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 X_7,10,8,11 X₃₉₄₈ 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
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.

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, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q\leftrightarrow q^{-1}} ): {K11n16, ...}

Vassiliev invariants

V₂ and V₃:

(3, 7)

V_2,1 through V_6,9:

V_2,1

V_3,1

V_4,1

V_4,2

V_4,3

V_5,1

V_5,2

V_5,3

V_5,4

V_6,1

V_6,2

V_6,3

V_6,4

V_6,5

V_6,6

V_6,7

V_6,8

V_6,9

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 12}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 56}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 72}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 270}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 42}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 672}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{4304}{3}}

{\frac {704}{3}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 216}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 288}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1568}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3240}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 504}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{78751}{10}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{138}{5}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{48742}{15}}

{\frac {577}{6}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t^rq^j} are shown, along with their alternating sums Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \chi} (fixed Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j} , alternation over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} ). The squares with yellow highlighting are those on the "critical diagonals", where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j-2r=s+1} or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j-2r=s-1} , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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

15

3

1

-2

13

3

1

2

11

3

0

9

3

0

7

2

3

1

5

2

3

-1

3

1

3

2

1

-1

1

Integral Khovanov Homology

(db, data source)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i=3}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i=5}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-2}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}}
$r=-1$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=0}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}\oplus{\mathbb Z}_2}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=1}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=2}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=3}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=4}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=5}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}\oplus{\mathbb Z}_2^{3}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=6}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}\oplus{\mathbb Z}_2}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=7}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}_2}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}}

The Coloured Jones Polynomials

The Coloured Jones Polynomials (in the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n+1} -dimensional representation of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle sl(2)} )

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J_n}
2	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{25}-2 q^{24}+q^{23}+3 q^{22}-8 q^{21}+6 q^{20}+7 q^{19}-20 q^{18}+13 q^{17}+14 q^{16}-32 q^{15}+15 q^{14}+21 q^{13}-35 q^{12}+10 q^{11}+25 q^{10}-30 q^9+2 q^8+24 q^7-19 q^6-4 q^5+17 q^4-8 q^3-5 q^2+7 q-1-2 q^{-1} + q^{-2} }
3	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^{48}+2 q^{47}-q^{46}-q^{44}+3 q^{43}-3 q^{42}-q^{41}+5 q^{40}+2 q^{39}-14 q^{38}+q^{37}+23 q^{36}+2 q^{35}-40 q^{34}-5 q^{33}+57 q^{32}+10 q^{31}-67 q^{30}-24 q^{29}+79 q^{28}+32 q^{27}-77 q^{26}-46 q^{25}+75 q^{24}+51 q^{23}-62 q^{22}-61 q^{21}+51 q^{20}+63 q^{19}-34 q^{18}-68 q^{17}+22 q^{16}+64 q^{15}-3 q^{14}-63 q^{13}-8 q^{12}+53 q^{11}+22 q^{10}-44 q^9-26 q^8+27 q^7+32 q^6-17 q^5-25 q^4+4 q^3+20 q^2+q-11-4 q^{-1} +6 q^{-2} +2 q^{-3} - q^{-4} -2 q^{-5} + q^{-6} }
4	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{78}-2 q^{77}+q^{76}-2 q^{74}+6 q^{73}-6 q^{72}+3 q^{71}-2 q^{70}-7 q^{69}+19 q^{68}-10 q^{67}+4 q^{66}-14 q^{65}-21 q^{64}+52 q^{63}+5 q^{62}+3 q^{61}-61 q^{60}-66 q^{59}+111 q^{58}+68 q^{57}+29 q^{56}-144 q^{55}-177 q^{54}+155 q^{53}+175 q^{52}+117 q^{51}-210 q^{50}-326 q^{49}+139 q^{48}+251 q^{47}+235 q^{46}-200 q^{45}-431 q^{44}+76 q^{43}+245 q^{42}+314 q^{41}-135 q^{40}-445 q^{39}+20 q^{38}+175 q^{37}+329 q^{36}-56 q^{35}-393 q^{34}-18 q^{33}+84 q^{32}+306 q^{31}+21 q^{30}-308 q^{29}-51 q^{28}-16 q^{27}+263 q^{26}+97 q^{25}-201 q^{24}-73 q^{23}-113 q^{22}+186 q^{21}+146 q^{20}-75 q^{19}-51 q^{18}-177 q^{17}+75 q^{16}+134 q^{15}+27 q^{14}+14 q^{13}-164 q^{12}-21 q^{11}+61 q^{10}+56 q^9+71 q^8-89 q^7-48 q^6-7 q^5+24 q^4+69 q^3-20 q^2-22 q-23-6 q^{-1} +32 q^{-2} +2 q^{-3} -9 q^{-5} -8 q^{-6} +7 q^{-7} + q^{-8} +2 q^{-9} - q^{-10} -2 q^{-11} + q^{-12} }
5	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^{115}+2 q^{114}-q^{113}+2 q^{111}-3 q^{110}-3 q^{109}+6 q^{108}-2 q^{106}+4 q^{105}-8 q^{104}-7 q^{103}+15 q^{102}+9 q^{101}-4 q^{100}-9 q^{99}-26 q^{98}-10 q^{97}+41 q^{96}+56 q^{95}+8 q^{94}-63 q^{93}-114 q^{92}-46 q^{91}+120 q^{90}+211 q^{89}+105 q^{88}-164 q^{87}-365 q^{86}-227 q^{85}+212 q^{84}+551 q^{83}+407 q^{82}-198 q^{81}-767 q^{80}-664 q^{79}+141 q^{78}+957 q^{77}+954 q^{76}-5 q^{75}-1088 q^{74}-1248 q^{73}-203 q^{72}+1160 q^{71}+1499 q^{70}+411 q^{69}-1123 q^{68}-1663 q^{67}-648 q^{66}+1043 q^{65}+1761 q^{64}+796 q^{63}-906 q^{62}-1745 q^{61}-934 q^{60}+770 q^{59}+1701 q^{58}+976 q^{57}-635 q^{56}-1584 q^{55}-1012 q^{54}+508 q^{53}+1479 q^{52}+997 q^{51}-383 q^{50}-1336 q^{49}-1005 q^{48}+256 q^{47}+1206 q^{46}+984 q^{45}-108 q^{44}-1037 q^{43}-988 q^{42}-49 q^{41}+869 q^{40}+941 q^{39}+217 q^{38}-642 q^{37}-900 q^{36}-367 q^{35}+421 q^{34}+777 q^{33}+489 q^{32}-167 q^{31}-634 q^{30}-548 q^{29}-45 q^{28}+420 q^{27}+541 q^{26}+231 q^{25}-218 q^{24}-447 q^{23}-327 q^{22}-q^{21}+312 q^{20}+359 q^{19}+134 q^{18}-141 q^{17}-288 q^{16}-236 q^{15}-q^{14}+199 q^{13}+225 q^{12}+103 q^{11}-69 q^{10}-188 q^9-145 q^8-10 q^7+102 q^6+134 q^5+67 q^4-38 q^3-91 q^2-74 q-13+49 q^{-1} +61 q^{-2} +23 q^{-3} -11 q^{-4} -33 q^{-5} -30 q^{-6} -2 q^{-7} +19 q^{-8} +13 q^{-9} +6 q^{-10} -11 q^{-12} -6 q^{-13} +3 q^{-14} +2 q^{-15} + q^{-16} +2 q^{-17} - q^{-18} -2 q^{-19} + q^{-20} }
6	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{159}-2 q^{158}+q^{157}-2 q^{155}+3 q^{154}+3 q^{152}-9 q^{151}+4 q^{150}+5 q^{149}-9 q^{148}+7 q^{147}+q^{146}+2 q^{145}-22 q^{144}+14 q^{143}+23 q^{142}-17 q^{141}+7 q^{140}-8 q^{139}-20 q^{138}-50 q^{137}+49 q^{136}+91 q^{135}+4 q^{134}+7 q^{133}-74 q^{132}-140 q^{131}-147 q^{130}+126 q^{129}+324 q^{128}+197 q^{127}+80 q^{126}-272 q^{125}-576 q^{124}-535 q^{123}+171 q^{122}+880 q^{121}+896 q^{120}+532 q^{119}-526 q^{118}-1563 q^{117}-1660 q^{116}-215 q^{115}+1649 q^{114}+2360 q^{113}+1889 q^{112}-320 q^{111}-2866 q^{110}-3714 q^{109}-1619 q^{108}+1932 q^{107}+4143 q^{106}+4224 q^{105}+980 q^{104}-3574 q^{103}-5994 q^{102}-3962 q^{101}+1053 q^{100}+5152 q^{99}+6605 q^{98}+3142 q^{97}-3008 q^{96}-7301 q^{95}-6185 q^{94}-650 q^{93}+4821 q^{92}+7869 q^{91}+5049 q^{90}-1629 q^{89}-7217 q^{88}-7257 q^{87}-2117 q^{86}+3715 q^{85}+7801 q^{84}+5887 q^{83}-398 q^{82}-6372 q^{81}-7173 q^{80}-2778 q^{79}+2663 q^{78}+7061 q^{77}+5837 q^{76}+329 q^{75}-5445 q^{74}-6593 q^{73}-2950 q^{72}+1847 q^{71}+6211 q^{70}+5538 q^{69}+906 q^{68}-4520 q^{67}-5963 q^{66}-3170 q^{65}+929 q^{64}+5266 q^{63}+5298 q^{62}+1723 q^{61}-3306 q^{60}-5211 q^{59}-3559 q^{58}-342 q^{57}+3938 q^{56}+4911 q^{55}+2725 q^{54}-1636 q^{53}-3997 q^{52}-3749 q^{51}-1783 q^{50}+2085 q^{49}+3939 q^{48}+3405 q^{47}+222 q^{46}-2163 q^{45}-3196 q^{44}-2783 q^{43}+36 q^{42}+2210 q^{41}+3149 q^{40}+1556 q^{39}-113 q^{38}-1741 q^{37}-2679 q^{36}-1399 q^{35}+225 q^{34}+1847 q^{33}+1679 q^{32}+1255 q^{31}-4 q^{30}-1467 q^{29}-1540 q^{28}-1026 q^{27}+263 q^{26}+705 q^{25}+1311 q^{24}+974 q^{23}-55 q^{22}-651 q^{21}-1006 q^{20}-551 q^{19}-358 q^{18}+482 q^{17}+801 q^{16}+553 q^{15}+208 q^{14}-284 q^{13}-370 q^{12}-649 q^{11}-190 q^{10}+160 q^9+328 q^8+364 q^7+185 q^6+75 q^5-326 q^4-247 q^3-156 q^2-12 q+110+165 q^{-1} +202 q^{-2} -36 q^{-3} -56 q^{-4} -104 q^{-5} -80 q^{-6} -44 q^{-7} +24 q^{-8} +103 q^{-9} +22 q^{-10} +24 q^{-11} -13 q^{-12} -24 q^{-13} -39 q^{-14} -16 q^{-15} +24 q^{-16} +3 q^{-17} +14 q^{-18} +5 q^{-19} +3 q^{-20} -11 q^{-21} -8 q^{-22} +5 q^{-23} -2 q^{-24} +2 q^{-25} + q^{-26} +2 q^{-27} - q^{-28} -2 q^{-29} + q^{-30} }
7	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^{210}+2 q^{209}-q^{208}+2 q^{206}-3 q^{205}+5 q^{201}-7 q^{200}+10 q^{198}-7 q^{197}-2 q^{195}+10 q^{193}-23 q^{192}-2 q^{191}+30 q^{190}+2 q^{189}+8 q^{188}-13 q^{187}-16 q^{186}-2 q^{185}-59 q^{184}-4 q^{183}+80 q^{182}+66 q^{181}+72 q^{180}-25 q^{179}-111 q^{178}-128 q^{177}-195 q^{176}-22 q^{175}+234 q^{174}+355 q^{173}+396 q^{172}+55 q^{171}-391 q^{170}-691 q^{169}-824 q^{168}-287 q^{167}+598 q^{166}+1339 q^{165}+1627 q^{164}+762 q^{163}-796 q^{162}-2263 q^{161}-2972 q^{160}-1774 q^{159}+793 q^{158}+3493 q^{157}+5011 q^{156}+3568 q^{155}-291 q^{154}-4852 q^{153}-7804 q^{152}-6380 q^{151}-985 q^{150}+5979 q^{149}+11075 q^{148}+10283 q^{147}+3487 q^{146}-6418 q^{145}-14563 q^{144}-15080 q^{143}-7200 q^{142}+5766 q^{141}+17506 q^{140}+20258 q^{139}+12093 q^{138}-3700 q^{137}-19423 q^{136}-25223 q^{135}-17618 q^{134}+357 q^{133}+19939 q^{132}+29168 q^{131}+23038 q^{130}+3952 q^{129}-18853 q^{128}-31698 q^{127}-27838 q^{126}-8519 q^{125}+16685 q^{124}+32629 q^{123}+31204 q^{122}+12699 q^{121}-13680 q^{120}-32121 q^{119}-33229 q^{118}-16054 q^{117}+10735 q^{116}+30705 q^{115}+33766 q^{114}+18191 q^{113}-8068 q^{112}-28724 q^{111}-33353 q^{110}-19364 q^{109}+6072 q^{108}+26814 q^{107}+32294 q^{106}+19632 q^{105}-4667 q^{104}-24989 q^{103}-31027 q^{102}-19553 q^{101}+3673 q^{100}+23484 q^{99}+29786 q^{98}+19309 q^{97}-2816 q^{96}-22038 q^{95}-28674 q^{94}-19251 q^{93}+1810 q^{92}+20571 q^{91}+27645 q^{90}+19425 q^{89}-448 q^{88}-18811 q^{87}-26599 q^{86}-19868 q^{85}-1347 q^{84}+16644 q^{83}+25314 q^{82}+20431 q^{81}+3602 q^{80}-13898 q^{79}-23673 q^{78}-20966 q^{77}-6153 q^{76}+10627 q^{75}+21432 q^{74}+21134 q^{73}+8861 q^{72}-6795 q^{71}-18529 q^{70}-20805 q^{69}-11345 q^{68}+2718 q^{67}+14866 q^{66}+19571 q^{65}+13324 q^{64}+1475 q^{63}-10618 q^{62}-17429 q^{61}-14381 q^{60}-5224 q^{59}+5978 q^{58}+14230 q^{57}+14294 q^{56}+8187 q^{55}-1397 q^{54}-10277 q^{53}-12873 q^{52}-9924 q^{51}-2639 q^{50}+5923 q^{49}+10292 q^{48}+10201 q^{47}+5568 q^{46}-1731 q^{45}-6854 q^{44}-9031 q^{43}-7126 q^{42}-1705 q^{41}+3248 q^{40}+6739 q^{39}+7095 q^{38}+3871 q^{37}+47 q^{36}-3839 q^{35}-5856 q^{34}-4688 q^{33}-2293 q^{32}+1104 q^{31}+3757 q^{30}+4129 q^{29}+3389 q^{28}+1072 q^{27}-1580 q^{26}-2836 q^{25}-3283 q^{24}-2149 q^{23}-197 q^{22}+1173 q^{21}+2391 q^{20}+2332 q^{19}+1230 q^{18}+150 q^{17}-1205 q^{16}-1771 q^{15}-1450 q^{14}-957 q^{13}+153 q^{12}+942 q^{11}+1134 q^{10}+1164 q^9+478 q^8-209 q^7-592 q^6-925 q^5-656 q^4-253 q^3+75 q^2+539 q+567+390 q^{-1} +192 q^{-2} -194 q^{-3} -300 q^{-4} -314 q^{-5} -301 q^{-6} -38 q^{-7} +117 q^{-8} +195 q^{-9} +232 q^{-10} +85 q^{-11} +17 q^{-12} -43 q^{-13} -148 q^{-14} -102 q^{-15} -57 q^{-16} +4 q^{-17} +74 q^{-18} +41 q^{-19} +40 q^{-20} +37 q^{-21} -15 q^{-22} -27 q^{-23} -34 q^{-24} -22 q^{-25} +13 q^{-26} + q^{-27} +5 q^{-28} +16 q^{-29} +5 q^{-30} +2 q^{-31} -8 q^{-32} -8 q^{-33} +3 q^{-34} -2 q^{-36} +2 q^{-37} + q^{-38} +2 q^{-39} - q^{-40} -2 q^{-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.

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9_35

9_37

9 36

Contents

Knot presentations

Three dimensional invariants

Four dimensional invariants

Polynomial invariants

"Similar" Knots (within the Atlas)

Vassiliev invariants

Khovanov Homology

The Coloured Jones Polynomials

Computer Talk

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