9 11: Difference between revisions

Revision as of 20:14, 28 August 2005

9_10

9_12

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9 11 Quick Notes

9 11 Further Notes and Views

Knot presentations

Planar diagram presentation	X₁₄₂₅ X_9,12,10,13 X_3,11,4,10 X_11,3,12,2 X_13,1,14,18 X_5,15,6,14 X_7,17,8,16 X_15,7,16,6 X_17,9,18,8
Gauss code	-1, 4, -3, 1, -6, 8, -7, 9, -2, 3, -4, 2, -5, 6, -8, 7, -9, 5
Dowker-Thistlethwaite code	4 10 14 16 12 2 18 6 8
Conway Notation	[4122]

Three dimensional invariants

Symmetry type	Reversible
Unknotting number	2
3-genus	3
Bridge index	2
Super bridge index	$\{4,6\}$
Nakanishi index	1
Maximal Thurston-Bennequin number	[1][-12]
Hyperbolic Volume	8.28859
A-Polynomial	See Data:9 11/A-polynomial

[edit Notes for 9 11'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 11's four dimensional invariants]

Polynomial invariants

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

Further Quantum Invariants

Further quantum knot invariants for 9_11.

A1 Invariants.

Weight	Invariant
1	$q-q^{-1}+q^{-3}-q^{-5}+2q^{-7}+q^{-9}+q^{-13}-2q^{-15}+q^{-17}-q^{-19}$
2	$q^{6}-q^{4}-2q^{2}+3+q^{-2}-4q^{-4}+3q^{-6}+4q^{-8}-4q^{-10}+q^{-12}+4q^{-14}-4q^{-16}-q^{-18}+4q^{-20}-2q^{-24}+2q^{-26}+3q^{-28}-2q^{-30}-4q^{-32}+4q^{-34}-q^{-36}-5q^{-38}+4q^{-40}-q^{-42}-3q^{-44}+3q^{-46}-q^{-50}+q^{-52}$
3	$q^{15}-q^{13}-2q^{11}+4q^{7}+3q^{5}-5q^{3}-6q+4q^{-1}+9q^{-3}-11q^{-7}-4q^{-9}+11q^{-11}+8q^{-13}-7q^{-15}-10q^{-17}+4q^{-19}+13q^{-21}+q^{-23}-9q^{-25}-3q^{-27}+10q^{-29}+4q^{-31}-8q^{-33}-8q^{-35}+8q^{-37}+8q^{-39}-7q^{-41}-9q^{-43}+4q^{-45}+10q^{-47}-8q^{-51}-6q^{-53}+6q^{-55}+8q^{-57}-q^{-59}-14q^{-61}-2q^{-63}+10q^{-65}+5q^{-67}-9q^{-69}-4q^{-71}+6q^{-73}+2q^{-75}-3q^{-77}+3q^{-81}-q^{-83}-2q^{-85}+2q^{-87}+2q^{-89}-q^{-91}-q^{-93}+q^{-97}-q^{-99}$
4	$q^{28}-q^{26}-2q^{24}+q^{20}+6q^{18}+q^{16}-5q^{14}-7q^{12}-7q^{10}+12q^{8}+13q^{6}+3q^{4}-10q^{2}-24+16q^{-4}+23q^{-6}+9q^{-8}-29q^{-10}-21q^{-12}-4q^{-14}+25q^{-16}+33q^{-18}-5q^{-20}-21q^{-22}-31q^{-24}+35q^{-28}+21q^{-30}+3q^{-32}-36q^{-34}-28q^{-36}+14q^{-38}+30q^{-40}+30q^{-42}-20q^{-44}-37q^{-46}-6q^{-48}+23q^{-50}+36q^{-52}-9q^{-54}-36q^{-56}-13q^{-58}+20q^{-60}+30q^{-62}-7q^{-64}-32q^{-66}-16q^{-68}+19q^{-70}+29q^{-72}+3q^{-74}-25q^{-76}-27q^{-78}+6q^{-80}+22q^{-82}+24q^{-84}+q^{-86}-29q^{-88}-25q^{-90}-6q^{-92}+38q^{-94}+35q^{-96}-11q^{-98}-37q^{-100}-38q^{-102}+22q^{-104}+47q^{-106}+14q^{-108}-18q^{-110}-45q^{-112}-q^{-114}+28q^{-116}+17q^{-118}+5q^{-120}-26q^{-122}-7q^{-124}+7q^{-126}+5q^{-128}+11q^{-130}-8q^{-132}-2q^{-134}-2q^{-136}-3q^{-138}+7q^{-140}-2q^{-142}+q^{-144}-q^{-146}-3q^{-148}+2q^{-150}-q^{-152}+q^{-154}-q^{-158}+q^{-160}$
5	$q^{45}-q^{43}-2q^{41}+q^{37}+3q^{35}+4q^{33}+q^{31}-7q^{29}-9q^{27}-5q^{25}+3q^{23}+15q^{21}+18q^{19}+5q^{17}-18q^{15}-28q^{13}-20q^{11}+5q^{9}+35q^{7}+41q^{5}+16q^{3}-27q-53q^{-1}-43q^{-3}-q^{-5}+51q^{-7}+67q^{-9}+35q^{-11}-27q^{-13}-69q^{-15}-66q^{-17}-16q^{-19}+51q^{-21}+84q^{-23}+59q^{-25}-9q^{-27}-73q^{-29}-89q^{-31}-43q^{-33}+41q^{-35}+100q^{-37}+92q^{-39}+10q^{-41}-83q^{-43}-118q^{-45}-67q^{-47}+48q^{-49}+132q^{-51}+111q^{-53}-7q^{-55}-120q^{-57}-139q^{-59}-44q^{-61}+98q^{-63}+156q^{-65}+73q^{-67}-67q^{-69}-145q^{-71}-97q^{-73}+40q^{-75}+135q^{-77}+98q^{-79}-23q^{-81}-112q^{-83}-94q^{-85}+12q^{-87}+99q^{-89}+81q^{-91}-16q^{-93}-90q^{-95}-73q^{-97}+16q^{-99}+86q^{-101}+69q^{-103}-18q^{-105}-88q^{-107}-75q^{-109}+4q^{-111}+75q^{-113}+87q^{-115}+31q^{-117}-52q^{-119}-90q^{-121}-74q^{-123}+5q^{-125}+91q^{-127}+114q^{-129}+55q^{-131}-56q^{-133}-148q^{-135}-122q^{-137}+18q^{-139}+149q^{-141}+166q^{-143}+51q^{-145}-127q^{-147}-199q^{-149}-96q^{-151}+86q^{-153}+190q^{-155}+132q^{-157}-37q^{-159}-167q^{-161}-141q^{-163}-q^{-165}+127q^{-167}+128q^{-169}+23q^{-171}-82q^{-173}-104q^{-175}-34q^{-177}+51q^{-179}+74q^{-181}+33q^{-183}-25q^{-185}-47q^{-187}-29q^{-189}+8q^{-191}+28q^{-193}+22q^{-195}-q^{-197}-15q^{-199}-14q^{-201}-4q^{-203}+6q^{-205}+9q^{-207}+4q^{-209}-4q^{-211}-3q^{-213}-q^{-215}-q^{-217}+2q^{-219}+2q^{-221}-q^{-223}+q^{-227}-q^{-229}+q^{-233}-q^{-235}$

A2 Invariants.

Weight	Invariant
1,0	$1-q^{-8}+2q^{-10}+2q^{-14}+q^{-16}+q^{-20}-q^{-22}-q^{-26}-q^{-28}$
1,1	$q^{4}-2q^{2}+6-12q^{-2}+19q^{-4}-30q^{-6}+40q^{-8}-44q^{-10}+49q^{-12}-44q^{-14}+38q^{-16}-18q^{-18}+3q^{-20}+18q^{-22}-34q^{-24}+54q^{-26}-68q^{-28}+74q^{-30}-76q^{-32}+74q^{-34}-63q^{-36}+50q^{-38}-32q^{-40}+14q^{-42}-2q^{-44}-10q^{-46}+14q^{-48}-22q^{-50}+25q^{-52}-26q^{-54}+24q^{-56}-28q^{-58}+26q^{-60}-22q^{-62}+18q^{-64}-14q^{-66}+11q^{-68}-6q^{-70}+4q^{-72}-2q^{-74}+q^{-76}$
2,0	$q^{4}-1-q^{-2}+q^{-4}+q^{-6}-2q^{-8}+4q^{-12}+3q^{-14}-q^{-16}+q^{-18}+2q^{-20}-q^{-22}-q^{-24}+q^{-28}-2q^{-30}+3q^{-32}+3q^{-34}+q^{-38}+4q^{-40}-4q^{-44}-q^{-46}-2q^{-50}-4q^{-52}-q^{-54}-q^{-58}+q^{-66}+q^{-68}+q^{-70}$

A3 Invariants.

Weight	Invariant
0,1,0	$1-q^{-2}+q^{-4}+q^{-6}-2q^{-8}+3q^{-10}+q^{-12}-2q^{-14}+3q^{-16}-4q^{-20}+3q^{-22}+3q^{-24}-2q^{-26}+5q^{-28}+4q^{-30}+2q^{-32}-q^{-34}-6q^{-40}-3q^{-42}+2q^{-44}-4q^{-46}-2q^{-48}+5q^{-50}-2q^{-52}-2q^{-54}+3q^{-56}-q^{-60}+q^{-62}$
1,0,0	$q^{-1}+q^{-5}-q^{-7}+q^{-9}-q^{-11}+q^{-13}+q^{-17}+q^{-19}+q^{-21}+2q^{-23}+2q^{-27}-q^{-29}-2q^{-33}-q^{-35}-q^{-37}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{-2}+q^{-8}+q^{-14}+q^{-16}+2q^{-20}+2q^{-22}-q^{-24}-q^{-26}+2q^{-28}+2q^{-30}-4q^{-32}+2q^{-34}+6q^{-36}+2q^{-38}+q^{-40}+7q^{-42}+4q^{-44}+q^{-48}-5q^{-52}-6q^{-54}-2q^{-56}-4q^{-58}-6q^{-60}-q^{-62}+2q^{-64}-2q^{-66}-q^{-68}+2q^{-70}+2q^{-72}+q^{-78}+q^{-80}$
1,0,0,0	$q^{-2}+q^{-6}+q^{-12}-q^{-14}+q^{-16}-q^{-18}+q^{-20}+q^{-24}+q^{-26}+2q^{-28}+2q^{-30}+q^{-32}+2q^{-34}-q^{-36}-2q^{-40}-2q^{-42}-q^{-44}-q^{-46}$

B2 Invariants.

Weight

Invariant

0,1

1-q^{-2}+3q^{-4}-3q^{-6}+4q^{-8}-5q^{-10}+5q^{-12}-4q^{-14}+3q^{-16}-2q^{-18}+3q^{-22}-5q^{-24}+8q^{-26}-7q^{-28}+10q^{-30}-8q^{-32}+9q^{-34}-6q^{-36}+4q^{-38}-2q^{-40}-q^{-42}+2q^{-44}-4q^{-46}+4q^{-48}-5q^{-50}+4q^{-52}-4q^{-54}+3q^{-56}-2q^{-58}+q^{-60}-q^{-62}

1,0

q^{2}-q^{-2}-q^{-4}+2q^{-6}+2q^{-8}-2q^{-10}-3q^{-12}+q^{-14}+5q^{-16}+2q^{-18}-4q^{-20}-3q^{-22}+3q^{-24}+5q^{-26}-5q^{-30}-3q^{-32}+3q^{-34}+4q^{-36}-3q^{-40}+2q^{-42}+4q^{-44}+2q^{-46}-2q^{-48}+q^{-50}+4q^{-52}+2q^{-54}-4q^{-56}-3q^{-58}+2q^{-60}+2q^{-62}-3q^{-64}-5q^{-66}-q^{-68}+3q^{-70}+q^{-72}-4q^{-74}-4q^{-76}+q^{-78}+5q^{-80}+q^{-82}-3q^{-84}-3q^{-86}+q^{-88}+3q^{-90}+q^{-92}-q^{-94}-q^{-96}+q^{-100}

D4 Invariants.

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

G2 Invariants.

Weight

Invariant

1,0

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

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

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

Out[4]= $-t^{3}+5t^{2}-7t+7-7t^{-1}+5t^{-2}-t^{-3}$

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

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

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

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

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

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

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

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

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

Vassiliev invariants

V₂ and V₃:

(4, 9)

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

16

72

128

{\frac {1160}{3}}

{\frac {208}{3}}

1152

2288

384

392

{\frac {2048}{3}}

2592

{\frac {18560}{3}}

{\frac {3328}{3}}

{\frac {204662}{15}}

-{\frac {5408}{15}}

{\frac {284288}{45}}

{\frac {1450}{9}}

{\frac {13382}{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 11. 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

2

1

11

3

0

9

3

2

1

7

1

3

2

5

2

3

-1

3

1

2

1

-1

1

Integral Khovanov Homology

(db, data source)

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

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, 11]]
Out[2]=	9
In[3]:=	PD[Knot[9, 11]]
Out[3]=	PD[X[1, 4, 2, 5], X[9, 12, 10, 13], X[3, 11, 4, 10], X[11, 3, 12, 2], X[13, 1, 14, 18], X[5, 15, 6, 14], X[7, 17, 8, 16], X[15, 7, 16, 6], X[17, 9, 18, 8]]
In[4]:=	GaussCode[Knot[9, 11]]
Out[4]=	GaussCode[-1, 4, -3, 1, -6, 8, -7, 9, -2, 3, -4, 2, -5, 6, -8, 7, -9, 5]
In[5]:=	BR[Knot[9, 11]]
Out[5]=	BR[4, {1, 1, 1, 1, -2, 1, 3, -2, 3}]
In[6]:=	alex = Alexander[Knot[9, 11]][t]
Out[6]=	-3 5 7 2 3 7 - t + -- - - - 7 t + 5 t - t 2 t t
In[7]:=	Conway[Knot[9, 11]][z]
Out[7]=	2 4 6 1 + 4 z - z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[9, 11], Knot[11, NonAlternating, 95]}
In[9]:=	{KnotDet[Knot[9, 11]], KnotSignature[Knot[9, 11]]}
Out[9]=	{33, 4}
In[10]:=	J=Jones[Knot[9, 11]][q]
Out[10]=	2 3 4 5 6 7 8 9 1 - 2 q + 3 q - 4 q + 6 q - 5 q + 5 q - 4 q + 2 q - q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] \|\| (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[9, 11]}
In[12]:=	A2Invariant[Knot[9, 11]][q]
Out[12]=	8 10 14 16 20 22 26 28 1 - q + 2 q + 2 q + q + q - q - q - q
In[13]:=	Kauffman[Knot[9, 11]][a, z]
Out[13]=	2 2 2 -2 3 -4 -2 z 2 z 2 z 2 z z z 4 z 6 z -- - -- - a - a - --- + --- + --- - --- - -- - --- + ---- + ---- + 8 6 11 9 7 5 3 10 8 6 a a a a a a a a a a 2 2 3 3 3 3 3 4 4 4 5 z 4 z z 3 z 3 z 9 z 8 z 2 z 4 z 7 z ---- + ---- + --- - ---- - ---- + ---- + ---- + ---- - ---- - ---- - 4 2 11 9 7 5 3 10 8 6 a a a a a a a a a a 4 4 5 5 5 5 6 6 6 6 7 5 z 4 z 3 z z 12 z 8 z 3 z z z z 2 z ---- - ---- + ---- - -- - ----- - ---- + ---- + -- - -- + -- + ---- + 4 2 9 7 5 3 8 6 4 2 7 a a a a a a a a a a a 7 7 8 8 4 z 2 z z z ---- + ---- + -- + -- 5 3 6 4 a a a a
In[14]:=	{Vassiliev[2][Knot[9, 11]], Vassiliev[3][Knot[9, 11]]}
Out[14]=	{0, 9}
In[15]:=	Kh[Knot[9, 11]][q, t]
Out[15]=	3 3 5 1 q q 5 7 7 2 9 2 2 q + 2 q + ---- + - + -- + 3 q t + 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 2 q t + 3 q t + 3 q t + 2 q t + q t + 3 q t + 15 6 17 6 19 7 q t + q t + q t

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-{| align=left
-|- valign=top
-|[[Image:{{PAGENAME}}.gif]]
-|{{Rolfsen Knot Site Links|n=9|k=11|KnotilusURL=http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/-1,4,-3,1,-6,8,-7,9,-2,3,-4,2,-5,6,-8,7,-9,5/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>1</td><td bgcolor=yellow>&nbsp;</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>-1</td></tr>
 <tr align=center><td>-1</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  + q   t</nowiki></pre></td></tr>
 </table>
+ [[Category:Knot Page]]

9 11: Difference between revisions

Revision as of 20:14, 28 August 2005

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