9 32

9_31

9_33

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9 32 Further Notes and Views

Knot presentations

Planar diagram presentation	X1425 X13,18,14,1 X3948 X9,3,10,2 X7,15,8,14 X15,11,16,10 X5,12,6,13 X11,17,12,16 X17,7,18,6
Gauss code	-1, 4, -3, 1, -7, 9, -5, 3, -4, 6, -8, 7, -2, 5, -6, 8, -9, 2
Dowker-Thistlethwaite code	4 8 12 14 2 16 18 10 6
Conway Notation	[.21.20]

Minimum Braid Representative: Length is 9, width is 4. Braid index is 4.

A Morse Link Presentation:

Three dimensional invariants

Symmetry type Chiral Unknotting number 2 3-genus 3 Bridge index 3 Super bridge index $\{4,6\}$ Nakanishi index 1 Maximal Thurston-Bennequin number [-2][-9] Hyperbolic Volume 13.0999 A-Polynomial See Data:9 32/A-polynomial

[edit Notes for 9 32'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 9 32's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{3}-6t^{2}+14t-17+14t^{-1}-6t^{-2}+t^{-3}$
Conway polynomial	$z^{6}-z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 59, 2 }
Jones polynomial	$q^{7}-3q^{6}+6q^{5}-9q^{4}+10q^{3}-10q^{2}+9q-6+4q^{-1}-q^{-2}$
HOMFLY-PT polynomial (db, data sources)	$z^{6}a^{-2}+3z^{4}a^{-2}-2z^{4}a^{-4}-z^{4}+3z^{2}a^{-2}-4z^{2}a^{-4}+z^{2}a^{-6}-z^{2}+a^{-2}-2a^{-4}+a^{-6}+1$
Kauffman polynomial (db, data sources)	$2z^{8}a^{-2}+2z^{8}a^{-4}+5z^{7}a^{-1}+10z^{7}a^{-3}+5z^{7}a^{-5}+6z^{6}a^{-2}+7z^{6}a^{-4}+5z^{6}a^{-6}+4z^{6}+az^{5}-9z^{5}a^{-1}-18z^{5}a^{-3}-5z^{5}a^{-5}+3z^{5}a^{-7}-19z^{4}a^{-2}-18z^{4}a^{-4}-6z^{4}a^{-6}+z^{4}a^{-8}-8z^{4}-az^{3}+3z^{3}a^{-1}+9z^{3}a^{-3}+2z^{3}a^{-5}-3z^{3}a^{-7}+10z^{2}a^{-2}+12z^{2}a^{-4}+4z^{2}a^{-6}-z^{2}a^{-8}+3z^{2}-za^{-1}-2za^{-3}+za^{-7}-a^{-2}-2a^{-4}-a^{-6}+1$
The A2 invariant	$-q^{6}+2q^{4}+1+3q^{-2}-2q^{-4}+2q^{-6}-2q^{-8}-2q^{-14}+2q^{-16}-q^{-18}+q^{-22}$
The G2 invariant	$q^{32}-3q^{30}+7q^{28}-13q^{26}+13q^{24}-9q^{22}-6q^{20}+30q^{18}-50q^{16}+66q^{14}-56q^{12}+17q^{10}+39q^{8}-93q^{6}+126q^{4}-112q^{2}+58+22q^{-2}-92q^{-4}+126q^{-6}-106q^{-8}+48q^{-10}+29q^{-12}-83q^{-14}+89q^{-16}-47q^{-18}-23q^{-20}+92q^{-22}-122q^{-24}+101q^{-26}-35q^{-28}-53q^{-30}+131q^{-32}-173q^{-34}+158q^{-36}-91q^{-38}-6q^{-40}+98q^{-42}-157q^{-44}+157q^{-46}-103q^{-48}+19q^{-50}+58q^{-52}-102q^{-54}+89q^{-56}-33q^{-58}-39q^{-60}+90q^{-62}-94q^{-64}+49q^{-66}+22q^{-68}-90q^{-70}+125q^{-72}-111q^{-74}+63q^{-76}+q^{-78}-59q^{-80}+88q^{-82}-86q^{-84}+64q^{-86}-26q^{-88}-5q^{-90}+25q^{-92}-33q^{-94}+30q^{-96}-21q^{-98}+12q^{-100}-2q^{-102}-4q^{-104}+5q^{-106}-6q^{-108}+4q^{-110}-2q^{-112}+q^{-114}$

Further Quantum Invariants

Further quantum knot invariants for 9_32.

A1 Invariants.

Weight	Invariant
1	$-q^{5}+3q^{3}-2q+3q^{-1}-q^{-3}+q^{-7}-3q^{-9}+3q^{-11}-2q^{-13}+q^{-15}$
2	$q^{16}-3q^{14}-2q^{12}+11q^{10}-4q^{8}-14q^{6}+18q^{4}+5q^{2}-23+13q^{-2}+13q^{-4}-19q^{-6}+q^{-8}+13q^{-10}-5q^{-12}-10q^{-14}+6q^{-16}+13q^{-18}-17q^{-20}-5q^{-22}+25q^{-24}-13q^{-26}-13q^{-28}+19q^{-30}-3q^{-32}-9q^{-34}+6q^{-36}-2q^{-40}+q^{-42}$
3	$-q^{33}+3q^{31}+2q^{29}-7q^{27}-10q^{25}+8q^{23}+30q^{21}-5q^{19}-48q^{17}-18q^{15}+66q^{13}+54q^{11}-67q^{9}-93q^{7}+50q^{5}+124q^{3}-14q-143q^{-1}-23q^{-3}+139q^{-5}+56q^{-7}-117q^{-9}-80q^{-11}+93q^{-13}+90q^{-15}-59q^{-17}-93q^{-19}+26q^{-21}+87q^{-23}+13q^{-25}-83q^{-27}-52q^{-29}+69q^{-31}+93q^{-33}-48q^{-35}-125q^{-37}+16q^{-39}+147q^{-41}+21q^{-43}-146q^{-45}-52q^{-47}+117q^{-49}+75q^{-51}-83q^{-53}-76q^{-55}+46q^{-57}+60q^{-59}-16q^{-61}-39q^{-63}+3q^{-65}+21q^{-67}-q^{-69}-8q^{-71}+3q^{-75}-2q^{-79}+q^{-81}$
4	$q^{56}-3q^{54}-2q^{52}+7q^{50}+6q^{48}+6q^{46}-24q^{44}-31q^{42}+11q^{40}+45q^{38}+83q^{36}-29q^{34}-139q^{32}-103q^{30}+35q^{28}+282q^{26}+170q^{24}-155q^{22}-379q^{20}-280q^{18}+334q^{16}+561q^{14}+220q^{12}-448q^{10}-794q^{8}-75q^{6}+684q^{4}+799q^{2}-33-989q^{-2}-661q^{-4}+321q^{-6}+1039q^{-8}+514q^{-10}-697q^{-12}-919q^{-14}-162q^{-16}+832q^{-18}+751q^{-20}-263q^{-22}-793q^{-24}-418q^{-26}+472q^{-28}+692q^{-30}+74q^{-32}-546q^{-34}-518q^{-36}+131q^{-38}+573q^{-40}+390q^{-42}-277q^{-44}-629q^{-46}-287q^{-48}+418q^{-50}+776q^{-52}+124q^{-54}-670q^{-56}-794q^{-58}+49q^{-60}+1005q^{-62}+652q^{-64}-376q^{-66}-1079q^{-68}-494q^{-70}+763q^{-72}+929q^{-74}+169q^{-76}-815q^{-78}-774q^{-80}+197q^{-82}+673q^{-84}+469q^{-86}-269q^{-88}-552q^{-90}-138q^{-92}+212q^{-94}+330q^{-96}+32q^{-98}-191q^{-100}-111q^{-102}-5q^{-104}+105q^{-106}+40q^{-108}-32q^{-110}-20q^{-112}-16q^{-114}+18q^{-116}+7q^{-118}-6q^{-120}+q^{-122}-3q^{-124}+3q^{-126}-2q^{-130}+q^{-132}$
5	$-q^{85}+3q^{83}+2q^{81}-7q^{79}-6q^{77}-2q^{75}+10q^{73}+25q^{71}+25q^{69}-20q^{67}-73q^{65}-74q^{63}-4q^{61}+121q^{59}+203q^{57}+134q^{55}-145q^{53}-410q^{51}-392q^{49}-14q^{47}+561q^{45}+873q^{43}+484q^{41}-523q^{39}-1376q^{37}-1291q^{35}-44q^{33}+1653q^{31}+2363q^{29}+1179q^{27}-1344q^{25}-3271q^{23}-2809q^{21}+185q^{19}+3609q^{17}+4532q^{15}+1734q^{13}-3011q^{11}-5824q^{9}-4017q^{7}+1422q^{5}+6227q^{3}+6163q+822q^{-1}-5611q^{-3}-7593q^{-5}-3196q^{-7}+4095q^{-9}+8075q^{-11}+5192q^{-13}-2147q^{-15}-7620q^{-17}-6426q^{-19}+244q^{-21}+6455q^{-23}+6841q^{-25}+1297q^{-27}-5029q^{-29}-6557q^{-31}-2264q^{-33}+3612q^{-35}+5830q^{-37}+2780q^{-39}-2434q^{-41}-5011q^{-43}-2947q^{-45}+1501q^{-47}+4259q^{-49}+3079q^{-51}-734q^{-53}-3704q^{-55}-3331q^{-57}-62q^{-59}+3284q^{-61}+3846q^{-63}+1053q^{-65}-2839q^{-67}-4548q^{-69}-2406q^{-71}+2149q^{-73}+5300q^{-75}+4045q^{-77}-1004q^{-79}-5743q^{-81}-5840q^{-83}-687q^{-85}+5612q^{-87}+7407q^{-89}+2741q^{-91}-4635q^{-93}-8311q^{-95}-4861q^{-97}+2869q^{-99}+8250q^{-101}+6537q^{-103}-679q^{-105}-7065q^{-107}-7315q^{-109}-1521q^{-111}+5098q^{-113}+7029q^{-115}+3096q^{-117}-2832q^{-119}-5760q^{-121}-3789q^{-123}+819q^{-125}+4014q^{-127}+3582q^{-129}+506q^{-131}-2296q^{-133}-2751q^{-135}-1078q^{-137}+971q^{-139}+1769q^{-141}+1069q^{-143}-215q^{-145}-942q^{-147}-753q^{-149}-98q^{-151}+396q^{-153}+432q^{-155}+150q^{-157}-139q^{-159}-200q^{-161}-91q^{-163}+29q^{-165}+73q^{-167}+46q^{-169}-29q^{-173}-16q^{-175}+5q^{-177}+4q^{-179}+2q^{-181}+3q^{-183}-2q^{-185}-3q^{-187}+3q^{-189}-2q^{-193}+q^{-195}$

A2 Invariants.

Weight	Invariant
1,0	$-q^{6}+2q^{4}+1+3q^{-2}-2q^{-4}+2q^{-6}-2q^{-8}-2q^{-14}+2q^{-16}-q^{-18}+q^{-22}$
1,1	$q^{20}-6q^{18}+20q^{16}-50q^{14}+97q^{12}-170q^{10}+268q^{8}-368q^{6}+464q^{4}-522q^{2}+534-462q^{-2}+323q^{-4}-120q^{-6}-132q^{-8}+394q^{-10}-641q^{-12}+832q^{-14}-960q^{-16}+998q^{-18}-942q^{-20}+802q^{-22}-590q^{-24}+342q^{-26}-79q^{-28}-154q^{-30}+334q^{-32}-448q^{-34}+491q^{-36}-470q^{-38}+408q^{-40}-330q^{-42}+247q^{-44}-170q^{-46}+110q^{-48}-70q^{-50}+40q^{-52}-20q^{-54}+10q^{-56}-4q^{-58}+q^{-60}$
2,0	$q^{18}-2q^{16}-3q^{14}+4q^{12}+4q^{10}-3q^{8}-5q^{6}+7q^{4}+10q^{2}-7-5q^{-2}+10q^{-4}+q^{-6}-8q^{-8}-2q^{-10}+6q^{-12}-4q^{-14}-4q^{-16}+5q^{-18}-6q^{-22}+5q^{-24}+8q^{-26}-10q^{-28}-2q^{-30}+9q^{-32}+3q^{-34}-8q^{-36}-2q^{-38}+9q^{-40}-6q^{-44}-q^{-46}+2q^{-48}-q^{-52}+q^{-56}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{14}-3q^{12}+q^{10}+5q^{8}-10q^{6}+7q^{4}+10q^{2}-14+11q^{-2}+11q^{-4}-17q^{-6}+4q^{-8}+8q^{-10}-10q^{-12}-3q^{-14}+4q^{-16}+3q^{-18}-5q^{-20}-5q^{-22}+14q^{-24}-6q^{-26}-12q^{-28}+19q^{-30}-5q^{-32}-11q^{-34}+14q^{-36}-2q^{-38}-7q^{-40}+5q^{-42}-2q^{-46}+q^{-48}$
1,0,0	$-q^{7}+2q^{5}-q^{3}+3q+3q^{-3}-q^{-5}+q^{-7}-q^{-11}-2q^{-15}+q^{-17}-3q^{-19}+2q^{-21}-q^{-23}+q^{-25}+q^{-29}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{16}-2q^{14}-q^{12}+4q^{10}-3q^{8}-4q^{6}+9q^{4}+6q^{2}-7+4q^{-2}+16q^{-4}-13q^{-8}+7q^{-10}+10q^{-12}-17q^{-14}-11q^{-16}+13q^{-18}-6q^{-20}-13q^{-22}+11q^{-24}+8q^{-26}-10q^{-28}+q^{-30}+14q^{-32}-4q^{-34}-12q^{-36}+8q^{-38}+9q^{-40}-11q^{-42}-4q^{-44}+10q^{-46}+q^{-48}-6q^{-50}+3q^{-54}-2q^{-58}+q^{-62}$
1,0,0,0	$-q^{8}+2q^{6}-q^{4}+2q^{2}+2+3q^{-4}-q^{-6}+2q^{-8}-q^{-10}+q^{-12}-q^{-14}-2q^{-18}-q^{-20}-3q^{-24}+2q^{-26}-q^{-28}+q^{-30}+q^{-32}+q^{-36}$

B2 Invariants.

Weight	Invariant
0,1	$-q^{14}+3q^{12}-7q^{10}+11q^{8}-14q^{6}+19q^{4}-18q^{2}+18-13q^{-2}+9q^{-4}+q^{-6}-10q^{-8}+20q^{-10}-28q^{-12}+33q^{-14}-36q^{-16}+33q^{-18}-29q^{-20}+21q^{-22}-12q^{-24}+2q^{-26}+6q^{-28}-13q^{-30}+17q^{-32}-19q^{-34}+18q^{-36}-14q^{-38}+11q^{-40}-7q^{-42}+4q^{-44}-2q^{-46}+q^{-48}$
1,0	$q^{24}-3q^{20}-3q^{18}+4q^{16}+8q^{14}-2q^{12}-12q^{10}-5q^{8}+16q^{6}+14q^{4}-9q^{2}-18+2q^{-2}+21q^{-4}+10q^{-6}-16q^{-8}-14q^{-10}+8q^{-12}+14q^{-14}-4q^{-16}-14q^{-18}-q^{-20}+12q^{-22}+3q^{-24}-13q^{-26}-6q^{-28}+11q^{-30}+10q^{-32}-9q^{-34}-13q^{-36}+7q^{-38}+17q^{-40}-2q^{-42}-19q^{-44}-5q^{-46}+18q^{-48}+13q^{-50}-11q^{-52}-18q^{-54}+2q^{-56}+16q^{-58}+6q^{-60}-8q^{-62}-9q^{-64}+q^{-66}+6q^{-68}+2q^{-70}-2q^{-72}-2q^{-74}+q^{-78}$

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{18}-3q^{16}+4q^{14}-6q^{12}+9q^{10}-12q^{8}+14q^{6}-13q^{4}+18q^{2}-12+13q^{-2}-7q^{-4}+9q^{-6}-q^{-8}-7q^{-10}+10q^{-12}-14q^{-14}+19q^{-16}-26q^{-18}+24q^{-20}-26q^{-22}+28q^{-24}-26q^{-26}+19q^{-28}-19q^{-30}+15q^{-32}-7q^{-34}+q^{-38}-6q^{-40}+14q^{-42}-12q^{-44}+13q^{-46}-15q^{-48}+16q^{-50}-10q^{-52}+8q^{-54}-9q^{-56}+6q^{-58}-3q^{-60}+2q^{-62}-2q^{-64}+q^{-66}$

G2 Invariants.

Weight

Invariant

1,0

$q^{32}-3q^{30}+7q^{28}-13q^{26}+13q^{24}-9q^{22}-6q^{20}+30q^{18}-50q^{16}+66q^{14}-56q^{12}+17q^{10}+39q^{8}-93q^{6}+126q^{4}-112q^{2}+58+22q^{-2}-92q^{-4}+126q^{-6}-106q^{-8}+48q^{-10}+29q^{-12}-83q^{-14}+89q^{-16}-47q^{-18}-23q^{-20}+92q^{-22}-122q^{-24}+101q^{-26}-35q^{-28}-53q^{-30}+131q^{-32}-173q^{-34}+158q^{-36}-91q^{-38}-6q^{-40}+98q^{-42}-157q^{-44}+157q^{-46}-103q^{-48}+19q^{-50}+58q^{-52}-102q^{-54}+89q^{-56}-33q^{-58}-39q^{-60}+90q^{-62}-94q^{-64}+49q^{-66}+22q^{-68}-90q^{-70}+125q^{-72}-111q^{-74}+63q^{-76}+q^{-78}-59q^{-80}+88q^{-82}-86q^{-84}+64q^{-86}-26q^{-88}-5q^{-90}+25q^{-92}-33q^{-94}+30q^{-96}-21q^{-98}+12q^{-100}-2q^{-102}-4q^{-104}+5q^{-106}-6q^{-108}+4q^{-110}-2q^{-112}+q^{-114}$

.

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["9 32"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^{3}-6t^{2}+14t-17+14t^{-1}-6t^{-2}+t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 59, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^{7}-3q^{6}+6q^{5}-9q^{4}+10q^{3}-10q^{2}+9q-6+4q^{-1}-q^{-2}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$2z^{8}a^{-2}+2z^{8}a^{-4}+5z^{7}a^{-1}+10z^{7}a^{-3}+5z^{7}a^{-5}+6z^{6}a^{-2}+7z^{6}a^{-4}+5z^{6}a^{-6}+4z^{6}+az^{5}-9z^{5}a^{-1}-18z^{5}a^{-3}-5z^{5}a^{-5}+3z^{5}a^{-7}-19z^{4}a^{-2}-18z^{4}a^{-4}-6z^{4}a^{-6}+z^{4}a^{-8}-8z^{4}-az^{3}+3z^{3}a^{-1}+9z^{3}a^{-3}+2z^{3}a^{-5}-3z^{3}a^{-7}+10z^{2}a^{-2}+12z^{2}a^{-4}+4z^{2}a^{-6}-z^{2}a^{-8}+3z^{2}-za^{-1}-2za^{-3}+za^{-7}-a^{-2}-2a^{-4}-a^{-6}+1$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11n52, K11n124, ...}

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

Vassiliev invariants

V2 and V3:

(-1, -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 $-4$ $-16$ $8$ $-{\frac {62}{3}}$ ${\frac {14}{3}}$ $64$ ${\frac {224}{3}}$ ${\frac {128}{3}}$ $16$ $-{\frac {32}{3}}$ $128$ ${\frac {248}{3}}$ $-{\frac {56}{3}}$ ${\frac {12689}{30}}$ ${\frac {74}{5}}$ ${\frac {9778}{45}}$ ${\frac {79}{18}}$ ${\frac {689}{30}}$

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

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

Integral Khovanov Homology (db, data source)

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

The Coloured Jones Polynomials

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

$n$	$J_{n}$
2	$q^{20}-3q^{19}+2q^{18}+7q^{17}-18q^{16}+8q^{15}+29q^{14}-50q^{13}+8q^{12}+67q^{11}-80q^{10}-4q^{9}+97q^{8}-87q^{7}-20q^{6}+102q^{5}-69q^{4}-32q^{3}+82q^{2}-37q-32+46q^{-1}-9q^{-2}-19q^{-3}+14q^{-4}+q^{-5}-4q^{-6}+q^{-7}$
3	$q^{39}-3q^{38}+2q^{37}+3q^{36}-2q^{35}-11q^{34}+9q^{33}+25q^{32}-20q^{31}-53q^{30}+32q^{29}+101q^{28}-34q^{27}-175q^{26}+25q^{25}+259q^{24}+8q^{23}-344q^{22}-69q^{21}+426q^{20}+134q^{19}-475q^{18}-210q^{17}+503q^{16}+275q^{15}-499q^{14}-331q^{13}+472q^{12}+371q^{11}-425q^{10}-392q^{9}+353q^{8}+405q^{7}-276q^{6}-389q^{5}+180q^{4}+368q^{3}-103q^{2}-306q+18+248q^{-1}+26q^{-2}-168q^{-3}-56q^{-4}+105q^{-5}+52q^{-6}-47q^{-7}-44q^{-8}+21q^{-9}+22q^{-10}-4q^{-11}-9q^{-12}-q^{-13}+4q^{-14}-q^{-15}$
4	$q^{64}-3q^{63}+2q^{62}+3q^{61}-6q^{60}+5q^{59}-10q^{58}+15q^{57}+14q^{56}-40q^{55}+q^{54}-22q^{53}+87q^{52}+79q^{51}-150q^{50}-105q^{49}-102q^{48}+310q^{47}+377q^{46}-268q^{45}-455q^{44}-516q^{43}+593q^{42}+1115q^{41}-64q^{40}-931q^{39}-1487q^{38}+552q^{37}+2099q^{36}+696q^{35}-1097q^{34}-2744q^{33}-33q^{32}+2802q^{31}+1724q^{30}-744q^{29}-3700q^{28}-876q^{27}+2926q^{26}+2518q^{25}-92q^{24}-4058q^{23}-1581q^{22}+2584q^{21}+2870q^{20}+575q^{19}-3875q^{18}-2023q^{17}+1935q^{16}+2842q^{15}+1195q^{14}-3257q^{13}-2243q^{12}+1045q^{11}+2467q^{10}+1725q^{9}-2243q^{8}-2162q^{7}+51q^{6}+1710q^{5}+1947q^{4}-1032q^{3}-1637q^{2}-667q+728+1619q^{-1}-76q^{-2}-805q^{-3}-782q^{-4}-31q^{-5}+900q^{-6}+270q^{-7}-137q^{-8}-441q^{-9}-258q^{-10}+286q^{-11}+171q^{-12}+87q^{-13}-116q^{-14}-146q^{-15}+39q^{-16}+33q^{-17}+51q^{-18}-6q^{-19}-34q^{-20}+q^{-21}-q^{-22}+9q^{-23}+q^{-24}-4q^{-25}+q^{-26}$
5	$q^{95}-3q^{94}+2q^{93}+3q^{92}-6q^{91}+q^{90}+6q^{89}-4q^{88}+4q^{87}+4q^{86}-27q^{85}-12q^{84}+35q^{83}+42q^{82}+31q^{81}-40q^{80}-147q^{79}-121q^{78}+96q^{77}+331q^{76}+313q^{75}-76q^{74}-641q^{73}-776q^{72}-93q^{71}+1058q^{70}+1597q^{69}+624q^{68}-1439q^{67}-2825q^{66}-1766q^{65}+1513q^{64}+4399q^{63}+3700q^{62}-1007q^{61}-6020q^{60}-6374q^{59}-458q^{58}+7327q^{57}+9628q^{56}+2926q^{55}-7951q^{54}-12993q^{53}-6252q^{52}+7577q^{51}+16014q^{50}+10142q^{49}-6238q^{48}-18374q^{47}-13982q^{46}+4127q^{45}+19690q^{44}+17518q^{43}-1572q^{42}-20169q^{41}-20281q^{40}-1026q^{39}+19787q^{38}+22257q^{37}+3477q^{36}-18914q^{35}-23432q^{34}-5581q^{33}+17645q^{32}+23966q^{31}+7369q^{30}-16121q^{29}-23994q^{28}-8927q^{27}+14376q^{26}+23593q^{25}+10339q^{24}-12308q^{23}-22814q^{22}-11685q^{21}+9928q^{20}+21529q^{19}+12916q^{18}-7094q^{17}-19764q^{16}-13903q^{15}+4052q^{14}+17236q^{13}+14444q^{12}-768q^{11}-14220q^{10}-14289q^{9}-2159q^{8}+10566q^{7}+13250q^{6}+4705q^{5}-6881q^{4}-11406q^{3}-6139q^{2}+3275q+8853+6687q^{-1}-448q^{-2}-6065q^{-3}-6075q^{-4}-1530q^{-5}+3414q^{-6}+4880q^{-7}+2365q^{-8}-1320q^{-9}-3277q^{-10}-2453q^{-11}-10q^{-12}+1886q^{-13}+1903q^{-14}+607q^{-15}-754q^{-16}-1269q^{-17}-720q^{-18}+189q^{-19}+656q^{-20}+522q^{-21}+99q^{-22}-262q^{-23}-331q^{-24}-123q^{-25}+81q^{-26}+144q^{-27}+81q^{-28}+3q^{-29}-52q^{-30}-54q^{-31}-q^{-32}+19q^{-33}+11q^{-34}+4q^{-35}+q^{-36}-9q^{-37}-q^{-38}+4q^{-39}-q^{-40}$
6	$q^{132}-3q^{131}+2q^{130}+3q^{129}-6q^{128}+q^{127}+2q^{126}+12q^{125}-15q^{124}-6q^{123}+17q^{122}-30q^{121}+4q^{120}+31q^{119}+65q^{118}-36q^{117}-75q^{116}-17q^{115}-136q^{114}+21q^{113}+227q^{112}+387q^{111}+45q^{110}-325q^{109}-451q^{108}-838q^{107}-198q^{106}+889q^{105}+1910q^{104}+1288q^{103}-320q^{102}-1953q^{101}-4001q^{100}-2719q^{99}+1130q^{98}+5982q^{97}+7019q^{96}+3609q^{95}-2908q^{94}-11755q^{93}-12929q^{92}-5042q^{91}+9892q^{90}+20073q^{89}+19301q^{88}+5530q^{87}-19619q^{86}-34059q^{85}-27884q^{84}+1648q^{83}+33577q^{82}+49646q^{81}+35307q^{80}-12472q^{79}-56251q^{78}-68728q^{77}-31404q^{76}+29748q^{75}+81352q^{74}+84964q^{73}+21832q^{72}-59742q^{71}-110550q^{70}-84404q^{69}-1980q^{68}+93434q^{67}+134097q^{66}+74956q^{65}-35893q^{64}-131885q^{63}-134970q^{62}-50512q^{61}+79231q^{60}+161902q^{59}+124029q^{58}+2940q^{57}-127781q^{56}-164451q^{55}-94293q^{54}+51048q^{53}+165275q^{52}+153502q^{51}+38125q^{50}-109645q^{49}-171876q^{48}-121540q^{47}+23969q^{46}+154579q^{45}+164354q^{44}+61828q^{43}-89099q^{42}-166670q^{41}-135004q^{40}+2412q^{39}+138590q^{38}+165053q^{37}+78198q^{36}-67952q^{35}-155281q^{34}-142225q^{33}-18612q^{32}+117220q^{31}+159998q^{30}+93712q^{29}-40914q^{28}-135979q^{27}-145520q^{26}-44187q^{25}+85033q^{24}+145596q^{23}+108204q^{22}-4583q^{21}-102796q^{20}-138786q^{19}-71117q^{18}+40159q^{17}+114625q^{16}+112292q^{15}+34544q^{14}-54804q^{13}-112907q^{12}-86365q^{11}-7813q^{10}+66676q^{9}+94702q^{8}+59951q^{7}-4087q^{6}-68014q^{5}-77020q^{4}-39655q^{3}+16095q^{2}+56492q+58074+28574q^{-1}-20644q^{-2}-45896q^{-3}-42500q^{-4}-15598q^{-5}+16165q^{-6}+34152q^{-7}+32323q^{-8}+7842q^{-9}-13086q^{-10}-24431q^{-11}-20201q^{-12}-6222q^{-13}+9095q^{-14}+17942q^{-15}+12261q^{-16}+3608q^{-17}-6096q^{-18}-10190q^{-19}-8666q^{-20}-2333q^{-21}+4520q^{-22}+5656q^{-23}+4916q^{-24}+1259q^{-25}-1819q^{-26}-3733q^{-27}-2765q^{-28}-291q^{-29}+799q^{-30}+1798q^{-31}+1315q^{-32}+512q^{-33}-660q^{-34}-888q^{-35}-437q^{-36}-245q^{-37}+233q^{-38}+334q^{-39}+324q^{-40}-11q^{-41}-118q^{-42}-77q^{-43}-110q^{-44}-10q^{-45}+27q^{-46}+73q^{-47}+4q^{-48}-12q^{-49}+4q^{-50}-16q^{-51}-4q^{-52}-q^{-53}+9q^{-54}+q^{-55}-4q^{-56}+q^{-57}$
7	$q^{175}-3q^{174}+2q^{173}+3q^{172}-6q^{171}+q^{170}+2q^{169}+8q^{168}+q^{167}-25q^{166}+7q^{165}+14q^{164}-14q^{163}+10q^{162}+13q^{161}+39q^{160}-122q^{158}-44q^{157}+18q^{156}+25q^{155}+151q^{154}+148q^{153}+179q^{152}-44q^{151}-557q^{150}-553q^{149}-334q^{148}+145q^{147}+1000q^{146}+1347q^{145}+1318q^{144}+214q^{143}-2076q^{142}-3397q^{141}-3455q^{140}-1292q^{139}+3115q^{138}+6743q^{137}+8395q^{136}+5285q^{135}-3214q^{134}-12095q^{133}-17673q^{132}-14436q^{131}-342q^{130}+17614q^{129}+32371q^{128}+32798q^{127}+12676q^{126}-19664q^{125}-51986q^{124}-63534q^{123}-40070q^{122}+11067q^{121}+71544q^{120}+107217q^{119}+89212q^{118}+18600q^{117}-81855q^{116}-159932q^{115}-163410q^{114}-79204q^{113}+69749q^{112}+210456q^{111}+259411q^{110}+177698q^{109}-20488q^{108}-243093q^{107}-366766q^{106}-313090q^{105}-75419q^{104}+240410q^{103}+466810q^{102}+474676q^{101}+220283q^{100}-188080q^{99}-539814q^{98}-644240q^{97}-404663q^{96}+81887q^{95}+568344q^{94}+798380q^{93}+610234q^{92}+72472q^{91}-543085q^{90}-917388q^{89}-814225q^{88}-258583q^{87}+466948q^{86}+988216q^{85}+993376q^{84}+454830q^{83}-350667q^{82}-1007602q^{81}-1133393q^{80}-640204q^{79}+213254q^{78}+983011q^{77}+1226367q^{76}+797405q^{75}-72075q^{74}-926425q^{73}-1275408q^{72}-919017q^{71}-56996q^{70}+853575q^{69}+1288481q^{68}+1003430q^{67}+165485q^{66}-776310q^{65}-1277125q^{64}-1056639q^{63}-251190q^{62}+703612q^{61}+1252009q^{60}+1086812q^{59}+317114q^{58}-638364q^{57}-1220714q^{56}-1103145q^{55}-369907q^{54}+579062q^{53}+1187340q^{52}+1113068q^{51}+417185q^{50}-521047q^{49}-1151735q^{48}-1120943q^{47}-466067q^{46}+457494q^{45}+1110493q^{44}+1128133q^{43}+521580q^{42}-381833q^{41}-1057824q^{40}-1131910q^{39}-585697q^{38}+288357q^{37}+986871q^{36}+1126789q^{35}+656067q^{34}-174596q^{33}-890498q^{32}-1104431q^{31}-726912q^{30}+41884q^{29}+765105q^{28}+1055734q^{27}+786917q^{26}+102804q^{25}-609132q^{24}-972385q^{23}-824806q^{22}-247638q^{21}+429692q^{20}+850811q^{19}+826008q^{18}+376048q^{17}-237132q^{16}-692579q^{15}-783220q^{14}-471908q^{13}+51082q^{12}+508938q^{11}+692548q^{10}+519701q^{9}+110089q^{8}-315866q^{7}-562486q^{6}-514084q^{5}-227070q^{4}+136576q^{3}+407418q^{2}+456726q+289913+10048q^{-1}-249838q^{-2}-362267q^{-3}-297100q^{-4}-108674q^{-5}+110713q^{-6}+249742q^{-7}+259807q^{-8}+155997q^{-9}-7231q^{-10}-141472q^{-11}-194768q^{-12}-158264q^{-13}-54934q^{-14}+54368q^{-15}+123208q^{-16}+130419q^{-17}+77577q^{-18}+2657q^{-19}-60293q^{-20}-89229q^{-21}-73055q^{-22}-30554q^{-23}+16850q^{-24}+49846q^{-25}+53589q^{-26}+35681q^{-27}+7106q^{-28}-20034q^{-29}-31954q^{-30}-29028q^{-31}-15014q^{-32}+3251q^{-33}+14636q^{-34}+18046q^{-35}+13731q^{-36}+4079q^{-37}-4078q^{-38}-9107q^{-39}-9313q^{-40}-4946q^{-41}-389q^{-42}+3232q^{-43}+4792q^{-44}+3582q^{-45}+1738q^{-46}-545q^{-47}-2123q^{-48}-1942q^{-49}-1286q^{-50}-242q^{-51}+602q^{-52}+753q^{-53}+771q^{-54}+403q^{-55}-146q^{-56}-297q^{-57}-325q^{-58}-175q^{-59}+9q^{-60}+17q^{-61}+115q^{-62}+115q^{-63}+22q^{-64}-20q^{-65}-48q^{-66}-23q^{-67}+9q^{-68}-11q^{-69}+q^{-70}+16q^{-71}+4q^{-72}+q^{-73}-9q^{-74}-q^{-75}+4q^{-76}-q^{-77}$

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[9, 32]]
Out[2]=	PD[X[1, 4, 2, 5], X[13, 18, 14, 1], X[3, 9, 4, 8], X[9, 3, 10, 2], X[7, 15, 8, 14], X[15, 11, 16, 10], X[5, 12, 6, 13], X[11, 17, 12, 16], X[17, 7, 18, 6]]
In[3]:=	GaussCode[Knot[9, 32]]
Out[3]=	GaussCode[-1, 4, -3, 1, -7, 9, -5, 3, -4, 6, -8, 7, -2, 5, -6, 8, -9, 2]
In[4]:=	DTCode[Knot[9, 32]]
Out[4]=	DTCode[4, 8, 12, 14, 2, 16, 18, 10, 6]
In[5]:=	br = BR[Knot[9, 32]]
Out[5]=	BR[4, {1, 1, -2, 1, -2, 1, 3, -2, 3}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{4, 9}
In[7]:=	BraidIndex[Knot[9, 32]]
Out[7]=	4
In[8]:=	Show[DrawMorseLink[Knot[9, 32]]]
Out[8]=	-Graphics-
In[9]:=	(#[Knot[9, 32]]&) /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Chiral, 2, 3, 3, {4, 6}, 1}
In[10]:=	alex = Alexander[Knot[9, 32]][t]
Out[10]=	-3 6 14 2 3 -17 + t - -- + -- + 14 t - 6 t + t 2 t t
In[11]:=	Conway[Knot[9, 32]][z]
Out[11]=	2 6 1 - z + z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[9, 32], Knot[11, NonAlternating, 52], Knot[11, NonAlternating, 124]}
In[13]:=	{KnotDet[Knot[9, 32]], KnotSignature[Knot[9, 32]]}
Out[13]=	{59, 2}
In[14]:=	Jones[Knot[9, 32]][q]
Out[14]=	-2 4 2 3 4 5 6 7 -6 - q + - + 9 q - 10 q + 10 q - 9 q + 6 q - 3 q + q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[9, 32]}
In[16]:=	A2Invariant[Knot[9, 32]][q]
Out[16]=	-6 2 2 4 6 8 14 16 18 22 1 - q + -- + 3 q - 2 q + 2 q - 2 q - 2 q + 2 q - q + q 4 q
In[17]:=	HOMFLYPT[Knot[9, 32]][a, z]
Out[17]=	2 2 2 4 4 6 -6 2 -2 2 z 4 z 3 z 4 2 z 3 z z 1 + a - -- + a - z + -- - ---- + ---- - z - ---- + ---- + -- 4 6 4 2 4 2 2 a a a a a a a
In[18]:=	Kauffman[Knot[9, 32]][a, z]
Out[18]=	2 2 2 2 -6 2 -2 z 2 z z 2 z 4 z 12 z 10 z 1 - a - -- - a + -- - --- - - + 3 z - -- + ---- + ----- + ----- - 4 7 3 a 8 6 4 2 a a a a a a a 3 3 3 3 4 4 4 4 3 z 2 z 9 z 3 z 3 4 z 6 z 18 z 19 z ---- + ---- + ---- + ---- - a z - 8 z + -- - ---- - ----- - ----- + 7 5 3 a 8 6 4 2 a a a a a a a 5 5 5 5 6 6 6 3 z 5 z 18 z 9 z 5 6 5 z 7 z 6 z ---- - ---- - ----- - ---- + a z + 4 z + ---- + ---- + ---- + 7 5 3 a 6 4 2 a a a a a a 7 7 7 8 8 5 z 10 z 5 z 2 z 2 z ---- + ----- + ---- + ---- + ---- 5 3 a 4 2 a a a a
In[19]:=	{Vassiliev[2][Knot[9, 32]], Vassiliev[3][Knot[9, 32]]}
Out[19]=	{-1, -2}
In[20]:=	Kh[Knot[9, 32]][q, t]
Out[20]=	3 1 3 1 3 3 q 3 5 6 q + 4 q + ----- + ----- + ---- + --- + --- + 5 q t + 5 q t + 5 3 3 2 2 q t t q t q t q t 5 2 7 2 7 3 9 3 9 4 11 4 11 5 5 q t + 5 q t + 4 q t + 5 q t + 2 q t + 4 q t + q t + 13 5 15 6 2 q t + q t
In[21]:=	ColouredJones[Knot[9, 32], 2][q]
Out[21]=	-7 4 -5 14 19 9 46 2 3 -32 + q - -- + q + -- - -- - -- + -- - 37 q + 82 q - 32 q - 6 4 3 2 q q q q q 4 5 6 7 8 9 10 11 69 q + 102 q - 20 q - 87 q + 97 q - 4 q - 80 q + 67 q + 12 13 14 15 16 17 18 19 20 8 q - 50 q + 29 q + 8 q - 18 q + 7 q + 2 q - 3 q + q

See/edit the Rolfsen_Splice_Template.