9 36

9_35

9_37

Visit 9 36's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 36's page at Knotilus! Visit 9 36's page at the original Knot Atlas! 9 36 Quick Notes

9 36 Further Notes and Views

Knot presentations

Planar diagram presentation	X1425 X7,10,8,11 X3948 X9,3,10,2 X11,17,12,16 X5,15,6,14 X15,7,16,6 X13,1,14,18 X17,13,18,12
Gauss code	-1, 4, -3, 1, -6, 7, -2, 3, -4, 2, -5, 9, -8, 6, -7, 5, -9, 8
Dowker-Thistlethwaite code	4 8 14 10 2 16 18 6 12
Conway Notation	[22,3,2]

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

A Morse Link Presentation:

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 3 Bridge index 3 Super bridge index $\{4,7\}$ Nakanishi index 1 Maximal Thurston-Bennequin number [1][-12] Hyperbolic Volume 9.88458 A-Polynomial See Data:9 36/A-polynomial

[edit Notes for 9 36's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for 9 36's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{3}+5t^{2}-8t+9-8t^{-1}+5t^{-2}-t^{-3}$
Conway polynomial	$-z^{6}-z^{4}+3z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 37, 4 }
Jones polynomial	$-q^{9}+2q^{8}-4q^{7}+6q^{6}-6q^{5}+6q^{4}-5q^{3}+4q^{2}-2q+1$
HOMFLY-PT polynomial (db, data sources)	$-z^{6}a^{-4}+z^{4}a^{-2}-4z^{4}a^{-4}+2z^{4}a^{-6}+3z^{2}a^{-2}-5z^{2}a^{-4}+6z^{2}a^{-6}-z^{2}a^{-8}+2a^{-2}-3a^{-4}+4a^{-6}-2a^{-8}$
Kauffman polynomial (db, data sources)	$z^{8}a^{-4}+z^{8}a^{-6}+2z^{7}a^{-3}+5z^{7}a^{-5}+3z^{7}a^{-7}+z^{6}a^{-2}+z^{6}a^{-4}+4z^{6}a^{-6}+4z^{6}a^{-8}-7z^{5}a^{-3}-14z^{5}a^{-5}-4z^{5}a^{-7}+3z^{5}a^{-9}-4z^{4}a^{-2}-12z^{4}a^{-4}-17z^{4}a^{-6}-7z^{4}a^{-8}+2z^{4}a^{-10}+6z^{3}a^{-3}+9z^{3}a^{-5}-2z^{3}a^{-9}+z^{3}a^{-11}+5z^{2}a^{-2}+12z^{2}a^{-4}+15z^{2}a^{-6}+7z^{2}a^{-8}-z^{2}a^{-10}-za^{-3}-2za^{-5}+za^{-7}+za^{-9}-za^{-11}-2a^{-2}-3a^{-4}-4a^{-6}-2a^{-8}$
The A2 invariant	$1+q^{-4}+q^{-6}-q^{-8}+q^{-10}-2q^{-12}+q^{-14}+q^{-16}+q^{-18}+2q^{-20}-q^{-22}-q^{-26}-q^{-28}$
The G2 invariant	$q^{-2}-q^{-4}+4q^{-6}-5q^{-8}+5q^{-10}-2q^{-12}-4q^{-14}+14q^{-16}-17q^{-18}+19q^{-20}-11q^{-22}-2q^{-24}+18q^{-26}-27q^{-28}+28q^{-30}-17q^{-32}+q^{-34}+13q^{-36}-23q^{-38}+20q^{-40}-9q^{-42}-5q^{-44}+15q^{-46}-18q^{-48}+9q^{-50}+3q^{-52}-18q^{-54}+24q^{-56}-24q^{-58}+16q^{-60}+q^{-62}-18q^{-64}+32q^{-66}-33q^{-68}+28q^{-70}-9q^{-72}-10q^{-74}+25q^{-76}-28q^{-78}+24q^{-80}-7q^{-82}-7q^{-84}+19q^{-86}-16q^{-88}+6q^{-90}+6q^{-92}-16q^{-94}+18q^{-96}-11q^{-98}-2q^{-100}+11q^{-102}-18q^{-104}+20q^{-106}-15q^{-108}+4q^{-110}+3q^{-112}-12q^{-114}+12q^{-116}-13q^{-118}+10q^{-120}-5q^{-122}+q^{-124}+3q^{-126}-7q^{-128}+6q^{-130}-5q^{-132}+4q^{-134}-2q^{-136}+q^{-140}-2q^{-142}+2q^{-144}-q^{-146}+q^{-148}$

Further Quantum Invariants

Further quantum knot invariants for 9_36.

A1 Invariants.

Weight	Invariant
1	$q-q^{-1}+2q^{-3}-q^{-5}+q^{-7}+2q^{-13}-2q^{-15}+q^{-17}-q^{-19}$
2	$q^{6}-q^{4}-2q^{2}+4+q^{-2}-6q^{-4}+4q^{-6}+5q^{-8}-6q^{-10}+q^{-12}+7q^{-14}-4q^{-16}-3q^{-18}+5q^{-20}-4q^{-24}+q^{-26}+4q^{-28}-3q^{-30}-5q^{-32}+7q^{-34}-7q^{-38}+5q^{-40}+q^{-42}-4q^{-44}+2q^{-46}-q^{-50}+q^{-52}$
3	$q^{15}-q^{13}-2q^{11}+5q^{7}+3q^{5}-7q^{3}-8q+6q^{-1}+14q^{-3}-18q^{-7}-6q^{-9}+17q^{-11}+16q^{-13}-11q^{-15}-21q^{-17}+5q^{-19}+23q^{-21}+4q^{-23}-21q^{-25}-10q^{-27}+20q^{-29}+15q^{-31}-16q^{-33}-17q^{-35}+12q^{-37}+19q^{-39}-9q^{-41}-21q^{-43}+3q^{-45}+18q^{-47}+3q^{-49}-16q^{-51}-12q^{-53}+10q^{-55}+20q^{-57}-2q^{-59}-24q^{-61}-5q^{-63}+22q^{-65}+14q^{-67}-20q^{-69}-14q^{-71}+12q^{-73}+12q^{-75}-6q^{-77}-8q^{-79}+3q^{-81}+4q^{-83}-2q^{-85}-q^{-87}+q^{-89}+q^{-97}-q^{-99}$
4	$q^{28}-q^{26}-2q^{24}+q^{20}+7q^{18}+q^{16}-7q^{14}-9q^{12}-8q^{10}+17q^{8}+19q^{6}+5q^{4}-17q^{2}-39-2q^{-2}+28q^{-4}+44q^{-6}+18q^{-8}-51q^{-10}-49q^{-12}-17q^{-14}+51q^{-16}+78q^{-18}+3q^{-20}-54q^{-22}-83q^{-24}-10q^{-26}+86q^{-28}+73q^{-30}+8q^{-32}-94q^{-34}-82q^{-36}+29q^{-38}+93q^{-40}+71q^{-42}-55q^{-44}-104q^{-46}-27q^{-48}+70q^{-50}+92q^{-52}-15q^{-54}-91q^{-56}-48q^{-58}+52q^{-60}+85q^{-62}-77q^{-66}-54q^{-68}+37q^{-70}+75q^{-72}+23q^{-74}-56q^{-76}-71q^{-78}-q^{-80}+55q^{-82}+69q^{-84}+4q^{-86}-75q^{-88}-69q^{-90}-6q^{-92}+99q^{-94}+89q^{-96}-29q^{-98}-105q^{-100}-89q^{-102}+60q^{-104}+126q^{-106}+38q^{-108}-69q^{-110}-113q^{-112}-2q^{-114}+81q^{-116}+55q^{-118}-8q^{-120}-67q^{-122}-22q^{-124}+25q^{-126}+26q^{-128}+11q^{-130}-22q^{-132}-8q^{-134}+4q^{-136}+3q^{-138}+7q^{-140}-6q^{-142}-q^{-144}+q^{-146}-q^{-148}+3q^{-150}-2q^{-152}-q^{-158}+q^{-160}$
5	$q^{45}-q^{43}-2q^{41}+q^{37}+3q^{35}+5q^{33}+q^{31}-9q^{29}-11q^{27}-6q^{25}+5q^{23}+21q^{21}+25q^{19}+6q^{17}-27q^{15}-44q^{13}-34q^{11}+8q^{9}+59q^{7}+76q^{5}+35q^{3}-45q-106q^{-1}-100q^{-3}-15q^{-5}+100q^{-7}+164q^{-9}+110q^{-11}-40q^{-13}-176q^{-15}-207q^{-17}-84q^{-19}+125q^{-21}+269q^{-23}+221q^{-25}+2q^{-27}-242q^{-29}-333q^{-31}-173q^{-33}+140q^{-35}+375q^{-37}+336q^{-39}+30q^{-41}-322q^{-43}-450q^{-45}-221q^{-47}+200q^{-49}+482q^{-51}+381q^{-53}-35q^{-55}-433q^{-57}-485q^{-59}-128q^{-61}+338q^{-63}+519q^{-65}+253q^{-67}-222q^{-69}-494q^{-71}-330q^{-73}+118q^{-75}+436q^{-77}+348q^{-79}-47q^{-81}-372q^{-83}-329q^{-85}+8q^{-87}+313q^{-89}+296q^{-91}-3q^{-93}-278q^{-95}-265q^{-97}+8q^{-99}+260q^{-101}+253q^{-103}+5q^{-105}-247q^{-107}-268q^{-109}-46q^{-111}+216q^{-113}+294q^{-115}+133q^{-117}-138q^{-119}-318q^{-121}-258q^{-123}+15q^{-125}+301q^{-127}+383q^{-129}+166q^{-131}-219q^{-133}-481q^{-135}-364q^{-137}+81q^{-139}+496q^{-141}+531q^{-143}+115q^{-145}-430q^{-147}-633q^{-149}-289q^{-151}+295q^{-153}+616q^{-155}+423q^{-157}-124q^{-159}-530q^{-161}-459q^{-163}-22q^{-165}+380q^{-167}+414q^{-169}+112q^{-171}-228q^{-173}-320q^{-175}-139q^{-177}+112q^{-179}+213q^{-181}+116q^{-183}-39q^{-185}-118q^{-187}-82q^{-189}+6q^{-191}+60q^{-193}+48q^{-195}+q^{-197}-25q^{-199}-22q^{-201}-4q^{-203}+9q^{-205}+11q^{-207}+2q^{-209}-7q^{-211}-3q^{-213}+2q^{-215}+2q^{-219}+q^{-221}-3q^{-223}-q^{-225}+2q^{-227}+q^{-233}-q^{-235}$

A2 Invariants.

Weight	Invariant
1,0	$1+q^{-4}+q^{-6}-q^{-8}+q^{-10}-2q^{-12}+q^{-14}+q^{-16}+q^{-18}+2q^{-20}-q^{-22}-q^{-26}-q^{-28}$
1,1	$q^{4}-2q^{2}+8-16q^{-2}+29q^{-4}-44q^{-6}+60q^{-8}-74q^{-10}+80q^{-12}-72q^{-14}+62q^{-16}-32q^{-18}+3q^{-20}+38q^{-22}-70q^{-24}+100q^{-26}-123q^{-28}+132q^{-30}-134q^{-32}+118q^{-34}-96q^{-36}+64q^{-38}-30q^{-40}+28q^{-44}-44q^{-46}+52q^{-48}-52q^{-50}+46q^{-52}-44q^{-54}+34q^{-56}-28q^{-58}+24q^{-60}-20q^{-62}+16q^{-64}-12q^{-66}+9q^{-68}-6q^{-70}+4q^{-72}-2q^{-74}+q^{-76}$
2,0	$q^{4}-1+2q^{-4}+q^{-6}-2q^{-8}-q^{-10}+4q^{-12}+q^{-14}-2q^{-16}+2q^{-18}+3q^{-20}-q^{-24}+2q^{-26}+2q^{-28}-q^{-30}+2q^{-32}+q^{-34}-4q^{-36}-3q^{-38}+2q^{-40}-2q^{-42}-3q^{-44}+2q^{-46}+4q^{-48}+q^{-50}-2q^{-52}+q^{-54}-3q^{-58}-2q^{-60}-q^{-62}+q^{-66}+q^{-68}+q^{-70}$

A3 Invariants.

Weight	Invariant
0,1,0	$1-q^{-2}+2q^{-4}+2q^{-6}-2q^{-8}+4q^{-10}+q^{-12}-4q^{-14}+4q^{-16}-q^{-18}-5q^{-20}+4q^{-22}+2q^{-24}-3q^{-26}+3q^{-28}+3q^{-30}+q^{-32}-q^{-34}+q^{-36}+3q^{-38}-5q^{-40}-2q^{-42}+4q^{-44}-6q^{-46}-2q^{-48}+5q^{-50}-3q^{-52}-2q^{-54}+3q^{-56}-q^{-60}+q^{-62}$
1,0,0	$q^{-1}+2q^{-5}+2q^{-9}-q^{-11}-2q^{-15}-q^{-17}+q^{-21}+3q^{-23}+q^{-25}+3q^{-27}-q^{-29}-2q^{-33}-q^{-35}-q^{-37}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{-2}+q^{-6}+3q^{-8}+q^{-10}+q^{-12}+3q^{-14}-3q^{-18}+q^{-22}-3q^{-24}-q^{-26}+6q^{-28}+5q^{-30}-4q^{-32}+2q^{-34}+5q^{-36}-5q^{-38}-5q^{-40}+4q^{-42}+q^{-44}-q^{-46}+5q^{-48}+6q^{-50}-q^{-52}-2q^{-54}+2q^{-56}-3q^{-58}-8q^{-60}-2q^{-62}+q^{-64}-4q^{-66}-2q^{-68}+2q^{-70}+2q^{-72}+q^{-78}+q^{-80}$
1,0,0,0	$q^{-2}+2q^{-6}+q^{-8}+q^{-10}+2q^{-12}-q^{-14}-3q^{-18}-q^{-20}-2q^{-22}+q^{-26}+3q^{-28}+3q^{-30}+2q^{-32}+3q^{-34}-q^{-36}-2q^{-40}-2q^{-42}-q^{-44}-q^{-46}$

B2 Invariants.

Weight	Invariant
0,1	$1-q^{-2}+4q^{-4}-4q^{-6}+6q^{-8}-6q^{-10}+7q^{-12}-6q^{-14}+4q^{-16}-3q^{-18}-q^{-20}+4q^{-22}-8q^{-24}+11q^{-26}-11q^{-28}+13q^{-30}-11q^{-32}+11q^{-34}-7q^{-36}+5q^{-38}-q^{-40}-2q^{-42}+4q^{-44}-6q^{-46}+6q^{-48}-7q^{-50}+5q^{-52}-4q^{-54}+3q^{-56}-2q^{-58}+q^{-60}-q^{-62}$
1,0	$q^{2}-q^{-2}-q^{-4}+3q^{-6}+3q^{-8}-2q^{-10}-4q^{-12}+q^{-14}+6q^{-16}+3q^{-18}-6q^{-20}-5q^{-22}+4q^{-24}+7q^{-26}-7q^{-30}-3q^{-32}+5q^{-34}+5q^{-36}-2q^{-38}-4q^{-40}+2q^{-42}+4q^{-44}-4q^{-48}+5q^{-52}+2q^{-54}-5q^{-56}-3q^{-58}+4q^{-60}+5q^{-62}-3q^{-64}-6q^{-66}+6q^{-70}+q^{-72}-6q^{-74}-5q^{-76}+2q^{-78}+6q^{-80}-4q^{-84}-3q^{-86}+q^{-88}+3q^{-90}+q^{-92}-q^{-94}-q^{-96}+q^{-100}$

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{-2}-q^{-4}+3q^{-6}-2q^{-8}+6q^{-10}-3q^{-12}+6q^{-14}-4q^{-16}+7q^{-18}-5q^{-20}+2q^{-22}-4q^{-24}-q^{-28}-4q^{-30}+4q^{-32}-6q^{-34}+9q^{-36}-7q^{-38}+12q^{-40}-6q^{-42}+12q^{-44}-7q^{-46}+9q^{-48}-5q^{-50}+4q^{-52}-4q^{-54}-2q^{-56}-q^{-58}-4q^{-60}+3q^{-62}-6q^{-64}+3q^{-66}-5q^{-68}+6q^{-70}-4q^{-72}+2q^{-74}-3q^{-76}+3q^{-78}-q^{-80}+q^{-82}-q^{-84}+q^{-86}$

G2 Invariants.

Weight

Invariant

1,0

$q^{-2}-q^{-4}+4q^{-6}-5q^{-8}+5q^{-10}-2q^{-12}-4q^{-14}+14q^{-16}-17q^{-18}+19q^{-20}-11q^{-22}-2q^{-24}+18q^{-26}-27q^{-28}+28q^{-30}-17q^{-32}+q^{-34}+13q^{-36}-23q^{-38}+20q^{-40}-9q^{-42}-5q^{-44}+15q^{-46}-18q^{-48}+9q^{-50}+3q^{-52}-18q^{-54}+24q^{-56}-24q^{-58}+16q^{-60}+q^{-62}-18q^{-64}+32q^{-66}-33q^{-68}+28q^{-70}-9q^{-72}-10q^{-74}+25q^{-76}-28q^{-78}+24q^{-80}-7q^{-82}-7q^{-84}+19q^{-86}-16q^{-88}+6q^{-90}+6q^{-92}-16q^{-94}+18q^{-96}-11q^{-98}-2q^{-100}+11q^{-102}-18q^{-104}+20q^{-106}-15q^{-108}+4q^{-110}+3q^{-112}-12q^{-114}+12q^{-116}-13q^{-118}+10q^{-120}-5q^{-122}+q^{-124}+3q^{-126}-7q^{-128}+6q^{-130}-5q^{-132}+4q^{-134}-2q^{-136}+q^{-140}-2q^{-142}+2q^{-144}-q^{-146}+q^{-148}$

.

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 36"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t^{3}+5t^{2}-8t+9-8t^{-1}+5t^{-2}-t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 37, 4 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{8}a^{-4}+z^{8}a^{-6}+2z^{7}a^{-3}+5z^{7}a^{-5}+3z^{7}a^{-7}+z^{6}a^{-2}+z^{6}a^{-4}+4z^{6}a^{-6}+4z^{6}a^{-8}-7z^{5}a^{-3}-14z^{5}a^{-5}-4z^{5}a^{-7}+3z^{5}a^{-9}-4z^{4}a^{-2}-12z^{4}a^{-4}-17z^{4}a^{-6}-7z^{4}a^{-8}+2z^{4}a^{-10}+6z^{3}a^{-3}+9z^{3}a^{-5}-2z^{3}a^{-9}+z^{3}a^{-11}+5z^{2}a^{-2}+12z^{2}a^{-4}+15z^{2}a^{-6}+7z^{2}a^{-8}-z^{2}a^{-10}-za^{-3}-2za^{-5}+za^{-7}+za^{-9}-za^{-11}-2a^{-2}-3a^{-4}-4a^{-6}-2a^{-8}$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {...}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {K11n16, ...}

Vassiliev invariants

V2 and V3:

(3, 7)

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 $12$ $56$ $72$ $270$ $42$ $672$ ${\frac {4304}{3}}$ ${\frac {704}{3}}$ $216$ $288$ $1568$ $3240$ $504$ ${\frac {78751}{10}}$ ${\frac {138}{5}}$ ${\frac {48742}{15}}$ ${\frac {577}{6}}$ ${\frac {4191}{10}}$

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=$4 is the signature of 9 36. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

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

The Coloured Jones Polynomials

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

$n$	$J_{n}$
2	$q^{25}-2q^{24}+q^{23}+3q^{22}-8q^{21}+6q^{20}+7q^{19}-20q^{18}+13q^{17}+14q^{16}-32q^{15}+15q^{14}+21q^{13}-35q^{12}+10q^{11}+25q^{10}-30q^{9}+2q^{8}+24q^{7}-19q^{6}-4q^{5}+17q^{4}-8q^{3}-5q^{2}+7q-1-2q^{-1}+q^{-2}$
3	$-q^{48}+2q^{47}-q^{46}-q^{44}+3q^{43}-3q^{42}-q^{41}+5q^{40}+2q^{39}-14q^{38}+q^{37}+23q^{36}+2q^{35}-40q^{34}-5q^{33}+57q^{32}+10q^{31}-67q^{30}-24q^{29}+79q^{28}+32q^{27}-77q^{26}-46q^{25}+75q^{24}+51q^{23}-62q^{22}-61q^{21}+51q^{20}+63q^{19}-34q^{18}-68q^{17}+22q^{16}+64q^{15}-3q^{14}-63q^{13}-8q^{12}+53q^{11}+22q^{10}-44q^{9}-26q^{8}+27q^{7}+32q^{6}-17q^{5}-25q^{4}+4q^{3}+20q^{2}+q-11-4q^{-1}+6q^{-2}+2q^{-3}-q^{-4}-2q^{-5}+q^{-6}$
4	$q^{78}-2q^{77}+q^{76}-2q^{74}+6q^{73}-6q^{72}+3q^{71}-2q^{70}-7q^{69}+19q^{68}-10q^{67}+4q^{66}-14q^{65}-21q^{64}+52q^{63}+5q^{62}+3q^{61}-61q^{60}-66q^{59}+111q^{58}+68q^{57}+29q^{56}-144q^{55}-177q^{54}+155q^{53}+175q^{52}+117q^{51}-210q^{50}-326q^{49}+139q^{48}+251q^{47}+235q^{46}-200q^{45}-431q^{44}+76q^{43}+245q^{42}+314q^{41}-135q^{40}-445q^{39}+20q^{38}+175q^{37}+329q^{36}-56q^{35}-393q^{34}-18q^{33}+84q^{32}+306q^{31}+21q^{30}-308q^{29}-51q^{28}-16q^{27}+263q^{26}+97q^{25}-201q^{24}-73q^{23}-113q^{22}+186q^{21}+146q^{20}-75q^{19}-51q^{18}-177q^{17}+75q^{16}+134q^{15}+27q^{14}+14q^{13}-164q^{12}-21q^{11}+61q^{10}+56q^{9}+71q^{8}-89q^{7}-48q^{6}-7q^{5}+24q^{4}+69q^{3}-20q^{2}-22q-23-6q^{-1}+32q^{-2}+2q^{-3}-9q^{-5}-8q^{-6}+7q^{-7}+q^{-8}+2q^{-9}-q^{-10}-2q^{-11}+q^{-12}$
5	$-q^{115}+2q^{114}-q^{113}+2q^{111}-3q^{110}-3q^{109}+6q^{108}-2q^{106}+4q^{105}-8q^{104}-7q^{103}+15q^{102}+9q^{101}-4q^{100}-9q^{99}-26q^{98}-10q^{97}+41q^{96}+56q^{95}+8q^{94}-63q^{93}-114q^{92}-46q^{91}+120q^{90}+211q^{89}+105q^{88}-164q^{87}-365q^{86}-227q^{85}+212q^{84}+551q^{83}+407q^{82}-198q^{81}-767q^{80}-664q^{79}+141q^{78}+957q^{77}+954q^{76}-5q^{75}-1088q^{74}-1248q^{73}-203q^{72}+1160q^{71}+1499q^{70}+411q^{69}-1123q^{68}-1663q^{67}-648q^{66}+1043q^{65}+1761q^{64}+796q^{63}-906q^{62}-1745q^{61}-934q^{60}+770q^{59}+1701q^{58}+976q^{57}-635q^{56}-1584q^{55}-1012q^{54}+508q^{53}+1479q^{52}+997q^{51}-383q^{50}-1336q^{49}-1005q^{48}+256q^{47}+1206q^{46}+984q^{45}-108q^{44}-1037q^{43}-988q^{42}-49q^{41}+869q^{40}+941q^{39}+217q^{38}-642q^{37}-900q^{36}-367q^{35}+421q^{34}+777q^{33}+489q^{32}-167q^{31}-634q^{30}-548q^{29}-45q^{28}+420q^{27}+541q^{26}+231q^{25}-218q^{24}-447q^{23}-327q^{22}-q^{21}+312q^{20}+359q^{19}+134q^{18}-141q^{17}-288q^{16}-236q^{15}-q^{14}+199q^{13}+225q^{12}+103q^{11}-69q^{10}-188q^{9}-145q^{8}-10q^{7}+102q^{6}+134q^{5}+67q^{4}-38q^{3}-91q^{2}-74q-13+49q^{-1}+61q^{-2}+23q^{-3}-11q^{-4}-33q^{-5}-30q^{-6}-2q^{-7}+19q^{-8}+13q^{-9}+6q^{-10}-11q^{-12}-6q^{-13}+3q^{-14}+2q^{-15}+q^{-16}+2q^{-17}-q^{-18}-2q^{-19}+q^{-20}$
6	$q^{159}-2q^{158}+q^{157}-2q^{155}+3q^{154}+3q^{152}-9q^{151}+4q^{150}+5q^{149}-9q^{148}+7q^{147}+q^{146}+2q^{145}-22q^{144}+14q^{143}+23q^{142}-17q^{141}+7q^{140}-8q^{139}-20q^{138}-50q^{137}+49q^{136}+91q^{135}+4q^{134}+7q^{133}-74q^{132}-140q^{131}-147q^{130}+126q^{129}+324q^{128}+197q^{127}+80q^{126}-272q^{125}-576q^{124}-535q^{123}+171q^{122}+880q^{121}+896q^{120}+532q^{119}-526q^{118}-1563q^{117}-1660q^{116}-215q^{115}+1649q^{114}+2360q^{113}+1889q^{112}-320q^{111}-2866q^{110}-3714q^{109}-1619q^{108}+1932q^{107}+4143q^{106}+4224q^{105}+980q^{104}-3574q^{103}-5994q^{102}-3962q^{101}+1053q^{100}+5152q^{99}+6605q^{98}+3142q^{97}-3008q^{96}-7301q^{95}-6185q^{94}-650q^{93}+4821q^{92}+7869q^{91}+5049q^{90}-1629q^{89}-7217q^{88}-7257q^{87}-2117q^{86}+3715q^{85}+7801q^{84}+5887q^{83}-398q^{82}-6372q^{81}-7173q^{80}-2778q^{79}+2663q^{78}+7061q^{77}+5837q^{76}+329q^{75}-5445q^{74}-6593q^{73}-2950q^{72}+1847q^{71}+6211q^{70}+5538q^{69}+906q^{68}-4520q^{67}-5963q^{66}-3170q^{65}+929q^{64}+5266q^{63}+5298q^{62}+1723q^{61}-3306q^{60}-5211q^{59}-3559q^{58}-342q^{57}+3938q^{56}+4911q^{55}+2725q^{54}-1636q^{53}-3997q^{52}-3749q^{51}-1783q^{50}+2085q^{49}+3939q^{48}+3405q^{47}+222q^{46}-2163q^{45}-3196q^{44}-2783q^{43}+36q^{42}+2210q^{41}+3149q^{40}+1556q^{39}-113q^{38}-1741q^{37}-2679q^{36}-1399q^{35}+225q^{34}+1847q^{33}+1679q^{32}+1255q^{31}-4q^{30}-1467q^{29}-1540q^{28}-1026q^{27}+263q^{26}+705q^{25}+1311q^{24}+974q^{23}-55q^{22}-651q^{21}-1006q^{20}-551q^{19}-358q^{18}+482q^{17}+801q^{16}+553q^{15}+208q^{14}-284q^{13}-370q^{12}-649q^{11}-190q^{10}+160q^{9}+328q^{8}+364q^{7}+185q^{6}+75q^{5}-326q^{4}-247q^{3}-156q^{2}-12q+110+165q^{-1}+202q^{-2}-36q^{-3}-56q^{-4}-104q^{-5}-80q^{-6}-44q^{-7}+24q^{-8}+103q^{-9}+22q^{-10}+24q^{-11}-13q^{-12}-24q^{-13}-39q^{-14}-16q^{-15}+24q^{-16}+3q^{-17}+14q^{-18}+5q^{-19}+3q^{-20}-11q^{-21}-8q^{-22}+5q^{-23}-2q^{-24}+2q^{-25}+q^{-26}+2q^{-27}-q^{-28}-2q^{-29}+q^{-30}$
7	$-q^{210}+2q^{209}-q^{208}+2q^{206}-3q^{205}+5q^{201}-7q^{200}+10q^{198}-7q^{197}-2q^{195}+10q^{193}-23q^{192}-2q^{191}+30q^{190}+2q^{189}+8q^{188}-13q^{187}-16q^{186}-2q^{185}-59q^{184}-4q^{183}+80q^{182}+66q^{181}+72q^{180}-25q^{179}-111q^{178}-128q^{177}-195q^{176}-22q^{175}+234q^{174}+355q^{173}+396q^{172}+55q^{171}-391q^{170}-691q^{169}-824q^{168}-287q^{167}+598q^{166}+1339q^{165}+1627q^{164}+762q^{163}-796q^{162}-2263q^{161}-2972q^{160}-1774q^{159}+793q^{158}+3493q^{157}+5011q^{156}+3568q^{155}-291q^{154}-4852q^{153}-7804q^{152}-6380q^{151}-985q^{150}+5979q^{149}+11075q^{148}+10283q^{147}+3487q^{146}-6418q^{145}-14563q^{144}-15080q^{143}-7200q^{142}+5766q^{141}+17506q^{140}+20258q^{139}+12093q^{138}-3700q^{137}-19423q^{136}-25223q^{135}-17618q^{134}+357q^{133}+19939q^{132}+29168q^{131}+23038q^{130}+3952q^{129}-18853q^{128}-31698q^{127}-27838q^{126}-8519q^{125}+16685q^{124}+32629q^{123}+31204q^{122}+12699q^{121}-13680q^{120}-32121q^{119}-33229q^{118}-16054q^{117}+10735q^{116}+30705q^{115}+33766q^{114}+18191q^{113}-8068q^{112}-28724q^{111}-33353q^{110}-19364q^{109}+6072q^{108}+26814q^{107}+32294q^{106}+19632q^{105}-4667q^{104}-24989q^{103}-31027q^{102}-19553q^{101}+3673q^{100}+23484q^{99}+29786q^{98}+19309q^{97}-2816q^{96}-22038q^{95}-28674q^{94}-19251q^{93}+1810q^{92}+20571q^{91}+27645q^{90}+19425q^{89}-448q^{88}-18811q^{87}-26599q^{86}-19868q^{85}-1347q^{84}+16644q^{83}+25314q^{82}+20431q^{81}+3602q^{80}-13898q^{79}-23673q^{78}-20966q^{77}-6153q^{76}+10627q^{75}+21432q^{74}+21134q^{73}+8861q^{72}-6795q^{71}-18529q^{70}-20805q^{69}-11345q^{68}+2718q^{67}+14866q^{66}+19571q^{65}+13324q^{64}+1475q^{63}-10618q^{62}-17429q^{61}-14381q^{60}-5224q^{59}+5978q^{58}+14230q^{57}+14294q^{56}+8187q^{55}-1397q^{54}-10277q^{53}-12873q^{52}-9924q^{51}-2639q^{50}+5923q^{49}+10292q^{48}+10201q^{47}+5568q^{46}-1731q^{45}-6854q^{44}-9031q^{43}-7126q^{42}-1705q^{41}+3248q^{40}+6739q^{39}+7095q^{38}+3871q^{37}+47q^{36}-3839q^{35}-5856q^{34}-4688q^{33}-2293q^{32}+1104q^{31}+3757q^{30}+4129q^{29}+3389q^{28}+1072q^{27}-1580q^{26}-2836q^{25}-3283q^{24}-2149q^{23}-197q^{22}+1173q^{21}+2391q^{20}+2332q^{19}+1230q^{18}+150q^{17}-1205q^{16}-1771q^{15}-1450q^{14}-957q^{13}+153q^{12}+942q^{11}+1134q^{10}+1164q^{9}+478q^{8}-209q^{7}-592q^{6}-925q^{5}-656q^{4}-253q^{3}+75q^{2}+539q+567+390q^{-1}+192q^{-2}-194q^{-3}-300q^{-4}-314q^{-5}-301q^{-6}-38q^{-7}+117q^{-8}+195q^{-9}+232q^{-10}+85q^{-11}+17q^{-12}-43q^{-13}-148q^{-14}-102q^{-15}-57q^{-16}+4q^{-17}+74q^{-18}+41q^{-19}+40q^{-20}+37q^{-21}-15q^{-22}-27q^{-23}-34q^{-24}-22q^{-25}+13q^{-26}+q^{-27}+5q^{-28}+16q^{-29}+5q^{-30}+2q^{-31}-8q^{-32}-8q^{-33}+3q^{-34}-2q^{-36}+2q^{-37}+q^{-38}+2q^{-39}-q^{-40}-2q^{-41}+q^{-42}$

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, 36]]
Out[2]=	PD[X[1, 4, 2, 5], X[7, 10, 8, 11], X[3, 9, 4, 8], X[9, 3, 10, 2], X[11, 17, 12, 16], X[5, 15, 6, 14], X[15, 7, 16, 6], X[13, 1, 14, 18], X[17, 13, 18, 12]]
In[3]:=	GaussCode[Knot[9, 36]]
Out[3]=	GaussCode[-1, 4, -3, 1, -6, 7, -2, 3, -4, 2, -5, 9, -8, 6, -7, 5, -9, 8]
In[4]:=	DTCode[Knot[9, 36]]
Out[4]=	DTCode[4, 8, 14, 10, 2, 16, 18, 6, 12]
In[5]:=	br = BR[Knot[9, 36]]
Out[5]=	BR[4, {1, 1, 1, -2, 1, 1, 3, -2, 3}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{4, 9}
In[7]:=	BraidIndex[Knot[9, 36]]
Out[7]=	4
In[8]:=	Show[DrawMorseLink[Knot[9, 36]]]
Out[8]=	-Graphics-
In[9]:=	(#[Knot[9, 36]]&) /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 2, 3, 3, {4, 7}, 1}
In[10]:=	alex = Alexander[Knot[9, 36]][t]
Out[10]=	-3 5 8 2 3 9 - t + -- - - - 8 t + 5 t - t 2 t t
In[11]:=	Conway[Knot[9, 36]][z]
Out[11]=	2 4 6 1 + 3 z - z - z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[9, 36]}
In[13]:=	{KnotDet[Knot[9, 36]], KnotSignature[Knot[9, 36]]}
Out[13]=	{37, 4}
In[14]:=	Jones[Knot[9, 36]][q]
Out[14]=	2 3 4 5 6 7 8 9 1 - 2 q + 4 q - 5 q + 6 q - 6 q + 6 q - 4 q + 2 q - q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[9, 36], Knot[11, NonAlternating, 16]}
In[16]:=	A2Invariant[Knot[9, 36]][q]
Out[16]=	4 6 8 10 12 14 16 18 20 22 26 1 + q + q - q + q - 2 q + q + q + q + 2 q - q - q - 28 q
In[17]:=	HOMFLYPT[Knot[9, 36]][a, z]
Out[17]=	2 2 2 2 4 4 4 6 -2 4 3 2 z 6 z 5 z 3 z 2 z 4 z z z -- + -- - -- + -- - -- + ---- - ---- + ---- + ---- - ---- + -- - -- 8 6 4 2 8 6 4 2 6 4 2 4 a a a a a a a a a a a a
In[18]:=	Kauffman[Knot[9, 36]][a, z]
Out[18]=	2 2 2 -2 4 3 2 z z z 2 z z z 7 z 15 z -- - -- - -- - -- - --- + -- + -- - --- - -- - --- + ---- + ----- + 8 6 4 2 11 9 7 5 3 10 8 6 a a a a a a a a a a a a 2 2 3 3 3 3 4 4 4 12 z 5 z z 2 z 9 z 6 z 2 z 7 z 17 z ----- + ---- + --- - ---- + ---- + ---- + ---- - ---- - ----- - 4 2 11 9 5 3 10 8 6 a a a a a a a a a 4 4 5 5 5 5 6 6 6 6 12 z 4 z 3 z 4 z 14 z 7 z 4 z 4 z z z ----- - ---- + ---- - ---- - ----- - ---- + ---- + ---- + -- + -- + 4 2 9 7 5 3 8 6 4 2 a a a a a a a a a a 7 7 7 8 8 3 z 5 z 2 z z z ---- + ---- + ---- + -- + -- 7 5 3 6 4 a a a a a
In[19]:=	{Vassiliev[2][Knot[9, 36]], Vassiliev[3][Knot[9, 36]]}
Out[19]=	{3, 7}
In[20]:=	Kh[Knot[9, 36]][q, t]
Out[20]=	3 3 5 1 q q 5 7 7 2 9 2 3 q + 2 q + ---- + - + -- + 3 q t + 2 q t + 3 q t + 3 q t + 2 t t q t 9 3 11 3 11 4 13 4 13 5 15 5 3 q t + 3 q t + 3 q t + 3 q t + q t + 3 q t + 15 6 17 6 19 7 q t + q t + q t
In[21]:=	ColouredJones[Knot[9, 36], 2][q]
Out[21]=	-2 2 2 3 4 5 6 7 -1 + q - - + 7 q - 5 q - 8 q + 17 q - 4 q - 19 q + 24 q + q 8 9 10 11 12 13 14 15 2 q - 30 q + 25 q + 10 q - 35 q + 21 q + 15 q - 32 q + 16 17 18 19 20 21 22 23 14 q + 13 q - 20 q + 7 q + 6 q - 8 q + 3 q + q - 24 25 2 q + q

See/edit the Rolfsen_Splice_Template.