{26}+q^{22}-3q^{20}+q^{18}+4q^{16}-2q^{14}+2q^{12}+5q^{10}-2q^{8}-q^{6}+q^{4}-2q^{2}-2-2q^{-2}+q^{-4}-q^{-6}-2q^{-8}+5q^{-10}+2q^{-12}-2q^{-14}+4q^{-16}+q^{-18}-3q^{-20}+q^{-22}-q^{-26}+q^{-28}$
1,0,0	$q^{15}+2q^{11}+q^{7}-q^{5}-q-q^{-1}-q^{-5}+q^{-7}+2q^{-11}+q^{-15}$
1,0,1	$q^{46}-2q^{44}+3q^{42}-3q^{40}-q^{38}+8q^{36}-9q^{34}+13q^{32}-3q^{30}-9q^{28}+18q^{26}-27q^{24}+19q^{22}-2q^{20}-20q^{18}+36q^{16}-34q^{14}+16q^{12}+5q^{10}-22q^{8}+21q^{6}-9q^{4}+4q^{2}+9+4q^{-2}-9q^{-4}+21q^{-6}-22q^{-8}+5q^{-10}+16q^{-12}-34q^{-14}+36q^{-16}-20q^{-18}-2q^{-20}+19q^{-22}-27q^{-24}+18q^{-26}-9q^{-28}-3q^{-30}+13q^{-32}-9q^{-34}+8q^{-36}-q^{-38}-3q^{-40}+3q^{-42}-2q^{-44}+q^{-46}$

A4 Invariants.

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

B2 Invariants.

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

D4 Invariants.

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

G2 Invariants.

Weight

Invariant

1,0

$q^{66}-q^{64}+2q^{62}-3q^{60}+q^{58}-3q^{54}+6q^{52}-6q^{50}+7q^{48}-5q^{46}+6q^{42}-10q^{40}+12q^{38}-8q^{36}+4q^{34}+3q^{32}-6q^{30}+11q^{28}-6q^{26}+2q^{24}+4q^{22}-7q^{20}+5q^{18}-8q^{14}+11q^{12}-10q^{10}+9q^{8}-3q^{6}-10q^{4}+14q^{2}-17+14q^{-2}-10q^{-4}-3q^{-6}+9q^{-8}-10q^{-10}+11q^{-12}-8q^{-14}+5q^{-18}-7q^{-20}+4q^{-22}+2q^{-24}-6q^{-26}+11q^{-28}-6q^{-30}+3q^{-32}+4q^{-34}-8q^{-36}+12q^{-38}-10q^{-40}+6q^{-42}-5q^{-46}+7q^{-48}-6q^{-50}+6q^{-52}-3q^{-54}+q^{-58}-3q^{-60}+2q^{-62}-q^{-64}+q^{-66}$

.

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 9"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t^{3}+3t^{2}-5t+7-5t^{-1}+3t^{-2}-t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

$-z^{6}-3z^{4}-2z^{2}+1$

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]=

{ 25, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$az^{7}+z^{7}a^{-1}+2a^{2}z^{6}+2z^{6}a^{-2}+4z^{6}+2a^{3}z^{5}+2z^{5}a^{-3}+a^{4}z^{4}-4a^{2}z^{4}-4z^{4}a^{-2}+z^{4}a^{-4}-10z^{4}-4a^{3}z^{3}-az^{3}-z^{3}a^{-1}-4z^{3}a^{-3}-2a^{4}z^{2}+4a^{2}z^{2}+4z^{2}a^{-2}-2z^{2}a^{-4}+12z^{2}+a^{3}z+az+za^{-1}+za^{-3}-2a^{2}-2a^{-2}-3$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_155, K11n37, ...}

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

Vassiliev invariants

V2 and V3:

(-2, 0)

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 $-8$ $0$ $32$ ${\frac {212}{3}}$ ${\frac {124}{3}}$ $0$ $0$ $0$ $0$ $-{\frac {256}{3}}$ $0$ $-{\frac {1696}{3}}$ $-{\frac {992}{3}}$ $-{\frac {8071}{15}}$ ${\frac {1668}{5}}$ $-{\frac {37804}{45}}$ ${\frac {1063}{9}}$ $-{\frac {3271}{15}}$

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

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

Integral Khovanov Homology (db, data source)

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

The Coloured Jones Polynomials

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

$n$	$J_{n}$
2	$q^{12}-2q^{11}+5q^{9}-6q^{8}-2q^{7}+12q^{6}-10q^{5}-7q^{4}+20q^{3}-12q^{2}-11q+25-11q^{-1}-12q^{-2}+20q^{-3}-7q^{-4}-10q^{-5}+12q^{-6}-2q^{-7}-6q^{-8}+5q^{-9}-2q^{-11}+q^{-12}$
3	$q^{24}-2q^{23}+2q^{21}+2q^{20}-5q^{19}-3q^{18}+7q^{17}+7q^{16}-11q^{15}-11q^{14}+12q^{13}+19q^{12}-13q^{11}-27q^{10}+12q^{9}+36q^{8}-10q^{7}-44q^{6}+7q^{5}+51q^{4}-5q^{3}-54q^{2}+59-54q^{-2}-5q^{-3}+51q^{-4}+7q^{-5}-44q^{-6}-10q^{-7}+36q^{-8}+12q^{-9}-27q^{-10}-13q^{-11}+19q^{-12}+12q^{-13}-11q^{-14}-11q^{-15}+7q^{-16}+7q^{-17}-3q^{-18}-5q^{-19}+2q^{-20}+2q^{-21}-2q^{-23}+q^{-24}$
4	$q^{40}-2q^{39}+2q^{37}-q^{36}+3q^{35}-7q^{34}+q^{33}+7q^{32}-3q^{31}+8q^{30}-19q^{29}-2q^{28}+17q^{27}+q^{26}+20q^{25}-39q^{24}-16q^{23}+21q^{22}+12q^{21}+53q^{20}-54q^{19}-44q^{18}+4q^{17}+20q^{16}+105q^{15}-53q^{14}-72q^{13}-31q^{12}+15q^{11}+159q^{10}-40q^{9}-89q^{8}-64q^{7}+q^{6}+197q^{5}-25q^{4}-93q^{3}-86q^{2}-13q+213-13q^{-1}-86q^{-2}-93q^{-3}-25q^{-4}+197q^{-5}+q^{-6}-64q^{-7}-89q^{-8}-40q^{-9}+159q^{-10}+15q^{-11}-31q^{-12}-72q^{-13}-53q^{-14}+105q^{-15}+20q^{-16}+4q^{-17}-44q^{-18}-54q^{-19}+53q^{-20}+12q^{-21}+21q^{-22}-16q^{-23}-39q^{-24}+20q^{-25}+q^{-26}+17q^{-27}-2q^{-28}-19q^{-29}+8q^{-30}-3q^{-31}+7q^{-32}+q^{-33}-7q^{-34}+3q^{-35}-q^{-36}+2q^{-37}-2q^{-39}+q^{-40}$
5	$q^{60}-2q^{59}+2q^{57}-q^{56}+q^{54}-3q^{53}+6q^{51}-5q^{49}-2q^{48}-5q^{47}+2q^{46}+14q^{45}+9q^{44}-7q^{43}-16q^{42}-19q^{41}-3q^{40}+28q^{39}+34q^{38}+12q^{37}-24q^{36}-55q^{35}-37q^{34}+19q^{33}+67q^{32}+70q^{31}+9q^{30}-78q^{29}-110q^{28}-45q^{27}+71q^{26}+147q^{25}+100q^{24}-52q^{23}-182q^{22}-156q^{21}+19q^{20}+204q^{19}+216q^{18}+21q^{17}-217q^{16}-268q^{15}-64q^{14}+221q^{13}+312q^{12}+101q^{11}-218q^{10}-341q^{9}-138q^{8}+211q^{7}+370q^{6}+158q^{5}-206q^{4}-372q^{3}-183q^{2}+188q+393+188q^{-1}-183q^{-2}-372q^{-3}-206q^{-4}+158q^{-5}+370q^{-6}+211q^{-7}-138q^{-8}-341q^{-9}-218q^{-10}+101q^{-11}+312q^{-12}+221q^{-13}-64q^{-14}-268q^{-15}-217q^{-16}+21q^{-17}+216q^{-18}+204q^{-19}+19q^{-20}-156q^{-21}-182q^{-22}-52q^{-23}+100q^{-24}+147q^{-25}+71q^{-26}-45q^{-27}-110q^{-28}-78q^{-29}+9q^{-30}+70q^{-31}+67q^{-32}+19q^{-33}-37q^{-34}-55q^{-35}-24q^{-36}+12q^{-37}+34q^{-38}+28q^{-39}-3q^{-40}-19q^{-41}-16q^{-42}-7q^{-43}+9q^{-44}+14q^{-45}+2q^{-46}-5q^{-47}-2q^{-48}-5q^{-49}+6q^{-51}-3q^{-53}+q^{-54}-q^{-56}+2q^{-57}-2q^{-59}+q^{-60}$
6	$q^{84}-2q^{83}+2q^{81}-q^{80}-2q^{78}+5q^{77}-4q^{76}-q^{75}+8q^{74}-4q^{73}-4q^{72}-9q^{71}+12q^{70}-5q^{69}+2q^{68}+23q^{67}-5q^{66}-13q^{65}-31q^{64}+15q^{63}-13q^{62}+8q^{61}+60q^{60}+15q^{59}-13q^{58}-68q^{57}-3q^{56}-57q^{55}-9q^{54}+107q^{53}+79q^{52}+42q^{51}-73q^{50}-19q^{49}-163q^{48}-108q^{47}+93q^{46}+146q^{45}+172q^{44}+32q^{43}+60q^{42}-272q^{41}-302q^{40}-68q^{39}+107q^{38}+298q^{37}+249q^{36}+310q^{35}-265q^{34}-491q^{33}-357q^{32}-103q^{31}+302q^{30}+470q^{29}+687q^{28}-100q^{27}-566q^{26}-651q^{25}-416q^{24}+164q^{23}+590q^{22}+1052q^{21}+143q^{20}-522q^{19}-847q^{18}-696q^{17}-31q^{16}+605q^{15}+1305q^{14}+348q^{13}-430q^{12}-936q^{11}-867q^{10}-188q^{9}+570q^{8}+1436q^{7}+466q^{6}-349q^{5}-959q^{4}-941q^{3}-283q^{2}+525q+1477+525q^{-1}-283q^{-2}-941q^{-3}-959q^{-4}-349q^{-5}+466q^{-6}+1436q^{-7}+570q^{-8}-188q^{-9}-867q^{-10}-936q^{-11}-430q^{-12}+348q^{-13}+1305q^{-14}+605q^{-15}-31q^{-16}-696q^{-17}-847q^{-18}-522q^{-19}+143q^{-20}+1052q^{-21}+590q^{-22}+164q^{-23}-416q^{-24}-651q^{-25}-566q^{-26}-100q^{-27}+687q^{-28}+470q^{-29}+302q^{-30}-103q^{-31}-357q^{-32}-491q^{-33}-265q^{-34}+310q^{-35}+249q^{-36}+298q^{-37}+107q^{-38}-68q^{-39}-302q^{-40}-272q^{-41}+60q^{-42}+32q^{-43}+172q^{-44}+146q^{-45}+93q^{-46}-108q^{-47}-163q^{-48}-19q^{-49}-73q^{-50}+42q^{-51}+79q^{-52}+107q^{-53}-9q^{-54}-57q^{-55}-3q^{-56}-68q^{-57}-13q^{-58}+15q^{-59}+60q^{-60}+8q^{-61}-13q^{-62}+15q^{-63}-31q^{-64}-13q^{-65}-5q^{-66}+23q^{-67}+2q^{-68}-5q^{-69}+12q^{-70}-9q^{-71}-4q^{-72}-4q^{-73}+8q^{-74}-q^{-75}-4q^{-76}+5q^{-77}-2q^{-78}-q^{-80}+2q^{-81}-2q^{-83}+q^{-84}$
7	$q^{112}-2q^{111}+2q^{109}-q^{108}-2q^{106}+2q^{105}+4q^{104}-5q^{103}+q^{102}+4q^{101}-4q^{100}-2q^{99}-8q^{98}+q^{97}+16q^{96}-3q^{95}+5q^{94}+8q^{93}-11q^{92}-6q^{91}-29q^{90}-11q^{89}+32q^{88}+10q^{87}+28q^{86}+27q^{85}-15q^{84}-12q^{83}-71q^{82}-61q^{81}+24q^{80}+16q^{79}+79q^{78}+92q^{77}+27q^{76}+21q^{75}-113q^{74}-156q^{73}-70q^{72}-69q^{71}+85q^{70}+191q^{69}+158q^{68}+185q^{67}-27q^{66}-203q^{65}-227q^{64}-336q^{63}-117q^{62}+140q^{61}+270q^{60}+503q^{59}+329q^{58}+30q^{57}-227q^{56}-661q^{55}-601q^{54}-292q^{53}+72q^{52}+736q^{51}+878q^{50}+659q^{49}+227q^{48}-711q^{47}-1129q^{46}-1075q^{45}-631q^{44}+549q^{43}+1283q^{42}+1492q^{41}+1138q^{40}-257q^{39}-1345q^{38}-1870q^{37}-1666q^{36}-122q^{35}+1287q^{34}+2159q^{33}+2185q^{32}+554q^{31}-1140q^{30}-2362q^{29}-2642q^{28}-984q^{27}+942q^{26}+2477q^{25}+3009q^{24}+1367q^{23}-721q^{22}-2511q^{21}-3289q^{20}-1694q^{19}+513q^{18}+2513q^{17}+3488q^{16}+1926q^{15}-347q^{14}-2473q^{13}-3596q^{12}-2106q^{11}+194q^{10}+2437q^{9}+3692q^{8}+2217q^{7}-126q^{6}-2389q^{5}-3694q^{4}-2294q^{3}+13q^{2}+2343q+3751+2343q^{-1}+13q^{-2}-2294q^{-3}-3694q^{-4}-2389q^{-5}-126q^{-6}+2217q^{-7}+3692q^{-8}+2437q^{-9}+194q^{-10}-2106q^{-11}-3596q^{-12}-2473q^{-13}-347q^{-14}+1926q^{-15}+3488q^{-16}+2513q^{-17}+513q^{-18}-1694q^{-19}-3289q^{-20}-2511q^{-21}-721q^{-22}+1367q^{-23}+3009q^{-24}+2477q^{-25}+942q^{-26}-984q^{-27}-2642q^{-28}-2362q^{-29}-1140q^{-30}+554q^{-31}+2185q^{-32}+2159q^{-33}+1287q^{-34}-122q^{-35}-1666q^{-36}-1870q^{-37}-1345q^{-38}-257q^{-39}+1138q^{-40}+1492q^{-41}+1283q^{-42}+549q^{-43}-631q^{-44}-1075q^{-45}-1129q^{-46}-711q^{-47}+227q^{-48}+659q^{-49}+878q^{-50}+736q^{-51}+72q^{-52}-292q^{-53}-601q^{-54}-661q^{-55}-227q^{-56}+30q^{-57}+329q^{-58}+503q^{-59}+270q^{-60}+140q^{-61}-117q^{-62}-336q^{-63}-227q^{-64}-203q^{-65}-27q^{-66}+185q^{-67}+158q^{-68}+191q^{-69}+85q^{-70}-69q^{-71}-70q^{-72}-156q^{-73}-113q^{-74}+21q^{-75}+27q^{-76}+92q^{-77}+79q^{-78}+16q^{-79}+24q^{-80}-61q^{-81}-71q^{-82}-12q^{-83}-15q^{-84}+27q^{-85}+28q^{-86}+10q^{-87}+32q^{-88}-11q^{-89}-29q^{-90}-6q^{-91}-11q^{-92}+8q^{-93}+5q^{-94}-3q^{-95}+16q^{-96}+q^{-97}-8q^{-98}-2q^{-99}-4q^{-100}+4q^{-101}+q^{-102}-5q^{-103}+4q^{-104}+2q^{-105}-2q^{-106}-q^{-108}+2q^{-109}-2q^{-111}+q^{-112}$

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, 9]]
Out[2]=	PD[X[6, 2, 7, 1], X[14, 8, 15, 7], X[10, 3, 11, 4], X[2, 13, 3, 14], X[12, 5, 13, 6], X[4, 11, 5, 12], X[16, 10, 1, 9], X[8, 16, 9, 15]]
In[3]:=	GaussCode[Knot[8, 9]]
Out[3]=	GaussCode[1, -4, 3, -6, 5, -1, 2, -8, 7, -3, 6, -5, 4, -2, 8, -7]
In[4]:=	DTCode[Knot[8, 9]]
Out[4]=	DTCode[6, 10, 12, 14, 16, 4, 2, 8]
In[5]:=	br = BR[Knot[8, 9]]
Out[5]=	BR[3, {-1, -1, -1, 2, -1, 2, 2, 2}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{3, 8}
In[7]:=	BraidIndex[Knot[8, 9]]
Out[7]=	3
In[8]:=	Show[DrawMorseLink[Knot[8, 9]]]
Out[8]=	-Graphics-
In[9]:=	(#[Knot[8, 9]]&) /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{FullyAmphicheiral, 1, 3, 2, {3, 6}, 1}
In[10]:=	alex = Alexander[Knot[8, 9]][t]
Out[10]=	-3 3 5 2 3 7 - t + -- - - - 5 t + 3 t - t 2 t t
In[11]:=	Conway[Knot[8, 9]][z]
Out[11]=	2 4 6 1 - 2 z - 3 z - z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[8, 9], Knot[10, 155], Knot[11, NonAlternating, 37]}
In[13]:=	{KnotDet[Knot[8, 9]], KnotSignature[Knot[8, 9]]}
Out[13]=	{25, 0}
In[14]:=	Jones[Knot[8, 9]][q]
Out[14]=	-4 2 3 4 2 3 4 5 + q - -- + -- - - - 4 q + 3 q - 2 q + q 3 2 q q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[8, 9]}
In[16]:=	A2Invariant[Knot[8, 9]][q]
Out[16]=	-12 -8 -4 -2 2 4 8 12 -1 + q + q - q + q + q - q + q + q
In[17]:=	HOMFLYPT[Knot[8, 9]][a, z]
Out[17]=	2 4 2 2 2 3 z 2 2 4 z 2 4 6 -3 + -- + 2 a - 8 z + ---- + 3 a z - 5 z + -- + a z - z 2 2 2 a a a
In[18]:=	Kauffman[Knot[8, 9]][a, z]
Out[18]=	2 2 2 2 z z 3 2 2 z 4 z 2 2 -3 - -- - 2 a + -- + - + a z + a z + 12 z - ---- + ---- + 4 a z - 2 3 a 4 2 a a a a 3 3 4 4 4 2 4 z z 3 3 3 4 z 4 z 2 4 2 a z - ---- - -- - a z - 4 a z - 10 z + -- - ---- - 4 a z + 3 a 4 2 a a a 5 6 7 4 4 2 z 3 5 6 2 z 2 6 z 7 a z + ---- + 2 a z + 4 z + ---- + 2 a z + -- + a z 3 2 a a a
In[19]:=	{Vassiliev[2][Knot[8, 9]], Vassiliev[3][Knot[8, 9]]}
Out[19]=	{-2, 0}
In[20]:=	Kh[Knot[8, 9]][q, t]
Out[20]=	3 1 1 1 2 1 2 2 - + 3 q + ----- + ----- + ----- + ----- + ----- + ---- + --- + 2 q t + q 9 4 7 3 5 3 5 2 3 2 3 q t q t q t q t q t q t q t 3 3 2 5 2 5 3 7 3 9 4 2 q t + q t + 2 q t + q t + q t + q t
In[21]:=	ColouredJones[Knot[8, 9], 2][q]
Out[21]=	-12 2 5 6 2 12 10 7 20 12 11 25 + q - --- + -- - -- - -- + -- - -- - -- + -- - -- - -- - 11 q - 11 9 8 7 6 5 4 3 2 q q q q q q q q q q 2 3 4 5 6 7 8 9 11 12 q + 20 q - 7 q - 10 q + 12 q - 2 q - 6 q + 5 q - 2 q + 12 q

See/edit the Rolfsen_Splice_Template. Back to the top.

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