9 10: Difference between revisions

Revision as of 19:16, 28 August 2005

9_9

9_11

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9 10 Quick Notes

9 10 Further Notes and Views

Knot presentations

Planar diagram presentation	X₈₂₉₁ X_12,4,13,3 X_18,10,1,9 X_10,18,11,17 X_16,8,17,7 X_2,12,3,11 X_4,16,5,15 X_14,6,15,5 X_6,14,7,13
Gauss code	1, -6, 2, -7, 8, -9, 5, -1, 3, -4, 6, -2, 9, -8, 7, -5, 4, -3
Dowker-Thistlethwaite code	8 12 14 16 18 2 6 4 10
Conway Notation	[333]

Three dimensional invariants

Symmetry type	Reversible
Unknotting number	$\{2,3\}$
3-genus	2
Bridge index	2
Super bridge index	$\{4,6\}$
Nakanishi index	1
Maximal Thurston-Bennequin number	[3][-14]
Hyperbolic Volume	8.77346
A-Polynomial	See Data:9 10/A-polynomial

[edit Notes for 9 10's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for 9 10's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$4t^{2}-8t+9-8t^{-1}+4t^{-2}$
Conway polynomial	$4z^{4}+8z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 33, 4 }
Jones polynomial	$-q^{11}+q^{10}-3q^{9}+5q^{8}-5q^{7}+6q^{6}-5q^{5}+4q^{4}-2q^{3}+q^{2}$
HOMFLY-PT polynomial (db, data sources)	$z^{4}a^{-4}+2z^{4}a^{-6}+z^{4}a^{-8}+2z^{2}a^{-4}+5z^{2}a^{-6}+2z^{2}a^{-8}-z^{2}a^{-10}+2a^{-6}+a^{-8}-2a^{-10}$
Kauffman polynomial (db, data sources)	$z^{8}a^{-8}+z^{8}a^{-10}+2z^{7}a^{-7}+3z^{7}a^{-9}+z^{7}a^{-11}+3z^{6}a^{-6}-z^{6}a^{-8}-3z^{6}a^{-10}+z^{6}a^{-12}+2z^{5}a^{-5}-3z^{5}a^{-7}-7z^{5}a^{-9}-z^{5}a^{-11}+z^{5}a^{-13}+z^{4}a^{-4}-7z^{4}a^{-6}+3z^{4}a^{-8}+9z^{4}a^{-10}-2z^{4}a^{-12}-3z^{3}a^{-5}+3z^{3}a^{-7}+9z^{3}a^{-9}-z^{3}a^{-11}-4z^{3}a^{-13}-2z^{2}a^{-4}+7z^{2}a^{-6}-2z^{2}a^{-8}-11z^{2}a^{-10}-4za^{-9}+4za^{-13}-2a^{-6}+a^{-8}+2a^{-10}$
The A2 invariant	$q^{-6}-q^{-8}+q^{-10}+2q^{-16}+2q^{-20}+q^{-22}+q^{-24}+q^{-26}-2q^{-28}-q^{-30}-q^{-32}-q^{-34}$
The G2 invariant	$q^{-30}-q^{-32}+2q^{-34}-3q^{-36}+2q^{-38}-q^{-40}-2q^{-42}+7q^{-44}-9q^{-46}+11q^{-48}-8q^{-50}+3q^{-52}+5q^{-54}-13q^{-56}+21q^{-58}-19q^{-60}+12q^{-62}-2q^{-64}-10q^{-66}+18q^{-68}-17q^{-70}+14q^{-72}-2q^{-74}-7q^{-76}+13q^{-78}-9q^{-80}-q^{-82}+12q^{-84}-19q^{-86}+18q^{-88}-9q^{-90}-4q^{-92}+21q^{-94}-28q^{-96}+31q^{-98}-20q^{-100}+6q^{-102}+11q^{-104}-21q^{-106}+26q^{-108}-18q^{-110}+12q^{-112}+2q^{-114}-10q^{-116}+15q^{-118}-10q^{-120}-2q^{-122}+9q^{-124}-16q^{-126}+10q^{-128}-3q^{-130}-12q^{-132}+18q^{-134}-20q^{-136}+15q^{-138}-10q^{-140}-9q^{-142}+13q^{-144}-16q^{-146}+13q^{-148}-9q^{-150}+2q^{-152}+3q^{-154}-4q^{-156}+6q^{-158}-6q^{-160}+4q^{-162}-q^{-164}+q^{-168}-2q^{-170}+2q^{-172}+q^{-176}$

Further Quantum Invariants

Further quantum knot invariants for 9_10.

A1 Invariants.

Weight	Invariant
1	$q^{-3}-q^{-5}+2q^{-7}-q^{-9}+q^{-11}+q^{-13}+2q^{-17}-2q^{-19}-q^{-23}$
2	$q^{-6}-q^{-8}+4q^{-12}-2q^{-14}-3q^{-16}+7q^{-18}-q^{-20}-6q^{-22}+7q^{-24}+q^{-26}-5q^{-28}+2q^{-30}+2q^{-32}-3q^{-36}+4q^{-38}+3q^{-40}-7q^{-42}+q^{-44}+5q^{-46}-7q^{-48}-2q^{-50}+4q^{-52}-3q^{-54}-q^{-56}+2q^{-58}+q^{-64}$
3	$q^{-9}-q^{-11}+2q^{-15}+2q^{-17}-2q^{-19}-3q^{-21}+4q^{-23}+7q^{-25}-4q^{-27}-10q^{-29}+2q^{-31}+17q^{-33}+2q^{-35}-20q^{-37}-6q^{-39}+21q^{-41}+12q^{-43}-20q^{-45}-14q^{-47}+17q^{-49}+15q^{-51}-9q^{-53}-14q^{-55}+3q^{-57}+9q^{-59}+3q^{-61}-9q^{-63}-9q^{-65}+6q^{-67}+15q^{-69}-2q^{-71}-20q^{-73}+q^{-75}+20q^{-77}+3q^{-79}-23q^{-81}-9q^{-83}+16q^{-85}+13q^{-87}-16q^{-89}-13q^{-91}+8q^{-93}+14q^{-95}-2q^{-97}-10q^{-99}+q^{-101}+7q^{-103}+2q^{-105}-3q^{-107}+q^{-111}+q^{-113}-q^{-115}-q^{-123}$
4	$q^{-12}-q^{-14}+2q^{-18}+2q^{-22}-3q^{-24}-q^{-26}+5q^{-28}+5q^{-32}-8q^{-34}-6q^{-36}+9q^{-38}+6q^{-40}+14q^{-42}-17q^{-44}-24q^{-46}+q^{-48}+19q^{-50}+47q^{-52}-10q^{-54}-51q^{-56}-33q^{-58}+14q^{-60}+83q^{-62}+23q^{-64}-54q^{-66}-70q^{-68}-17q^{-70}+90q^{-72}+55q^{-74}-28q^{-76}-73q^{-78}-42q^{-80}+56q^{-82}+55q^{-84}+6q^{-86}-43q^{-88}-44q^{-90}+11q^{-92}+35q^{-94}+26q^{-96}-10q^{-98}-34q^{-100}-28q^{-102}+15q^{-104}+44q^{-106}+17q^{-108}-28q^{-110}-58q^{-112}+60q^{-116}+42q^{-118}-19q^{-120}-84q^{-122}-19q^{-124}+59q^{-126}+62q^{-128}+5q^{-130}-86q^{-132}-45q^{-134}+27q^{-136}+66q^{-138}+43q^{-140}-53q^{-142}-52q^{-144}-10q^{-146}+38q^{-148}+55q^{-150}-8q^{-152}-29q^{-154}-28q^{-156}+2q^{-158}+32q^{-160}+9q^{-162}-4q^{-164}-16q^{-166}-9q^{-168}+8q^{-170}+3q^{-172}+4q^{-174}-3q^{-176}-4q^{-178}+q^{-180}-2q^{-182}+2q^{-184}-q^{-188}+q^{-190}-q^{-192}+q^{-200}$
5	$q^{-15}-q^{-17}+2q^{-21}+q^{-27}-q^{-29}-q^{-31}+3q^{-33}+3q^{-35}-2q^{-37}-q^{-41}+4q^{-45}+6q^{-47}-q^{-49}-9q^{-51}-11q^{-53}-4q^{-55}+14q^{-57}+26q^{-59}+22q^{-61}-9q^{-63}-49q^{-65}-52q^{-67}-8q^{-69}+62q^{-71}+99q^{-73}+56q^{-75}-60q^{-77}-153q^{-79}-118q^{-81}+31q^{-83}+186q^{-85}+199q^{-87}+30q^{-89}-197q^{-91}-273q^{-93}-104q^{-95}+171q^{-97}+314q^{-99}+183q^{-101}-115q^{-103}-323q^{-105}-243q^{-107}+53q^{-109}+287q^{-111}+266q^{-113}+15q^{-115}-227q^{-117}-259q^{-119}-71q^{-121}+158q^{-123}+220q^{-125}+96q^{-127}-80q^{-129}-173q^{-131}-115q^{-133}+26q^{-135}+121q^{-137}+115q^{-139}+26q^{-141}-78q^{-143}-122q^{-145}-65q^{-147}+45q^{-149}+124q^{-151}+100q^{-153}-22q^{-155}-140q^{-157}-139q^{-159}+5q^{-161}+164q^{-163}+177q^{-165}+17q^{-167}-179q^{-169}-224q^{-171}-48q^{-173}+193q^{-175}+264q^{-177}+89q^{-179}-176q^{-181}-302q^{-183}-140q^{-185}+144q^{-187}+300q^{-189}+204q^{-191}-76q^{-193}-287q^{-195}-239q^{-197}+2q^{-199}+221q^{-201}+257q^{-203}+86q^{-205}-151q^{-207}-235q^{-209}-128q^{-211}+58q^{-213}+184q^{-215}+157q^{-217}+16q^{-219}-118q^{-221}-140q^{-223}-62q^{-225}+48q^{-227}+102q^{-229}+76q^{-231}-3q^{-233}-64q^{-235}-64q^{-237}-23q^{-239}+22q^{-241}+44q^{-243}+25q^{-245}-3q^{-247}-20q^{-249}-20q^{-251}-7q^{-253}+9q^{-255}+11q^{-257}+6q^{-259}+2q^{-261}-5q^{-263}-6q^{-265}+q^{-269}+q^{-271}+4q^{-273}-2q^{-277}-q^{-283}+q^{-285}+q^{-287}-q^{-295}$

A2 Invariants.

Weight	Invariant
1,0	$q^{-6}-q^{-8}+q^{-10}+2q^{-16}+2q^{-20}+q^{-22}+q^{-24}+q^{-26}-2q^{-28}-q^{-30}-q^{-32}-q^{-34}$
1,1	$q^{-12}-2q^{-14}+4q^{-16}-8q^{-18}+17q^{-20}-22q^{-22}+32q^{-24}-42q^{-26}+58q^{-28}-56q^{-30}+60q^{-32}-54q^{-34}+43q^{-36}-16q^{-38}-12q^{-40}+44q^{-42}-69q^{-44}+100q^{-46}-114q^{-48}+118q^{-50}-119q^{-52}+94q^{-54}-80q^{-56}+46q^{-58}-24q^{-60}-8q^{-62}+36q^{-64}-46q^{-66}+56q^{-68}-56q^{-70}+56q^{-72}-44q^{-74}+30q^{-76}-28q^{-78}+16q^{-80}-12q^{-82}+8q^{-84}-4q^{-86}+4q^{-88}+q^{-92}$
2,0	$q^{-12}-q^{-14}-q^{-16}+3q^{-18}+2q^{-20}-3q^{-22}-q^{-24}+4q^{-26}+3q^{-28}-2q^{-30}+q^{-32}+4q^{-34}-q^{-36}-2q^{-38}+2q^{-40}+q^{-42}+4q^{-46}+4q^{-48}+2q^{-54}-6q^{-58}-q^{-60}-3q^{-64}-6q^{-66}-3q^{-68}-q^{-72}-q^{-74}+2q^{-78}+q^{-80}+2q^{-82}+q^{-84}+q^{-86}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{-12}-q^{-14}+2q^{-18}-3q^{-20}+q^{-22}+6q^{-24}-4q^{-26}+q^{-28}+7q^{-30}-2q^{-32}+q^{-34}+6q^{-36}+q^{-38}+2q^{-44}-2q^{-46}-4q^{-48}+3q^{-50}-8q^{-54}+2q^{-56}-q^{-58}-7q^{-60}+q^{-62}-q^{-66}+2q^{-68}+2q^{-70}+q^{-74}$
1,0,0	$q^{-9}-q^{-11}+q^{-13}-q^{-15}+q^{-17}+2q^{-21}+q^{-23}+q^{-25}+2q^{-27}+q^{-29}+2q^{-31}+q^{-35}-2q^{-37}-q^{-39}-2q^{-41}-q^{-43}-q^{-45}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{-18}-q^{-20}+q^{-24}-q^{-26}-q^{-28}+3q^{-30}+2q^{-32}-2q^{-34}+q^{-36}+5q^{-38}+2q^{-40}-2q^{-42}+4q^{-44}+8q^{-46}+q^{-50}+6q^{-52}+3q^{-54}-q^{-56}+4q^{-58}+5q^{-60}-q^{-64}+2q^{-66}-4q^{-68}-11q^{-70}-7q^{-72}-5q^{-74}-10q^{-76}-8q^{-78}+2q^{-82}+q^{-84}+2q^{-86}+5q^{-88}+3q^{-90}+q^{-92}+q^{-94}+q^{-96}$
1,0,0,0	$q^{-12}-q^{-14}+q^{-16}-q^{-18}+q^{-22}+2q^{-26}+q^{-28}+2q^{-30}+q^{-32}+2q^{-34}+q^{-36}+2q^{-38}+q^{-40}+q^{-44}-2q^{-46}-q^{-48}-2q^{-50}-2q^{-52}-q^{-54}-q^{-56}$

B2 Invariants.

Weight	Invariant
0,1	$q^{-12}-q^{-14}+2q^{-16}-4q^{-18}+5q^{-20}-5q^{-22}+6q^{-24}-4q^{-26}+5q^{-28}-q^{-30}+5q^{-34}-6q^{-36}+9q^{-38}-10q^{-40}+10q^{-42}-10q^{-44}+8q^{-46}-6q^{-48}+3q^{-50}-2q^{-54}+4q^{-56}-5q^{-58}+5q^{-60}-5q^{-62}+4q^{-64}-3q^{-66}+2q^{-68}-2q^{-70}-q^{-74}$
1,0	$q^{-18}-q^{-22}-q^{-24}+q^{-26}+3q^{-28}-4q^{-32}-2q^{-34}+4q^{-36}+6q^{-38}-q^{-40}-5q^{-42}-2q^{-44}+6q^{-46}+5q^{-48}-2q^{-50}-4q^{-52}+3q^{-54}+5q^{-56}+q^{-58}-3q^{-60}+4q^{-64}+2q^{-66}-3q^{-68}-3q^{-70}+2q^{-72}+3q^{-74}-2q^{-76}-4q^{-78}+q^{-80}+5q^{-82}-6q^{-86}-5q^{-88}+3q^{-90}+4q^{-92}-3q^{-94}-7q^{-96}-3q^{-98}+3q^{-100}+2q^{-102}-q^{-104}-2q^{-106}+2q^{-110}+2q^{-112}+q^{-120}$

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{-18}-q^{-20}+q^{-22}-2q^{-24}+3q^{-26}-4q^{-28}+4q^{-30}-3q^{-32}+6q^{-34}-3q^{-36}+4q^{-38}-q^{-40}+5q^{-42}+2q^{-44}-q^{-46}+5q^{-48}-2q^{-50}+9q^{-52}-5q^{-54}+8q^{-56}-7q^{-58}+9q^{-60}-7q^{-62}+5q^{-64}-8q^{-66}+2q^{-68}-3q^{-70}-q^{-72}-2q^{-74}-4q^{-76}+2q^{-78}-5q^{-80}+2q^{-82}-6q^{-84}+3q^{-86}-4q^{-88}+2q^{-90}-2q^{-92}+3q^{-94}+2q^{-98}+q^{-102}$

G2 Invariants.

Weight

Invariant

1,0

q^{-30}-q^{-32}+2q^{-34}-3q^{-36}+2q^{-38}-q^{-40}-2q^{-42}+7q^{-44}-9q^{-46}+11q^{-48}-8q^{-50}+3q^{-52}+5q^{-54}-13q^{-56}+21q^{-58}-19q^{-60}+12q^{-62}-2q^{-64}-10q^{-66}+18q^{-68}-17q^{-70}+14q^{-72}-2q^{-74}-7q^{-76}+13q^{-78}-9q^{-80}-q^{-82}+12q^{-84}-19q^{-86}+18q^{-88}-9q^{-90}-4q^{-92}+21q^{-94}-28q^{-96}+31q^{-98}-20q^{-100}+6q^{-102}+11q^{-104}-21q^{-106}+26q^{-108}-18q^{-110}+12q^{-112}+2q^{-114}-10q^{-116}+15q^{-118}-10q^{-120}-2q^{-122}+9q^{-124}-16q^{-126}+10q^{-128}-3q^{-130}-12q^{-132}+18q^{-134}-20q^{-136}+15q^{-138}-10q^{-140}-9q^{-142}+13q^{-144}-16q^{-146}+13q^{-148}-9q^{-150}+2q^{-152}+3q^{-154}-4q^{-156}+6q^{-158}-6q^{-160}+4q^{-162}-q^{-164}+q^{-168}-2q^{-170}+2q^{-172}+q^{-176}

.

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 10"];

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

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

Out[4]= $4t^{2}-8t+9-8t^{-1}+4t^{-2}$

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

Out[5]= $4z^{4}+8z^{2}+1$

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]= { 33, 4 }

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

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

Out[8]= $-q^{11}+q^{10}-3q^{9}+5q^{8}-5q^{7}+6q^{6}-5q^{5}+4q^{4}-2q^{3}+q^{2}$

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

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

Out[9]= $z^{4}a^{-4}+2z^{4}a^{-6}+z^{4}a^{-8}+2z^{2}a^{-4}+5z^{2}a^{-6}+2z^{2}a^{-8}-z^{2}a^{-10}+2a^{-6}+a^{-8}-2a^{-10}$

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

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

Out[10]= $z^{8}a^{-8}+z^{8}a^{-10}+2z^{7}a^{-7}+3z^{7}a^{-9}+z^{7}a^{-11}+3z^{6}a^{-6}-z^{6}a^{-8}-3z^{6}a^{-10}+z^{6}a^{-12}+2z^{5}a^{-5}-3z^{5}a^{-7}-7z^{5}a^{-9}-z^{5}a^{-11}+z^{5}a^{-13}+z^{4}a^{-4}-7z^{4}a^{-6}+3z^{4}a^{-8}+9z^{4}a^{-10}-2z^{4}a^{-12}-3z^{3}a^{-5}+3z^{3}a^{-7}+9z^{3}a^{-9}-z^{3}a^{-11}-4z^{3}a^{-13}-2z^{2}a^{-4}+7z^{2}a^{-6}-2z^{2}a^{-8}-11z^{2}a^{-10}-4za^{-9}+4za^{-13}-2a^{-6}+a^{-8}+2a^{-10}$

Vassiliev invariants

V₂ and V₃:

(8, 22)

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

32

176

512

{\frac {3856}{3}}

{\frac {656}{3}}

5632

{\frac {30752}{3}}

{\frac {5504}{3}}

1520

{\frac {16384}{3}}

15488

{\frac {123392}{3}}

{\frac {20992}{3}}

{\frac {1241164}{15}}

{\frac {6064}{15}}

{\frac {1589776}{45}}

{\frac {5108}{9}}

{\frac {72844}{15}}

V_2,1 through V_6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V₂ and V₃.

Khovanov Homology

The coefficients of the monomials

t^{r}q^{j}

are shown, along with their alternating sums

\chi

(fixed

j

, alternation over

r

). The squares with yellow highlighting are those on the "critical diagonals", where

j-2r=s+1

or

j-2r=s-1

, where

s=

4 is the signature of 9 10. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

0

1

2

3

4

5

6

7

8

9

χ

23

1

-1

21

0

19

3

1

-2

17

2

15

3

0

13

3

2

1

11

2

3

1

9

2

3

-1

7

2

5

1

2

-1

3

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=3$	$i=5$
$r=0$	${\mathbb {Z} }$	${\mathbb {Z} }$
$r=1$	${\mathbb {Z} }^{2}$
$r=2$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=3$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=4$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=5$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=6$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=7$	${\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=8$	${\mathbb {Z} }$
$r=9$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

${\textrm {Include}}({\textrm {ColouredJonesM.mhtml}})$

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 17, 2005, 14:44:34)...
In[2]:=	Crossings[Knot[9, 10]]
Out[2]=	9
In[3]:=	PD[Knot[9, 10]]
Out[3]=	PD[X[8, 2, 9, 1], X[12, 4, 13, 3], X[18, 10, 1, 9], X[10, 18, 11, 17], X[16, 8, 17, 7], X[2, 12, 3, 11], X[4, 16, 5, 15], X[14, 6, 15, 5], X[6, 14, 7, 13]]
In[4]:=	GaussCode[Knot[9, 10]]
Out[4]=	GaussCode[1, -6, 2, -7, 8, -9, 5, -1, 3, -4, 6, -2, 9, -8, 7, -5, 4, -3]
In[5]:=	BR[Knot[9, 10]]
Out[5]=	BR[4, {1, 1, 2, -1, 2, 2, 2, 2, 3, -2, 3}]
In[6]:=	alex = Alexander[Knot[9, 10]][t]
Out[6]=	4 8 2 9 + -- - - - 8 t + 4 t 2 t t
In[7]:=	Conway[Knot[9, 10]][z]
Out[7]=	2 4 1 + 8 z + 4 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[9, 10]}
In[9]:=	{KnotDet[Knot[9, 10]], KnotSignature[Knot[9, 10]]}
Out[9]=	{33, 4}
In[10]:=	J=Jones[Knot[9, 10]][q]
Out[10]=	2 3 4 5 6 7 8 9 10 11 q - 2 q + 4 q - 5 q + 6 q - 5 q + 5 q - 3 q + q - q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] \|\| (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[9, 10]}
In[12]:=	A2Invariant[Knot[9, 10]][q]
Out[12]=	6 8 10 16 20 22 24 26 28 30 32 q - q + q + 2 q + 2 q + q + q + q - 2 q - q - q - 34 q
In[13]:=	Kauffman[Knot[9, 10]][a, z]
Out[13]=	2 2 2 2 3 3 2 -8 2 4 z 4 z 11 z 2 z 7 z 2 z 4 z z --- + a - -- + --- - --- - ----- - ---- + ---- - ---- - ---- - --- + 10 6 13 9 10 8 6 4 13 11 a a a a a a a a a a 3 3 3 4 4 4 4 4 5 5 9 z 3 z 3 z 2 z 9 z 3 z 7 z z z z ---- + ---- - ---- - ---- + ---- + ---- - ---- + -- + --- - --- - 9 7 5 12 10 8 6 4 13 11 a a a a a a a a a a 5 5 5 6 6 6 6 7 7 7 7 z 3 z 2 z z 3 z z 3 z z 3 z 2 z ---- - ---- + ---- + --- - ---- - -- + ---- + --- + ---- + ---- + 9 7 5 12 10 8 6 11 9 7 a a a a a a a a a a 8 8 z z --- + -- 10 8 a a
In[14]:=	{Vassiliev[2][Knot[9, 10]], Vassiliev[3][Knot[9, 10]]}
Out[14]=	{0, 22}
In[15]:=	Kh[Knot[9, 10]][q, t]
Out[15]=	3 5 5 7 2 9 2 9 3 11 3 11 4 q + q + 2 q t + 2 q t + 2 q t + 3 q t + 2 q t + 3 q t + 13 4 13 5 15 5 15 6 17 6 19 7 3 q t + 2 q t + 3 q t + 3 q t + 2 q t + 3 q t + 19 8 23 9 q t + q t

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-|{{Rolfsen Knot Site Links|n=9|k=10|KnotilusURL=http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/1,-6,2,-7,8,-9,5,-1,3,-4,6,-2,9,-8,7,-5,4,-3/goTop.html}}
-|{{:{{PAGENAME}} Quick Notes}}
-|}
 <br style="clear:both" />
@@ Line 24: / Line 21: @@
 {{Vassiliev Invariants}}
-===[[Khovanov Homology]]===
+{{Khovanov Homology|table=<table border=1>
-The coefficients of the monomials <math>t^rq^j</math> are shown, along with their alternating sums <math>\chi</math> (fixed <math>j</math>, alternation over <math>r</math>). The squares with <font class=HLYellow>yellow</font> highlighting are those on the "critical diagonals", where <math>j-2r=s+1</math> or <math>j-2r=s+1</math>, where <math>s=</math>{{Data:{{PAGENAME}}/Signature}} is the signature of {{PAGENAME}}. Nonzero entries off the critical diagonals (if any exist) are highlighted in <font class=HLRed>red</font>.
-<center><table border=1>
 <tr align=center>
 <td width=14.2857%><table cellpadding=0 cellspacing=0>
@@ Line 47: / Line 40: @@
 <tr align=center><td>5</td><td bgcolor=yellow>1</td><td bgcolor=yellow>2</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>-1</td></tr>
 <tr align=center><td>3</td><td bgcolor=yellow>1</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>1</td></tr>
-</table></center>
+</table>}}
 {{Computer Talk Header}}
@@ Line 129: / Line 121: @@
   q   t  + q   t</nowiki></pre></td></tr>
 </table>
+ [[Category:Knot Page]]

9 10: Difference between revisions

Revision as of 19:16, 28 August 2005

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Four dimensional invariants

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