10 76: Difference between revisions

Revision as of 21:08, 29 August 2005

10_75

10_77

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10 76 Quick Notes

10 76 Further Notes and Views

Knot presentations

Planar diagram presentation	X₄₂₅₁ X_14,10,15,9 X_12,3,13,4 X_2,13,3,14 X_18,6,19,5 X_20,8,1,7 X_6,20,7,19 X_8,18,9,17 X_16,12,17,11 X_10,16,11,15
Gauss code	1, -4, 3, -1, 5, -7, 6, -8, 2, -10, 9, -3, 4, -2, 10, -9, 8, -5, 7, -6
Dowker-Thistlethwaite code	4 12 18 20 14 16 2 10 8 6
Conway Notation	[3,3,2++]

Minimum Braid Representative:

Length is 11, width is 4.

Braid index is 4.

A Morse Link Presentation:

Three dimensional invariants

Symmetry type	Reversible
Unknotting number	$\{2,3\}$
3-genus	3
Bridge index	3
Super bridge index	Missing
Nakanishi index	1
Maximal Thurston-Bennequin number	[1][-13]
Hyperbolic Volume	11.5129
A-Polynomial	See Data:10 76/A-polynomial

[edit Notes for 10 76'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 10 76's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-2t^{3}+7t^{2}-12t+15-12t^{-1}+7t^{-2}-2t^{-3}$
Conway polynomial	$-2z^{6}-5z^{4}-2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 57, 4 }
Jones polynomial	$q^{10}-3q^{9}+6q^{8}-8q^{7}+9q^{6}-10q^{5}+8q^{4}-6q^{3}+4q^{2}-q+1$
HOMFLY-PT polynomial (db, data sources)	$-z^{6}a^{-4}-z^{6}a^{-6}+z^{4}a^{-2}-4z^{4}a^{-4}-3z^{4}a^{-6}+z^{4}a^{-8}+4z^{2}a^{-2}-6z^{2}a^{-4}-2z^{2}a^{-6}+2z^{2}a^{-8}+4a^{-2}-4a^{-4}+a^{-8}$
Kauffman polynomial (db, data sources)	$z^{9}a^{-5}+z^{9}a^{-7}+z^{8}a^{-4}+4z^{8}a^{-6}+3z^{8}a^{-8}+z^{7}a^{-3}-2z^{7}a^{-5}+2z^{7}a^{-7}+5z^{7}a^{-9}+z^{6}a^{-2}-9z^{6}a^{-6}-3z^{6}a^{-8}+5z^{6}a^{-10}-2z^{5}a^{-3}+7z^{5}a^{-5}-2z^{5}a^{-7}-8z^{5}a^{-9}+3z^{5}a^{-11}-5z^{4}a^{-2}-7z^{4}a^{-4}+10z^{4}a^{-6}+4z^{4}a^{-8}-7z^{4}a^{-10}+z^{4}a^{-12}-2z^{3}a^{-3}-15z^{3}a^{-5}-3z^{3}a^{-7}+7z^{3}a^{-9}-3z^{3}a^{-11}+8z^{2}a^{-2}+9z^{2}a^{-4}-7z^{2}a^{-6}-4z^{2}a^{-8}+3z^{2}a^{-10}-z^{2}a^{-12}+4za^{-3}+8za^{-5}+2za^{-7}-2za^{-9}-4a^{-2}-4a^{-4}+a^{-8}$
The A2 invariant	$1+q^{-2}+2q^{-4}+3q^{-6}-q^{-8}+q^{-10}-3q^{-12}-2q^{-14}-2q^{-18}+2q^{-20}-q^{-22}+q^{-24}+q^{-26}-q^{-28}+q^{-30}$
The G2 invariant	$q^{-2}+3q^{-6}-2q^{-8}+3q^{-10}-q^{-12}+7q^{-16}-9q^{-18}+14q^{-20}-12q^{-22}+12q^{-24}+q^{-26}-12q^{-28}+31q^{-30}-40q^{-32}+46q^{-34}-33q^{-36}+q^{-38}+33q^{-40}-65q^{-42}+80q^{-44}-67q^{-46}+28q^{-48}+17q^{-50}-59q^{-52}+71q^{-54}-63q^{-56}+24q^{-58}+17q^{-60}-50q^{-62}+46q^{-64}-24q^{-66}-19q^{-68}+58q^{-70}-75q^{-72}+60q^{-74}-21q^{-76}-35q^{-78}+87q^{-80}-115q^{-82}+106q^{-84}-59q^{-86}-3q^{-88}+66q^{-90}-101q^{-92}+104q^{-94}-68q^{-96}+17q^{-98}+32q^{-100}-60q^{-102}+55q^{-104}-22q^{-106}-19q^{-108}+51q^{-110}-52q^{-112}+28q^{-114}+11q^{-116}-53q^{-118}+76q^{-120}-74q^{-122}+48q^{-124}-8q^{-126}-33q^{-128}+60q^{-130}-64q^{-132}+54q^{-134}-28q^{-136}+3q^{-138}+15q^{-140}-29q^{-142}+28q^{-144}-21q^{-146}+12q^{-148}-2q^{-150}-3q^{-152}+5q^{-154}-6q^{-156}+4q^{-158}-2q^{-160}+q^{-162}$

Further Quantum Invariants

Further quantum knot invariants for 10_76.

A1 Invariants.

Weight	Invariant
1	$q+3q^{-3}-2q^{-5}+2q^{-7}-2q^{-9}-q^{-11}+q^{-13}-2q^{-15}+3q^{-17}-2q^{-19}+q^{-21}$
2	$q^{6}+3-q^{-2}-3q^{-4}+8q^{-6}-2q^{-8}-11q^{-10}+12q^{-12}+3q^{-14}-18q^{-16}+9q^{-18}+9q^{-20}-14q^{-22}+2q^{-24}+10q^{-26}-3q^{-28}-7q^{-30}+4q^{-32}+10q^{-34}-12q^{-36}-4q^{-38}+18q^{-40}-11q^{-42}-9q^{-44}+15q^{-46}-4q^{-48}-7q^{-50}+6q^{-52}-2q^{-56}+q^{-58}$
3	$q^{15}+2q^{7}-q^{5}-q^{3}+q+6q^{-1}-3q^{-3}-10q^{-5}+21q^{-9}+3q^{-11}-29q^{-13}-18q^{-15}+36q^{-17}+37q^{-19}-36q^{-21}-54q^{-23}+25q^{-25}+73q^{-27}-9q^{-29}-78q^{-31}-12q^{-33}+79q^{-35}+25q^{-37}-67q^{-39}-43q^{-41}+58q^{-43}+47q^{-45}-37q^{-47}-50q^{-49}+17q^{-51}+50q^{-53}+4q^{-55}-46q^{-57}-31q^{-59}+38q^{-61}+52q^{-63}-23q^{-65}-74q^{-67}+10q^{-69}+83q^{-71}+9q^{-73}-82q^{-75}-22q^{-77}+70q^{-79}+33q^{-81}-53q^{-83}-34q^{-85}+34q^{-87}+26q^{-89}-17q^{-91}-20q^{-93}+9q^{-95}+13q^{-97}-5q^{-99}-6q^{-101}+q^{-103}+3q^{-105}-2q^{-109}+q^{-111}$
4	$q^{28}-q^{20}+2q^{18}-q^{16}+3q^{12}-2q^{10}+3q^{8}-6q^{6}-5q^{4}+8q^{2}+6+15q^{-2}-17q^{-4}-32q^{-6}-5q^{-8}+22q^{-10}+71q^{-12}+3q^{-14}-76q^{-16}-79q^{-18}-15q^{-20}+156q^{-22}+118q^{-24}-45q^{-26}-187q^{-28}-182q^{-30}+141q^{-32}+267q^{-34}+135q^{-36}-175q^{-38}-382q^{-40}-48q^{-42}+278q^{-44}+354q^{-46}+10q^{-48}-433q^{-50}-266q^{-52}+125q^{-54}+433q^{-56}+213q^{-58}-319q^{-60}-360q^{-62}-48q^{-64}+365q^{-66}+301q^{-68}-162q^{-70}-333q^{-72}-148q^{-74}+237q^{-76}+295q^{-78}-9q^{-80}-255q^{-82}-214q^{-84}+84q^{-86}+257q^{-88}+167q^{-90}-131q^{-92}-277q^{-94}-132q^{-96}+171q^{-98}+360q^{-100}+64q^{-102}-282q^{-104}-362q^{-106}-6q^{-108}+447q^{-110}+278q^{-112}-145q^{-114}-456q^{-116}-212q^{-118}+341q^{-120}+350q^{-122}+53q^{-124}-333q^{-126}-287q^{-128}+136q^{-130}+234q^{-132}+143q^{-134}-135q^{-136}-194q^{-138}+18q^{-140}+76q^{-142}+98q^{-144}-26q^{-146}-78q^{-148}+q^{-150}+8q^{-152}+39q^{-154}-5q^{-156}-25q^{-158}+5q^{-160}-3q^{-162}+12q^{-164}-7q^{-168}+2q^{-170}-2q^{-172}+3q^{-174}-2q^{-178}+q^{-180}$
5	$q^{45}-q^{37}-q^{35}+2q^{33}+3q^{27}-5q^{23}-2q^{19}-q^{17}+10q^{15}+10q^{13}-4q^{11}-9q^{9}-20q^{7}-18q^{5}+15q^{3}+43q+39q^{-1}+6q^{-3}-60q^{-5}-103q^{-7}-50q^{-9}+64q^{-11}+165q^{-13}+168q^{-15}-4q^{-17}-237q^{-19}-315q^{-21}-143q^{-23}+213q^{-25}+505q^{-27}+420q^{-29}-80q^{-31}-615q^{-33}-751q^{-35}-264q^{-37}+569q^{-39}+1085q^{-41}+751q^{-43}-298q^{-45}-1258q^{-47}-1298q^{-49}-243q^{-51}+1179q^{-53}+1780q^{-55}+920q^{-57}-819q^{-59}-2018q^{-61}-1629q^{-63}+196q^{-65}+1996q^{-67}+2188q^{-69}+519q^{-71}-1668q^{-73}-2508q^{-75}-1203q^{-77}+1184q^{-79}+2542q^{-81}+1713q^{-83}-602q^{-85}-2384q^{-87}-2002q^{-89}+130q^{-91}+2039q^{-93}+2073q^{-95}+269q^{-97}-1690q^{-99}-2005q^{-101}-479q^{-103}+1337q^{-105}+1828q^{-107}+644q^{-109}-1045q^{-111}-1666q^{-113}-744q^{-115}+779q^{-117}+1507q^{-119}+879q^{-121}-482q^{-123}-1387q^{-125}-1092q^{-127}+127q^{-129}+1251q^{-131}+1354q^{-133}+348q^{-135}-1024q^{-137}-1665q^{-139}-924q^{-141}+686q^{-143}+1876q^{-145}+1573q^{-147}-168q^{-149}-1950q^{-151}-2164q^{-153}-479q^{-155}+1770q^{-157}+2602q^{-159}+1171q^{-161}-1359q^{-163}-2742q^{-165}-1762q^{-167}+739q^{-169}+2579q^{-171}+2132q^{-173}-104q^{-175}-2104q^{-177}-2191q^{-179}-454q^{-181}+1477q^{-183}+1971q^{-185}+780q^{-187}-854q^{-189}-1530q^{-191}-867q^{-193}+345q^{-195}+1034q^{-197}+772q^{-199}-47q^{-201}-607q^{-203}-545q^{-205}-93q^{-207}+295q^{-209}+339q^{-211}+114q^{-213}-131q^{-215}-179q^{-217}-69q^{-219}+45q^{-221}+78q^{-223}+38q^{-225}-13q^{-227}-40q^{-229}-14q^{-231}+13q^{-233}+13q^{-235}+3q^{-237}-4q^{-239}-4q^{-241}-7q^{-243}+3q^{-245}+7q^{-247}-q^{-249}-2q^{-251}+q^{-253}-q^{-255}-2q^{-257}+3q^{-259}-2q^{-263}+q^{-265}$

A2 Invariants.

Weight	Invariant
1,0	$1+q^{-2}+2q^{-4}+3q^{-6}-q^{-8}+q^{-10}-3q^{-12}-2q^{-14}-2q^{-18}+2q^{-20}-q^{-22}+q^{-24}+q^{-26}-q^{-28}+q^{-30}$
1,1	$q^{4}+6-4q^{-2}+18q^{-4}-20q^{-6}+42q^{-8}-62q^{-10}+93q^{-12}-140q^{-14}+182q^{-16}-240q^{-18}+272q^{-20}-302q^{-22}+288q^{-24}-244q^{-26}+164q^{-28}-46q^{-30}-92q^{-32}+246q^{-34}-378q^{-36}+502q^{-38}-576q^{-40}+612q^{-42}-587q^{-44}+512q^{-46}-402q^{-48}+254q^{-50}-107q^{-52}-40q^{-54}+160q^{-56}-244q^{-58}+292q^{-60}-300q^{-62}+282q^{-64}-246q^{-66}+195q^{-68}-144q^{-70}+102q^{-72}-66q^{-74}+38q^{-76}-20q^{-78}+10q^{-80}-4q^{-82}+q^{-84}$
2,0	$q^{4}+q^{2}+1+2q^{-2}+5q^{-4}+3q^{-6}+q^{-10}+5q^{-12}-4q^{-14}-10q^{-16}-2q^{-18}-9q^{-22}-6q^{-24}+6q^{-26}+5q^{-28}-2q^{-30}+4q^{-32}+9q^{-34}-3q^{-36}-q^{-38}+7q^{-40}-6q^{-44}+2q^{-46}+4q^{-48}-6q^{-50}-5q^{-52}+5q^{-54}+3q^{-56}-6q^{-58}-q^{-60}+4q^{-62}-q^{-64}-2q^{-66}+2q^{-70}-q^{-74}+q^{-76}$

A3 Invariants.

Weight	Invariant
0,1,0	$1+3q^{-4}+4q^{-6}+2q^{-8}+6q^{-10}+7q^{-12}-7q^{-14}+2q^{-16}-20q^{-20}+4q^{-24}-14q^{-26}+5q^{-28}+12q^{-30}-4q^{-32}+q^{-34}+5q^{-36}+3q^{-38}-5q^{-40}-3q^{-42}+11q^{-44}-5q^{-46}-9q^{-48}+14q^{-50}-4q^{-52}-11q^{-54}+11q^{-56}-q^{-58}-6q^{-60}+5q^{-62}-2q^{-66}+q^{-68}$
1,0,0	$q^{-1}+q^{-3}+3q^{-5}+2q^{-7}+4q^{-9}-q^{-11}+q^{-13}-4q^{-15}-2q^{-17}-3q^{-19}-q^{-21}-q^{-25}+2q^{-27}-q^{-29}+2q^{-31}-q^{-33}+2q^{-35}-q^{-37}+q^{-39}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{-2}+q^{-4}+3q^{-6}+5q^{-8}+7q^{-10}+7q^{-12}+9q^{-14}+8q^{-16}+4q^{-18}-4q^{-20}-4q^{-22}-10q^{-24}-21q^{-26}-16q^{-28}-4q^{-30}-10q^{-32}-9q^{-34}+12q^{-36}+15q^{-38}+3q^{-40}+4q^{-42}+17q^{-44}+3q^{-46}-9q^{-48}+4q^{-50}+6q^{-52}-11q^{-54}-2q^{-56}+10q^{-58}-5q^{-60}-8q^{-62}+5q^{-64}+4q^{-66}-9q^{-68}-3q^{-70}+8q^{-72}+q^{-74}-6q^{-76}+2q^{-78}+4q^{-80}-2q^{-82}-q^{-84}+q^{-86}$
1,0,0,0	$q^{-2}+q^{-4}+3q^{-6}+3q^{-8}+3q^{-10}+4q^{-12}-q^{-14}+q^{-16}-4q^{-18}-3q^{-20}-3q^{-22}-3q^{-24}-q^{-26}-q^{-28}+q^{-30}-q^{-32}+2q^{-34}-q^{-36}+2q^{-38}+2q^{-44}-q^{-46}+q^{-48}$

B2 Invariants.

Weight

Invariant

0,1

1+3q^{-4}-2q^{-6}+6q^{-8}-6q^{-10}+11q^{-12}-11q^{-14}+14q^{-16}-14q^{-18}+12q^{-20}-10q^{-22}+2q^{-24}+2q^{-26}-11q^{-28}+16q^{-30}-24q^{-32}+27q^{-34}-29q^{-36}+27q^{-38}-23q^{-40}+17q^{-42}-9q^{-44}+3q^{-46}+5q^{-48}-10q^{-50}+14q^{-52}-15q^{-54}+15q^{-56}-13q^{-58}+10q^{-60}-7q^{-62}+4q^{-64}-2q^{-66}+q^{-68}

1,0

q^{2}+3q^{-6}+3q^{-8}+q^{-10}-2q^{-12}+q^{-14}+6q^{-16}+8q^{-18}-3q^{-20}-10q^{-22}-3q^{-24}+11q^{-26}+7q^{-28}-12q^{-30}-18q^{-32}-2q^{-34}+14q^{-36}+4q^{-38}-13q^{-40}-11q^{-42}+7q^{-44}+12q^{-46}-9q^{-50}+2q^{-52}+11q^{-54}+3q^{-56}-9q^{-58}-3q^{-60}+9q^{-62}+7q^{-64}-8q^{-66}-10q^{-68}+6q^{-70}+12q^{-72}-2q^{-74}-15q^{-76}-4q^{-78}+14q^{-80}+11q^{-82}-9q^{-84}-15q^{-86}+13q^{-90}+6q^{-92}-7q^{-94}-8q^{-96}+q^{-98}+6q^{-100}+2q^{-102}-2q^{-104}-2q^{-106}+q^{-110}

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{-2}+3q^{-6}+q^{-8}+7q^{-10}+q^{-12}+10q^{-14}-q^{-16}+12q^{-18}-9q^{-20}+8q^{-22}-15q^{-24}+5q^{-26}-18q^{-28}-11q^{-32}-6q^{-38}+12q^{-40}-8q^{-42}+20q^{-44}-19q^{-46}+22q^{-48}-20q^{-50}+23q^{-52}-20q^{-54}+16q^{-56}-14q^{-58}+13q^{-60}-4q^{-62}+q^{-64}-5q^{-68}+10q^{-70}-11q^{-72}+9q^{-74}-13q^{-76}+13q^{-78}-9q^{-80}+8q^{-82}-8q^{-84}+6q^{-86}-3q^{-88}+2q^{-90}-2q^{-92}+q^{-94}$

G2 Invariants.

Weight

Invariant

1,0

q^{-2}+3q^{-6}-2q^{-8}+3q^{-10}-q^{-12}+7q^{-16}-9q^{-18}+14q^{-20}-12q^{-22}+12q^{-24}+q^{-26}-12q^{-28}+31q^{-30}-40q^{-32}+46q^{-34}-33q^{-36}+q^{-38}+33q^{-40}-65q^{-42}+80q^{-44}-67q^{-46}+28q^{-48}+17q^{-50}-59q^{-52}+71q^{-54}-63q^{-56}+24q^{-58}+17q^{-60}-50q^{-62}+46q^{-64}-24q^{-66}-19q^{-68}+58q^{-70}-75q^{-72}+60q^{-74}-21q^{-76}-35q^{-78}+87q^{-80}-115q^{-82}+106q^{-84}-59q^{-86}-3q^{-88}+66q^{-90}-101q^{-92}+104q^{-94}-68q^{-96}+17q^{-98}+32q^{-100}-60q^{-102}+55q^{-104}-22q^{-106}-19q^{-108}+51q^{-110}-52q^{-112}+28q^{-114}+11q^{-116}-53q^{-118}+76q^{-120}-74q^{-122}+48q^{-124}-8q^{-126}-33q^{-128}+60q^{-130}-64q^{-132}+54q^{-134}-28q^{-136}+3q^{-138}+15q^{-140}-29q^{-142}+28q^{-144}-21q^{-146}+12q^{-148}-2q^{-150}-3q^{-152}+5q^{-154}-6q^{-156}+4q^{-158}-2q^{-160}+q^{-162}

.

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["10 76"];

In[4]:= Alexander[K][t]

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

Out[4]= $-2t^{3}+7t^{2}-12t+15-12t^{-1}+7t^{-2}-2t^{-3}$

In[5]:= Conway[K][z]

Out[5]= $-2z^{6}-5z^{4}-2z^{2}+1$

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]= { 57, 4 }

In[8]:= Jones[K][q]

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

Out[8]= $q^{10}-3q^{9}+6q^{8}-8q^{7}+9q^{6}-10q^{5}+8q^{4}-6q^{3}+4q^{2}-q+1$

In[9]:= HOMFLYPT[K][a, z]

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

Out[9]= $-z^{6}a^{-4}-z^{6}a^{-6}+z^{4}a^{-2}-4z^{4}a^{-4}-3z^{4}a^{-6}+z^{4}a^{-8}+4z^{2}a^{-2}-6z^{2}a^{-4}-2z^{2}a^{-6}+2z^{2}a^{-8}+4a^{-2}-4a^{-4}+a^{-8}$

In[10]:= Kauffman[K][a, z]

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

Out[10]= $z^{9}a^{-5}+z^{9}a^{-7}+z^{8}a^{-4}+4z^{8}a^{-6}+3z^{8}a^{-8}+z^{7}a^{-3}-2z^{7}a^{-5}+2z^{7}a^{-7}+5z^{7}a^{-9}+z^{6}a^{-2}-9z^{6}a^{-6}-3z^{6}a^{-8}+5z^{6}a^{-10}-2z^{5}a^{-3}+7z^{5}a^{-5}-2z^{5}a^{-7}-8z^{5}a^{-9}+3z^{5}a^{-11}-5z^{4}a^{-2}-7z^{4}a^{-4}+10z^{4}a^{-6}+4z^{4}a^{-8}-7z^{4}a^{-10}+z^{4}a^{-12}-2z^{3}a^{-3}-15z^{3}a^{-5}-3z^{3}a^{-7}+7z^{3}a^{-9}-3z^{3}a^{-11}+8z^{2}a^{-2}+9z^{2}a^{-4}-7z^{2}a^{-6}-4z^{2}a^{-8}+3z^{2}a^{-10}-z^{2}a^{-12}+4za^{-3}+8za^{-5}+2za^{-7}-2za^{-9}-4a^{-2}-4a^{-4}+a^{-8}$

"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}} ): {...}

Vassiliev invariants

V₂ and V₃:

(-2, -6)

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 -8}

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 -48}

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 32}

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{124}{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 \frac{172}{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 384}

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}

256

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 272}

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{256}{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 1152}

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{992}{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 -\frac{1376}{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 \frac{56369}{15}}

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{2468}{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{53396}{45}}

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{3535}{9}}

-{\frac {1711}{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 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 10 76. 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

8

χ

21

1

19

2

-2

17

4

1

3

15

4

2

-2

13

5

4

1

11

5

4

-1

9

3

5

-2

7

3

5

2

5

1

3

-2

3

1

4

3

1

0

-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}}
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}_2}	${\mathbb {Z} }$
$r=0$	${\mathbb {Z} }^{4}$	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}}
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^{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=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}^{5}\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}^{5}\oplus{\mathbb Z}_2^{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}^{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=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}^{4}\oplus{\mathbb Z}_2^{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}^{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=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}^{4}\oplus{\mathbb Z}_2^{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}^{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 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}^{2}\oplus{\mathbb Z}_2^{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}^{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 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}\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=8}	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^{28}-3 q^{27}+2 q^{26}+7 q^{25}-16 q^{24}+5 q^{23}+26 q^{22}-40 q^{21}+3 q^{20}+55 q^{19}-62 q^{18}-5 q^{17}+77 q^{16}-68 q^{15}-16 q^{14}+81 q^{13}-55 q^{12}-24 q^{11}+65 q^{10}-32 q^9-24 q^8+38 q^7-11 q^6-15 q^5+15 q^4-2 q^3-5 q^2+4 q- 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^{54}-3 q^{53}+2 q^{52}+3 q^{51}-q^{50}-10 q^{49}+3 q^{48}+21 q^{47}-5 q^{46}-39 q^{45}+6 q^{44}+64 q^{43}+3 q^{42}-107 q^{41}-13 q^{40}+150 q^{39}+40 q^{38}-199 q^{37}-73 q^{36}+241 q^{35}+114 q^{34}-272 q^{33}-157 q^{32}+292 q^{31}+189 q^{30}-286 q^{29}-226 q^{28}+277 q^{27}+239 q^{26}-240 q^{25}-259 q^{24}+210 q^{23}+252 q^{22}-156 q^{21}-248 q^{20}+109 q^{19}+228 q^{18}-64 q^{17}-194 q^{16}+18 q^{15}+162 q^{14}+5 q^{13}-112 q^{12}-30 q^{11}+83 q^{10}+23 q^9-39 q^8-31 q^7+29 q^6+12 q^5-7 q^4-13 q^3+8 q^2+2 q-4 q^{-1} +3 q^{-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^{88}-3 q^{87}+2 q^{86}+3 q^{85}-5 q^{84}+5 q^{83}-12 q^{82}+9 q^{81}+15 q^{80}-20 q^{79}+13 q^{78}-42 q^{77}+29 q^{76}+59 q^{75}-51 q^{74}+6 q^{73}-121 q^{72}+81 q^{71}+183 q^{70}-73 q^{69}-52 q^{68}-333 q^{67}+140 q^{66}+461 q^{65}+18 q^{64}-150 q^{63}-756 q^{62}+94 q^{61}+847 q^{60}+315 q^{59}-159 q^{58}-1309 q^{57}-150 q^{56}+1158 q^{55}+738 q^{54}+10 q^{53}-1762 q^{52}-506 q^{51}+1238 q^{50}+1084 q^{49}+306 q^{48}-1951 q^{47}-809 q^{46}+1093 q^{45}+1230 q^{44}+604 q^{43}-1861 q^{42}-982 q^{41}+795 q^{40}+1189 q^{39}+850 q^{38}-1557 q^{37}-1040 q^{36}+410 q^{35}+1004 q^{34}+1021 q^{33}-1094 q^{32}-976 q^{31}-3 q^{30}+692 q^{29}+1062 q^{28}-562 q^{27}-756 q^{26}-311 q^{25}+301 q^{24}+895 q^{23}-119 q^{22}-412 q^{21}-387 q^{20}-25 q^{19}+561 q^{18}+88 q^{17}-102 q^{16}-255 q^{15}-151 q^{14}+238 q^{13}+83 q^{12}+40 q^{11}-92 q^{10}-113 q^9+67 q^8+19 q^7+43 q^6-13 q^5-45 q^4+18 q^3-8 q^2+16 q+2-13 q^{-1} +9 q^{-2} -6 q^{-3} +3 q^{-4} + q^{-5} -4 q^{-6} +4 q^{-7} - q^{-8} - q^{-11} + q^{-12} }
5	Not Available
6	Not Available
7	Not Available

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[10, 76]]
Out[2]=	PD[X[4, 2, 5, 1], X[14, 10, 15, 9], X[12, 3, 13, 4], X[2, 13, 3, 14], X[18, 6, 19, 5], X[20, 8, 1, 7], X[6, 20, 7, 19], X[8, 18, 9, 17], X[16, 12, 17, 11], X[10, 16, 11, 15]]
In[3]:=	GaussCode[Knot[10, 76]]
Out[3]=	GaussCode[1, -4, 3, -1, 5, -7, 6, -8, 2, -10, 9, -3, 4, -2, 10, -9, 8, -5, 7, -6]
In[4]:=	DTCode[Knot[10, 76]]
Out[4]=	DTCode[4, 12, 18, 20, 14, 16, 2, 10, 8, 6]
In[5]:=	br = BR[Knot[10, 76]]
Out[5]=	BR[4, {1, 1, 1, 1, 2, -1, -3, 2, 2, 2, -3}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{4, 11}
In[7]:=	BraidIndex[Knot[10, 76]]
Out[7]=	4
In[8]:=	Show[DrawMorseLink[Knot[10, 76]]]

`Out[8]=`	`-Graphics-`
In[9]:=	(#[Knot[10, 76]]&) /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, {2, 3}, 3, 3, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 76]][t]
Out[10]=	2 7 12 2 3 15 - -- + -- - -- - 12 t + 7 t - 2 t 3 2 t t t
In[11]:=	Conway[Knot[10, 76]][z]
Out[11]=	2 4 6 1 - 2 z - 5 z - 2 z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 76]}
In[13]:=	{KnotDet[Knot[10, 76]], KnotSignature[Knot[10, 76]]}
Out[13]=	{57, 4}
In[14]:=	Jones[Knot[10, 76]][q]
Out[14]=	2 3 4 5 6 7 8 9 10 1 - q + 4 q - 6 q + 8 q - 10 q + 9 q - 8 q + 6 q - 3 q + q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] \|\| (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[10, 76]}
In[16]:=	A2Invariant[Knot[10, 76]][q]
Out[16]=	2 4 6 8 10 12 14 18 20 22 1 + q + 2 q + 3 q - q + q - 3 q - 2 q - 2 q + 2 q - q + 24 26 28 30 q + q - q + q
In[17]:=	HOMFLYPT[Knot[10, 76]][a, z]
Out[17]=	2 2 2 2 4 4 4 4 -8 4 4 2 z 2 z 6 z 4 z z 3 z 4 z z a - -- + -- + ---- - ---- - ---- + ---- + -- - ---- - ---- + -- - 4 2 8 6 4 2 8 6 4 2 a a a a a a a a a a 6 6 z z -- - -- 6 4 a a
In[18]:=	Kauffman[Knot[10, 76]][a, z]
Out[18]=	2 2 2 2 -8 4 4 2 z 2 z 8 z 4 z z 3 z 4 z 7 z a - -- - -- - --- + --- + --- + --- - --- + ---- - ---- - ---- + 4 2 9 7 5 3 12 10 8 6 a a a a a a a a a a 2 2 3 3 3 3 3 4 4 4 9 z 8 z 3 z 7 z 3 z 15 z 2 z z 7 z 4 z ---- + ---- - ---- + ---- - ---- - ----- - ---- + --- - ---- + ---- + 4 2 11 9 7 5 3 12 10 8 a a a a a a a a a a 4 4 4 5 5 5 5 5 6 10 z 7 z 5 z 3 z 8 z 2 z 7 z 2 z 5 z ----- - ---- - ---- + ---- - ---- - ---- + ---- - ---- + ---- - 6 4 2 11 9 7 5 3 10 a a a a a a a a a 6 6 6 7 7 7 7 8 8 8 9 3 z 9 z z 5 z 2 z 2 z z 3 z 4 z z z ---- - ---- + -- + ---- + ---- - ---- + -- + ---- + ---- + -- + -- + 8 6 2 9 7 5 3 8 6 4 7 a a a a a a a a a a a 9 z -- 5 a
In[19]:=	{Vassiliev[2][Knot[10, 76]], Vassiliev[3][Knot[10, 76]]}
Out[19]=	{-2, -6}
In[20]:=	Kh[Knot[10, 76]][q, t]
Out[20]=	3 3 5 1 q 5 7 7 2 9 2 9 3 4 q + q + ---- + -- + 3 q t + 3 q t + 5 q t + 3 q t + 5 q t + 2 t q t 11 3 11 4 13 4 13 5 15 5 15 6 5 q t + 4 q t + 5 q t + 4 q t + 4 q t + 2 q t + 17 6 17 7 19 7 21 8 4 q t + q t + 2 q t + q t
In[21]:=	ColouredJones[Knot[10, 76], 2][q]
Out[21]=	-2 1 2 3 4 5 6 7 8 q - - + 4 q - 5 q - 2 q + 15 q - 15 q - 11 q + 38 q - 24 q - q 9 10 11 12 13 14 15 32 q + 65 q - 24 q - 55 q + 81 q - 16 q - 68 q + 16 17 18 19 20 21 22 23 77 q - 5 q - 62 q + 55 q + 3 q - 40 q + 26 q + 5 q - 24 25 26 27 28 16 q + 7 q + 2 q - 3 q + q

See/edit the Rolfsen_Splice_Template.

Back to the top.

10_75

10_77

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+<!-- This page was generated from the splice template "Rolfsen_Splice_Template". Please do not edit! -->
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 </table>
+{| width=100%
-See/edit the [[Rolfsen_Splice_Template]].
+|align=left|See/edit the [[Rolfsen_Splice_Template]].
+Back to the [[#top|top]].
+|align=right|{{Knot Navigation Links|ext=gif}}
+|}
  [[Category:Knot Page]]

10 76: Difference between revisions

Revision as of 21:08, 29 August 2005

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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