10 56: Difference between revisions

Revision as of 20:05, 28 August 2005

10_55

10_57

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10 56 Quick Notes

10 56 Further Notes and Views

Knot presentations

Planar diagram presentation	X₄₂₅₁ X_10,4,11,3 X_12,6,13,5 X_18,14,19,13 X_16,7,17,8 X_6,17,7,18 X_20,16,1,15 X_14,20,15,19 X_8,12,9,11 X_2,10,3,9
Gauss code	1, -10, 2, -1, 3, -6, 5, -9, 10, -2, 9, -3, 4, -8, 7, -5, 6, -4, 8, -7
Dowker-Thistlethwaite code	4 10 12 16 2 8 18 20 6 14
Conway Notation	[221,3,2]

Three dimensional invariants

Symmetry type	Reversible
Unknotting number	2
3-genus	3
Bridge index	3
Super bridge index	Missing
Nakanishi index	1
Maximal Thurston-Bennequin number	[1][-13]
Hyperbolic Volume	12.3988
A-Polynomial	See Data:10 56/A-polynomial

[edit Notes for 10 56'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 56's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-2t^{3}+8t^{2}-14t+17-14t^{-1}+8t^{-2}-2t^{-3}$
Conway polynomial	$-2z^{6}-4z^{4}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 65, 4 }
Jones polynomial	$q^{10}-3q^{9}+6q^{8}-9q^{7}+10q^{6}-11q^{5}+10q^{4}-7q^{3}+5q^{2}-2q+1$
HOMFLY-PT polynomial (db, data sources)	$-z^{6}a^{-4}-z^{6}a^{-6}+z^{4}a^{-2}-3z^{4}a^{-4}-3z^{4}a^{-6}+z^{4}a^{-8}+3z^{2}a^{-2}-2z^{2}a^{-4}-3z^{2}a^{-6}+2z^{2}a^{-8}+2a^{-2}-2a^{-6}+a^{-8}$
Kauffman polynomial (db, data sources)	$z^{9}a^{-5}+z^{9}a^{-7}+2z^{8}a^{-4}+6z^{8}a^{-6}+4z^{8}a^{-8}+2z^{7}a^{-3}+3z^{7}a^{-5}+7z^{7}a^{-7}+6z^{7}a^{-9}+z^{6}a^{-2}-3z^{6}a^{-4}-14z^{6}a^{-6}-5z^{6}a^{-8}+5z^{6}a^{-10}-6z^{5}a^{-3}-13z^{5}a^{-5}-21z^{5}a^{-7}-11z^{5}a^{-9}+3z^{5}a^{-11}-4z^{4}a^{-2}-3z^{4}a^{-4}+12z^{4}a^{-6}+4z^{4}a^{-8}-6z^{4}a^{-10}+z^{4}a^{-12}+4z^{3}a^{-3}+11z^{3}a^{-5}+21z^{3}a^{-7}+11z^{3}a^{-9}-3z^{3}a^{-11}+5z^{2}a^{-2}+3z^{2}a^{-4}-7z^{2}a^{-6}-2z^{2}a^{-8}+2z^{2}a^{-10}-z^{2}a^{-12}-4za^{-5}-8za^{-7}-4za^{-9}-2a^{-2}+2a^{-6}+a^{-8}$
The A2 invariant	$1+q^{-4}+2q^{-6}-q^{-8}+3q^{-10}-q^{-12}-3q^{-18}+q^{-20}-2q^{-22}+q^{-24}+q^{-26}-q^{-28}+q^{-30}$
The G2 invariant	$q^{-2}-q^{-4}+4q^{-6}-5q^{-8}+6q^{-10}-4q^{-12}-q^{-14}+12q^{-16}-20q^{-18}+30q^{-20}-30q^{-22}+20q^{-24}+2q^{-26}-32q^{-28}+65q^{-30}-79q^{-32}+73q^{-34}-37q^{-36}-17q^{-38}+73q^{-40}-108q^{-42}+110q^{-44}-70q^{-46}+6q^{-48}+57q^{-50}-93q^{-52}+86q^{-54}-39q^{-56}-18q^{-58}+67q^{-60}-80q^{-62}+48q^{-64}+9q^{-66}-79q^{-68}+121q^{-70}-120q^{-72}+71q^{-74}+7q^{-76}-92q^{-78}+146q^{-80}-158q^{-82}+116q^{-84}-44q^{-86}-46q^{-88}+109q^{-90}-130q^{-92}+103q^{-94}-38q^{-96}-28q^{-98}+71q^{-100}-73q^{-102}+35q^{-104}+21q^{-106}-68q^{-108}+89q^{-110}-64q^{-112}+13q^{-114}+50q^{-116}-94q^{-118}+107q^{-120}-83q^{-122}+37q^{-124}+12q^{-126}-54q^{-128}+72q^{-130}-68q^{-132}+50q^{-134}-20q^{-136}-4q^{-138}+19q^{-140}-29q^{-142}+26q^{-144}-19q^{-146}+11q^{-148}-2q^{-150}-3q^{-152}+5q^{-154}-6q^{-156}+4q^{-158}-2q^{-160}+q^{-162}$

Further Quantum Invariants[hide]

Further quantum knot invariants for 10_56.

A1 Invariants.

Weight	Invariant
1	$q-q^{-1}+3q^{-3}-2q^{-5}+3q^{-7}-q^{-9}-q^{-11}+q^{-13}-3q^{-15}+3q^{-17}-2q^{-19}+q^{-21}$
2	$q^{6}-q^{4}-q^{2}+5-2q^{-2}-6q^{-4}+12q^{-6}-16q^{-10}+14q^{-12}+9q^{-14}-21q^{-16}+7q^{-18}+14q^{-20}-15q^{-22}-4q^{-24}+11q^{-26}-11q^{-30}+2q^{-32}+16q^{-34}-13q^{-36}-9q^{-38}+22q^{-40}-9q^{-42}-13q^{-44}+16q^{-46}-q^{-48}-8q^{-50}+5q^{-52}-2q^{-56}+q^{-58}$
3	$q^{15}-q^{13}-q^{11}+q^{9}+4q^{7}-2q^{5}-7q^{3}+2q+15q^{-1}-23q^{-5}-9q^{-7}+36q^{-9}+24q^{-11}-39q^{-13}-48q^{-15}+36q^{-17}+74q^{-19}-18q^{-21}-93q^{-23}-9q^{-25}+101q^{-27}+40q^{-29}-96q^{-31}-66q^{-33}+79q^{-35}+81q^{-37}-55q^{-39}-93q^{-41}+34q^{-43}+86q^{-45}-5q^{-47}-79q^{-49}-19q^{-51}+63q^{-53}+47q^{-55}-44q^{-57}-70q^{-59}+18q^{-61}+91q^{-63}+11q^{-65}-103q^{-67}-39q^{-69}+99q^{-71}+63q^{-73}-83q^{-75}-72q^{-77}+56q^{-79}+73q^{-81}-33q^{-83}-59q^{-85}+12q^{-87}+39q^{-89}-23q^{-93}-2q^{-95}+12q^{-97}-4q^{-101}+2q^{-105}-2q^{-109}+q^{-111}$
4	$q^{28}-q^{26}-q^{24}+q^{22}+4q^{18}-4q^{16}-5q^{14}+4q^{12}+3q^{10}+15q^{8}-12q^{6}-25q^{4}-q^{2}+16+59q^{-2}-4q^{-4}-70q^{-6}-60q^{-8}-6q^{-10}+150q^{-12}+97q^{-14}-64q^{-16}-183q^{-18}-171q^{-20}+163q^{-22}+285q^{-24}+140q^{-26}-196q^{-28}-447q^{-30}-76q^{-32}+330q^{-34}+472q^{-36}+77q^{-38}-549q^{-40}-451q^{-42}+66q^{-44}+622q^{-46}+471q^{-48}-333q^{-50}-639q^{-52}-307q^{-54}+470q^{-56}+670q^{-58}-5q^{-60}-555q^{-62}-503q^{-64}+215q^{-66}+613q^{-68}+210q^{-70}-357q^{-72}-501q^{-74}+q^{-76}+442q^{-78}+336q^{-80}-148q^{-82}-439q^{-84}-211q^{-86}+226q^{-88}+456q^{-90}+128q^{-92}-320q^{-94}-472q^{-96}-95q^{-98}+515q^{-100}+467q^{-102}-58q^{-104}-630q^{-106}-481q^{-108}+349q^{-110}+659q^{-112}+306q^{-114}-489q^{-116}-683q^{-118}+15q^{-120}+508q^{-122}+499q^{-124}-142q^{-126}-532q^{-128}-199q^{-130}+172q^{-132}+379q^{-134}+80q^{-136}-222q^{-138}-156q^{-140}-33q^{-142}+150q^{-144}+84q^{-146}-40q^{-148}-42q^{-150}-47q^{-152}+31q^{-154}+23q^{-156}-5q^{-158}+4q^{-160}-16q^{-162}+5q^{-164}+3q^{-166}-3q^{-168}+4q^{-170}-3q^{-172}+2q^{-174}-2q^{-178}+q^{-180}$
5	$q^{45}-q^{43}-q^{41}+q^{39}+2q^{33}-2q^{31}-4q^{29}+4q^{27}+6q^{25}+q^{23}-q^{21}-13q^{19}-18q^{17}+4q^{15}+34q^{13}+37q^{11}+10q^{9}-47q^{7}-91q^{5}-55q^{3}+55q+158q^{-1}+156q^{-3}-4q^{-5}-225q^{-7}-314q^{-9}-149q^{-11}+214q^{-13}+519q^{-15}+435q^{-17}-62q^{-19}-637q^{-21}-827q^{-23}-347q^{-25}+578q^{-27}+1225q^{-29}+958q^{-31}-169q^{-33}-1401q^{-35}-1708q^{-37}-617q^{-39}+1206q^{-41}+2328q^{-43}+1674q^{-45}-490q^{-47}-2576q^{-49}-2787q^{-51}-672q^{-53}+2279q^{-55}+3643q^{-57}+2059q^{-59}-1415q^{-61}-3997q^{-63}-3386q^{-65}+148q^{-67}+3776q^{-69}+4349q^{-71}+1221q^{-73}-3045q^{-75}-4789q^{-77}-2428q^{-79}+2023q^{-81}+4709q^{-83}+3267q^{-85}-997q^{-87}-4217q^{-89}-3627q^{-91}+95q^{-93}+3529q^{-95}+3648q^{-97}+510q^{-99}-2830q^{-101}-3367q^{-103}-897q^{-105}+2183q^{-107}+3071q^{-109}+1118q^{-111}-1684q^{-113}-2771q^{-115}-1338q^{-117}+1199q^{-119}+2594q^{-121}+1672q^{-123}-685q^{-125}-2452q^{-127}-2175q^{-129}-16q^{-131}+2274q^{-133}+2790q^{-135}+959q^{-137}-1893q^{-139}-3410q^{-141}-2108q^{-143}+1204q^{-145}+3818q^{-147}+3325q^{-149}-161q^{-151}-3825q^{-153}-4394q^{-155}-1116q^{-157}+3324q^{-159}+5039q^{-161}+2407q^{-163}-2327q^{-165}-5079q^{-167}-3463q^{-169}+1071q^{-171}+4497q^{-173}+3972q^{-175}+208q^{-177}-3418q^{-179}-3921q^{-181}-1180q^{-183}+2171q^{-185}+3312q^{-187}+1679q^{-189}-998q^{-191}-2423q^{-193}-1712q^{-195}+161q^{-197}+1511q^{-199}+1397q^{-201}+280q^{-203}-752q^{-205}-956q^{-207}-412q^{-209}+269q^{-211}+557q^{-213}+340q^{-215}-36q^{-217}-252q^{-219}-220q^{-221}-52q^{-223}+94q^{-225}+121q^{-227}+44q^{-229}-23q^{-231}-45q^{-233}-31q^{-235}-3q^{-237}+19q^{-239}+19q^{-241}-2q^{-243}-7q^{-245}-2q^{-247}-4q^{-249}+5q^{-253}+q^{-255}-3q^{-257}+2q^{-259}-2q^{-263}+q^{-265}$

A2 Invariants.

Weight	Invariant
1,0	$1+q^{-4}+2q^{-6}-q^{-8}+3q^{-10}-q^{-12}-3q^{-18}+q^{-20}-2q^{-22}+q^{-24}+q^{-26}-q^{-28}+q^{-30}$
1,1	$q^{4}-2q^{2}+8-16q^{-2}+35q^{-4}-58q^{-6}+102q^{-8}-152q^{-10}+223q^{-12}-294q^{-14}+366q^{-16}-418q^{-18}+434q^{-20}-408q^{-22}+318q^{-24}-180q^{-26}-5q^{-28}+210q^{-30}-420q^{-32}+610q^{-34}-751q^{-36}+832q^{-38}-844q^{-40}+784q^{-42}-661q^{-44}+488q^{-46}-284q^{-48}+80q^{-50}+107q^{-52}-252q^{-54}+348q^{-56}-396q^{-58}+395q^{-60}-360q^{-62}+306q^{-64}-246q^{-66}+187q^{-68}-134q^{-70}+92q^{-72}-60q^{-74}+36q^{-76}-20q^{-78}+10q^{-80}-4q^{-82}+q^{-84}$
2,0	$q^{4}-1+q^{-2}+4q^{-4}+q^{-6}-3q^{-8}+2q^{-10}+9q^{-12}-q^{-14}-8q^{-16}+3q^{-18}+6q^{-20}-7q^{-22}-6q^{-24}+5q^{-26}+3q^{-28}-6q^{-30}+q^{-32}+6q^{-34}-6q^{-36}-3q^{-38}+7q^{-40}-2q^{-42}-7q^{-44}+5q^{-46}+8q^{-48}-5q^{-50}-6q^{-52}+6q^{-54}+4q^{-56}-6q^{-58}-2q^{-60}+5q^{-62}-2q^{-66}-q^{-68}+q^{-70}-q^{-74}+q^{-76}$

A3 Invariants.

Weight	Invariant
0,1,0	$1-q^{-2}+2q^{-4}+3q^{-6}-3q^{-8}+5q^{-10}+6q^{-12}-9q^{-14}+9q^{-16}+7q^{-18}-15q^{-20}+9q^{-22}+7q^{-24}-18q^{-26}+q^{-28}+4q^{-30}-10q^{-32}-4q^{-34}+4q^{-36}+7q^{-38}-4q^{-40}-q^{-42}+16q^{-44}-7q^{-46}-10q^{-48}+17q^{-50}-7q^{-52}-11q^{-54}+12q^{-56}-2q^{-58}-6q^{-60}+5q^{-62}-2q^{-66}+q^{-68}$
1,0,0	$q^{-1}+2q^{-5}+3q^{-9}-q^{-11}+3q^{-13}-q^{-15}+q^{-17}-q^{-19}-q^{-21}-q^{-23}-3q^{-25}+q^{-27}-2q^{-29}+2q^{-31}-q^{-33}+2q^{-35}-q^{-37}+q^{-39}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{-2}+q^{-6}+3q^{-8}+2q^{-10}+q^{-12}+4q^{-14}+4q^{-16}+8q^{-22}+3q^{-24}-6q^{-26}+4q^{-28}+11q^{-30}-10q^{-32}-13q^{-34}+4q^{-36}-4q^{-38}-21q^{-40}-9q^{-42}+8q^{-44}-4q^{-46}-5q^{-48}+17q^{-50}+14q^{-52}-5q^{-54}+8q^{-56}+13q^{-58}-9q^{-60}-10q^{-62}+7q^{-64}+q^{-66}-12q^{-68}-2q^{-70}+9q^{-72}-q^{-74}-7q^{-76}+3q^{-78}+4q^{-80}-2q^{-82}-q^{-84}+q^{-86}$
1,0,0,0	$q^{-2}+2q^{-6}+q^{-8}+q^{-10}+3q^{-12}-q^{-14}+3q^{-16}-q^{-18}+q^{-20}-q^{-24}-q^{-26}-2q^{-28}-q^{-30}-3q^{-32}+q^{-34}-2q^{-36}+2q^{-38}+2q^{-44}-q^{-46}+q^{-48}$

B2 Invariants.

Weight

Invariant

0,1

1-q^{-2}+4q^{-4}-5q^{-6}+9q^{-8}-11q^{-10}+16q^{-12}-17q^{-14}+19q^{-16}-17q^{-18}+13q^{-20}-7q^{-22}-q^{-24}+10q^{-26}-19q^{-28}+28q^{-30}-34q^{-32}+36q^{-34}-36q^{-36}+31q^{-38}-26q^{-40}+15q^{-42}-6q^{-44}-3q^{-46}+10q^{-48}-15q^{-50}+19q^{-52}-19q^{-54}+18q^{-56}-14q^{-58}+10q^{-60}-7q^{-62}+4q^{-64}-2q^{-66}+q^{-68}

1,0

q^{2}-q^{-2}-q^{-4}+3q^{-6}+4q^{-8}-q^{-10}-6q^{-12}-q^{-14}+9q^{-16}+10q^{-18}-6q^{-20}-14q^{-22}+19q^{-26}+11q^{-28}-14q^{-30}-18q^{-32}+5q^{-34}+20q^{-36}+4q^{-38}-18q^{-40}-11q^{-42}+9q^{-44}+10q^{-46}-8q^{-48}-13q^{-50}+3q^{-52}+11q^{-54}-2q^{-56}-13q^{-58}+14q^{-62}+7q^{-64}-12q^{-66}-9q^{-68}+12q^{-70}+16q^{-72}-6q^{-74}-19q^{-76}-2q^{-78}+19q^{-80}+11q^{-82}-13q^{-84}-17q^{-86}+2q^{-88}+15q^{-90}+5q^{-92}-8q^{-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}-q^{-4}+3q^{-6}-2q^{-8}+7q^{-10}-5q^{-12}+10q^{-14}-8q^{-16}+15q^{-18}-13q^{-20}+15q^{-22}-13q^{-24}+15q^{-26}-10q^{-28}+7q^{-30}-3q^{-32}-q^{-34}+4q^{-36}-17q^{-38}+13q^{-40}-24q^{-42}+21q^{-44}-32q^{-46}+26q^{-48}-26q^{-50}+29q^{-52}-20q^{-54}+20q^{-56}-11q^{-58}+13q^{-60}-3q^{-64}+4q^{-66}-10q^{-68}+14q^{-70}-15q^{-72}+12q^{-74}-16q^{-76}+15q^{-78}-10q^{-80}+8q^{-82}-8q^{-84}+6q^{-86}-3q^{-88}+2q^{-90}-2q^{-92}+q^{-94}

G2 Invariants.

Weight

Invariant

1,0

q^{-2}-q^{-4}+4q^{-6}-5q^{-8}+6q^{-10}-4q^{-12}-q^{-14}+12q^{-16}-20q^{-18}+30q^{-20}-30q^{-22}+20q^{-24}+2q^{-26}-32q^{-28}+65q^{-30}-79q^{-32}+73q^{-34}-37q^{-36}-17q^{-38}+73q^{-40}-108q^{-42}+110q^{-44}-70q^{-46}+6q^{-48}+57q^{-50}-93q^{-52}+86q^{-54}-39q^{-56}-18q^{-58}+67q^{-60}-80q^{-62}+48q^{-64}+9q^{-66}-79q^{-68}+121q^{-70}-120q^{-72}+71q^{-74}+7q^{-76}-92q^{-78}+146q^{-80}-158q^{-82}+116q^{-84}-44q^{-86}-46q^{-88}+109q^{-90}-130q^{-92}+103q^{-94}-38q^{-96}-28q^{-98}+71q^{-100}-73q^{-102}+35q^{-104}+21q^{-106}-68q^{-108}+89q^{-110}-64q^{-112}+13q^{-114}+50q^{-116}-94q^{-118}+107q^{-120}-83q^{-122}+37q^{-124}+12q^{-126}-54q^{-128}+72q^{-130}-68q^{-132}+50q^{-134}-20q^{-136}-4q^{-138}+19q^{-140}-29q^{-142}+26q^{-144}-19q^{-146}+11q^{-148}-2q^{-150}-3q^{-152}+5q^{-154}-6q^{-156}+4q^{-158}-2q^{-160}+q^{-162}

.

Computer Talk[hide]

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 56"];

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

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

Out[4]= $-2t^{3}+8t^{2}-14t+17-14t^{-1}+8t^{-2}-2t^{-3}$

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

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

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]= { 65, 4 }

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

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

Out[8]= $q^{10}-3q^{9}+6q^{8}-9q^{7}+10q^{6}-11q^{5}+10q^{4}-7q^{3}+5q^{2}-2q+1$

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

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

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

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

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

Out[10]= $z^{9}a^{-5}+z^{9}a^{-7}+2z^{8}a^{-4}+6z^{8}a^{-6}+4z^{8}a^{-8}+2z^{7}a^{-3}+3z^{7}a^{-5}+7z^{7}a^{-7}+6z^{7}a^{-9}+z^{6}a^{-2}-3z^{6}a^{-4}-14z^{6}a^{-6}-5z^{6}a^{-8}+5z^{6}a^{-10}-6z^{5}a^{-3}-13z^{5}a^{-5}-21z^{5}a^{-7}-11z^{5}a^{-9}+3z^{5}a^{-11}-4z^{4}a^{-2}-3z^{4}a^{-4}+12z^{4}a^{-6}+4z^{4}a^{-8}-6z^{4}a^{-10}+z^{4}a^{-12}+4z^{3}a^{-3}+11z^{3}a^{-5}+21z^{3}a^{-7}+11z^{3}a^{-9}-3z^{3}a^{-11}+5z^{2}a^{-2}+3z^{2}a^{-4}-7z^{2}a^{-6}-2z^{2}a^{-8}+2z^{2}a^{-10}-z^{2}a^{-12}-4za^{-5}-8za^{-7}-4za^{-9}-2a^{-2}+2a^{-6}+a^{-8}$

Vassiliev invariants

V₂ and V₃:

(0, -2)

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

0

-16

0

-32

32

0

{\frac {224}{3}}

{\frac {320}{3}}

112

0

128

0

0

752

96

{\frac {1984}{3}}

{\frac {304}{3}}

16

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 10 56. 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

5

2

-3

13

5

4

1

11

6

5

-1

9

4

5

-1

7

3

6

3

5

2

4

-2

3

1

4

3

1

-1

1

Integral Khovanov Homology

(db, data source)

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

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[10, 56]]
Out[2]=	10
In[3]:=	PD[Knot[10, 56]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 4, 11, 3], X[12, 6, 13, 5], X[18, 14, 19, 13], X[16, 7, 17, 8], X[6, 17, 7, 18], X[20, 16, 1, 15], X[14, 20, 15, 19], X[8, 12, 9, 11], X[2, 10, 3, 9]]
In[4]:=	GaussCode[Knot[10, 56]]
Out[4]=	GaussCode[1, -10, 2, -1, 3, -6, 5, -9, 10, -2, 9, -3, 4, -8, 7, -5, 6, -4, 8, -7]
In[5]:=	BR[Knot[10, 56]]
Out[5]=	BR[4, {1, 1, 1, 2, -1, 2, -3, 2, 2, 2, -3}]
In[6]:=	alex = Alexander[Knot[10, 56]][t]
Out[6]=	2 8 14 2 3 17 - -- + -- - -- - 14 t + 8 t - 2 t 3 2 t t t
In[7]:=	Conway[Knot[10, 56]][z]
Out[7]=	4 6 1 - 4 z - 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 25], Knot[10, 56], Knot[11, Alternating, 140]}
In[9]:=	{KnotDet[Knot[10, 56]], KnotSignature[Knot[10, 56]]}
Out[9]=	{65, 4}
In[10]:=	J=Jones[Knot[10, 56]][q]
Out[10]=	2 3 4 5 6 7 8 9 10 1 - 2 q + 5 q - 7 q + 10 q - 11 q + 10 q - 9 q + 6 q - 3 q + q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] \|\| (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[10, 25], Knot[10, 56]}
In[12]:=	A2Invariant[Knot[10, 56]][q]
Out[12]=	4 6 8 10 12 18 20 22 24 26 1 + q + 2 q - q + 3 q - q - 3 q + q - 2 q + q + q - 28 30 q + q
In[13]:=	Kauffman[Knot[10, 56]][a, z]
Out[13]=	2 2 2 2 2 -8 2 2 4 z 8 z 4 z z 2 z 2 z 7 z 3 z a + -- - -- - --- - --- - --- - --- + ---- - ---- - ---- + ---- + 6 2 9 7 5 12 10 8 6 4 a a a a a a a a a a 2 3 3 3 3 3 4 4 4 5 z 3 z 11 z 21 z 11 z 4 z z 6 z 4 z ---- - ---- + ----- + ----- + ----- + ---- + --- - ---- + ---- + 2 11 9 7 5 3 12 10 8 a a a a a a a a a 4 4 4 5 5 5 5 5 6 12 z 3 z 4 z 3 z 11 z 21 z 13 z 6 z 5 z ----- - ---- - ---- + ---- - ----- - ----- - ----- - ---- + ---- - 6 4 2 11 9 7 5 3 10 a a a a a a a a a 6 6 6 6 7 7 7 7 8 8 5 z 14 z 3 z z 6 z 7 z 3 z 2 z 4 z 6 z ---- - ----- - ---- + -- + ---- + ---- + ---- + ---- + ---- + ---- + 8 6 4 2 9 7 5 3 8 6 a a a a a a a a a a 8 9 9 2 z z z ---- + -- + -- 4 7 5 a a a
In[14]:=	{Vassiliev[2][Knot[10, 56]], Vassiliev[3][Knot[10, 56]]}
Out[14]=	{0, -2}
In[15]:=	Kh[Knot[10, 56]][q, t]
Out[15]=	3 3 5 1 q q 5 7 7 2 9 2 4 q + 2 q + ---- + - + -- + 4 q t + 3 q t + 6 q t + 4 q t + 2 t t q t 9 3 11 3 11 4 13 4 13 5 15 5 5 q t + 6 q t + 5 q t + 5 q t + 4 q t + 5 q t + 15 6 17 6 17 7 19 7 21 8 2 q t + 4 q t + q t + 2 q t + q t

@@ Line 1: / Line 1: @@
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+<!-- -->
+<!--  -->
+<!-- -->
 <!-- provide an anchor so we can return to the top of the page -->
 <span id="top"></span>
+<!-- -->
 <!-- this relies on transclusion for next and previous links -->
 {{Knot Navigation Links|ext=gif}}
+{{Rolfsen Knot Page Header|n=10|k=56|KnotilusURL=http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/1,-10,2,-1,3,-6,5,-9,10,-2,9,-3,4,-8,7,-5,6,-4,8,-7/goTop.html}}
-{| align=left
-|- valign=top
-|[[Image:{{PAGENAME}}.gif]]
-|{{Rolfsen Knot Site Links|n=10|k=56|KnotilusURL=http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/1,-10,2,-1,3,-6,5,-9,10,-2,9,-3,4,-8,7,-5,6,-4,8,-7/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=13.3333%><table cellpadding=0 cellspacing=0>
@@ Line 48: / Line 41: @@
 <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>&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>&nbsp;</td><td>1</td></tr>
-</table></center>
+</table>}}
 {{Computer Talk Header}}
@@ Line 141: / Line 133: @@
 q   t  + 4 q   t  + q   t  + 2 q   t  + q   t</nowiki></pre></td></tr>
 </table>
+ [[Category:Knot Page]]

10 56: Difference between revisions

Revision as of 20:05, 28 August 2005

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