10 92

10_91

10_93

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10 92 Quick Notes

10 92 Further Notes and Views

Knot presentations

Planar diagram presentation	X4251 X10,4,11,3 X14,6,15,5 X20,16,1,15 X16,12,17,11 X18,7,19,8 X12,18,13,17 X6,19,7,20 X8,14,9,13 X2,10,3,9
Gauss code	1, -10, 2, -1, 3, -8, 6, -9, 10, -2, 5, -7, 9, -3, 4, -5, 7, -6, 8, -4
Dowker-Thistlethwaite code	4 10 14 18 2 16 8 20 12 6
Conway Notation	[.21.2.20]

Three dimensional invariants

Symmetry type Chiral Unknotting number 2 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index 1 Maximal Thurston-Bennequin number [1][-13] Hyperbolic Volume 14.8554 A-Polynomial See Data:10 92/A-polynomial

[edit Notes for 10 92'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 92's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-2t^{3}+10t^{2}-20t+25-20t^{-1}+10t^{-2}-2t^{-3}$
Conway polynomial	$-2z^{6}-2z^{4}+2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 89, 4 }
Jones polynomial	$q^{10}-4q^{9}+8q^{8}-12q^{7}+14q^{6}-15q^{5}+14q^{4}-10q^{3}+7q^{2}-3q+1$
HOMFLY-PT polynomial (db, data sources)	$-z^{6}a^{-4}-z^{6}a^{-6}+z^{4}a^{-2}-2z^{4}a^{-4}-2z^{4}a^{-6}+z^{4}a^{-8}+2z^{2}a^{-2}-z^{2}a^{-6}+z^{2}a^{-8}+a^{-2}+a^{-4}-a^{-6}$
Kauffman polynomial (db, data sources)	$2z^{9}a^{-5}+2z^{9}a^{-7}+4z^{8}a^{-4}+11z^{8}a^{-6}+7z^{8}a^{-8}+3z^{7}a^{-3}+5z^{7}a^{-5}+12z^{7}a^{-7}+10z^{7}a^{-9}+z^{6}a^{-2}-8z^{6}a^{-4}-22z^{6}a^{-6}-5z^{6}a^{-8}+8z^{6}a^{-10}-8z^{5}a^{-3}-22z^{5}a^{-5}-32z^{5}a^{-7}-14z^{5}a^{-9}+4z^{5}a^{-11}-3z^{4}a^{-2}+2z^{4}a^{-4}+10z^{4}a^{-6}-4z^{4}a^{-8}-8z^{4}a^{-10}+z^{4}a^{-12}+6z^{3}a^{-3}+18z^{3}a^{-5}+21z^{3}a^{-7}+7z^{3}a^{-9}-2z^{3}a^{-11}+3z^{2}a^{-2}+z^{2}a^{-4}-2z^{2}a^{-6}+2z^{2}a^{-8}+2z^{2}a^{-10}-za^{-3}-5za^{-5}-5za^{-7}-za^{-9}-a^{-2}+a^{-4}+a^{-6}$
The A2 invariant	$1-q^{-2}+q^{-4}+2q^{-6}-2q^{-8}+4q^{-10}-q^{-12}+q^{-14}+q^{-16}-3q^{-18}+2q^{-20}-3q^{-22}+q^{-24}+q^{-26}-2q^{-28}+q^{-30}$
The G2 invariant	$q^{-2}-2q^{-4}+6q^{-6}-10q^{-8}+13q^{-10}-12q^{-12}+3q^{-14}+19q^{-16}-45q^{-18}+75q^{-20}-88q^{-22}+65q^{-24}-7q^{-26}-85q^{-28}+181q^{-30}-229q^{-32}+207q^{-34}-97q^{-36}-66q^{-38}+227q^{-40}-319q^{-42}+300q^{-44}-165q^{-46}-29q^{-48}+202q^{-50}-276q^{-52}+226q^{-54}-72q^{-56}-100q^{-58}+223q^{-60}-234q^{-62}+120q^{-64}+62q^{-66}-250q^{-68}+358q^{-70}-329q^{-72}+175q^{-74}+56q^{-76}-283q^{-78}+422q^{-80}-428q^{-82}+287q^{-84}-63q^{-86}-176q^{-88}+333q^{-90}-352q^{-92}+236q^{-94}-38q^{-96}-143q^{-98}+230q^{-100}-197q^{-102}+55q^{-104}+115q^{-106}-235q^{-108}+256q^{-110}-158q^{-112}-6q^{-114}+173q^{-116}-275q^{-118}+278q^{-120}-193q^{-122}+61q^{-124}+66q^{-126}-157q^{-128}+184q^{-130}-156q^{-132}+99q^{-134}-28q^{-136}-26q^{-138}+54q^{-140}-65q^{-142}+54q^{-144}-34q^{-146}+16q^{-148}+q^{-150}-8q^{-152}+10q^{-154}-10q^{-156}+6q^{-158}-3q^{-160}+q^{-162}$

Further Quantum Invariants

Further quantum knot invariants for 10_92.

A1 Invariants.

Weight	Invariant
1	$q-2q^{-1}+4q^{-3}-3q^{-5}+4q^{-7}-q^{-9}-q^{-11}+2q^{-13}-4q^{-15}+4q^{-17}-3q^{-19}+q^{-21}$
2	$q^{6}-2q^{4}-q^{2}+9-6q^{-2}-13q^{-4}+24q^{-6}+q^{-8}-34q^{-10}+27q^{-12}+21q^{-14}-41q^{-16}+11q^{-18}+30q^{-20}-26q^{-22}-11q^{-24}+22q^{-26}+3q^{-28}-25q^{-30}+2q^{-32}+33q^{-34}-25q^{-36}-21q^{-38}+43q^{-40}-12q^{-42}-28q^{-44}+28q^{-46}+q^{-48}-15q^{-50}+8q^{-52}+q^{-54}-3q^{-56}+q^{-58}$
3	$q^{15}-2q^{13}-q^{11}+4q^{9}+6q^{7}-9q^{5}-19q^{3}+13q+44q^{-1}-3q^{-3}-77q^{-5}-33q^{-7}+111q^{-9}+92q^{-11}-114q^{-13}-172q^{-15}+85q^{-17}+247q^{-19}-14q^{-21}-291q^{-23}-75q^{-25}+293q^{-27}+168q^{-29}-258q^{-31}-233q^{-33}+192q^{-35}+269q^{-37}-116q^{-39}-282q^{-41}+44q^{-43}+259q^{-45}+34q^{-47}-229q^{-49}-106q^{-51}+176q^{-53}+181q^{-55}-110q^{-57}-243q^{-59}+20q^{-61}+288q^{-63}+77q^{-65}-295q^{-67}-168q^{-69}+262q^{-71}+233q^{-73}-192q^{-75}-251q^{-77}+108q^{-79}+229q^{-81}-42q^{-83}-174q^{-85}-q^{-87}+109q^{-89}+19q^{-91}-60q^{-93}-16q^{-95}+30q^{-97}+5q^{-99}-10q^{-101}-3q^{-103}+5q^{-105}+q^{-107}-3q^{-109}+q^{-111}$
4	$q^{28}-2q^{26}-q^{24}+4q^{22}+q^{20}+3q^{18}-15q^{16}-13q^{14}+21q^{12}+30q^{10}+38q^{8}-61q^{6}-117q^{4}-19q^{2}+116+269q^{-2}+28q^{-4}-330q^{-6}-394q^{-8}-75q^{-10}+664q^{-12}+656q^{-14}-103q^{-16}-949q^{-18}-1039q^{-20}+406q^{-22}+1467q^{-24}+1122q^{-26}-627q^{-28}-2170q^{-30}-1028q^{-32}+1205q^{-34}+2480q^{-36}+964q^{-38}-2076q^{-40}-2550q^{-42}-370q^{-44}+2572q^{-46}+2568q^{-48}-683q^{-50}-2837q^{-52}-1928q^{-54}+1465q^{-56}+2993q^{-58}+748q^{-60}-2054q^{-62}-2479q^{-64}+263q^{-66}+2457q^{-68}+1487q^{-70}-1081q^{-72}-2308q^{-74}-596q^{-76}+1663q^{-78}+1884q^{-80}-133q^{-82}-1952q^{-84}-1480q^{-86}+643q^{-88}+2233q^{-90}+1164q^{-92}-1199q^{-94}-2451q^{-96}-951q^{-98}+2029q^{-100}+2574q^{-102}+342q^{-104}-2637q^{-106}-2630q^{-108}+727q^{-110}+2948q^{-112}+2022q^{-114}-1453q^{-116}-3072q^{-118}-863q^{-120}+1790q^{-122}+2450q^{-124}+104q^{-126}-1962q^{-128}-1354q^{-130}+315q^{-132}+1520q^{-134}+670q^{-136}-631q^{-138}-778q^{-140}-264q^{-142}+507q^{-144}+394q^{-146}-67q^{-148}-204q^{-150}-174q^{-152}+96q^{-154}+98q^{-156}-q^{-158}-15q^{-160}-44q^{-162}+17q^{-164}+13q^{-166}-6q^{-168}+2q^{-170}-6q^{-172}+5q^{-174}+q^{-176}-3q^{-178}+q^{-180}$
5	$q^{45}-2q^{43}-q^{41}+4q^{39}+q^{37}-2q^{35}-3q^{33}-9q^{31}-5q^{29}+24q^{27}+36q^{25}+8q^{23}-40q^{21}-97q^{19}-89q^{17}+41q^{15}+232q^{13}+284q^{11}+74q^{9}-338q^{7}-670q^{5}-512q^{3}+244q+1159q^{-1}+1393q^{-3}+420q^{-5}-1360q^{-7}-2638q^{-9}-2020q^{-11}+671q^{-13}+3748q^{-15}+4487q^{-17}+1516q^{-19}-3666q^{-21}-7192q^{-23}-5449q^{-25}+1493q^{-27}+8853q^{-29}+10312q^{-31}+3344q^{-33}-7926q^{-35}-14714q^{-37}-10269q^{-39}+3624q^{-41}+16694q^{-43}+17663q^{-45}+3934q^{-47}-14899q^{-49}-23458q^{-51}-13256q^{-53}+9106q^{-55}+25834q^{-57}+22167q^{-59}-386q^{-61}-23990q^{-63}-28639q^{-65}-9350q^{-67}+18542q^{-69}+31428q^{-71}+17884q^{-73}-10903q^{-75}-30419q^{-77}-23837q^{-79}+2987q^{-81}+26597q^{-83}+26569q^{-85}+3629q^{-87}-21302q^{-89}-26442q^{-91}-8267q^{-93}+15967q^{-95}+24539q^{-97}+10786q^{-99}-11497q^{-101}-21849q^{-103}-11951q^{-105}+8041q^{-107}+19517q^{-109}+12632q^{-111}-5369q^{-113}-17777q^{-115}-13776q^{-117}+2565q^{-119}+16606q^{-121}+15936q^{-123}+1127q^{-125}-15198q^{-127}-19017q^{-129}-6419q^{-131}+12642q^{-133}+22214q^{-135}+13326q^{-137}-7954q^{-139}-24321q^{-141}-21034q^{-143}+851q^{-145}+23869q^{-147}+28018q^{-149}+8190q^{-151}-20034q^{-153}-32512q^{-155}-17554q^{-157}+12908q^{-159}+32986q^{-161}+25248q^{-163}-3741q^{-165}-29118q^{-167}-29445q^{-169}-5295q^{-171}+21820q^{-173}+29170q^{-175}+12189q^{-177}-13005q^{-179}-25026q^{-181}-15577q^{-183}+4974q^{-185}+18497q^{-187}+15350q^{-189}+806q^{-191}-11613q^{-193}-12610q^{-195}-3736q^{-197}+5992q^{-199}+8795q^{-201}+4272q^{-203}-2239q^{-205}-5256q^{-207}-3493q^{-209}+328q^{-211}+2711q^{-213}+2263q^{-215}+317q^{-217}-1146q^{-219}-1230q^{-221}-399q^{-223}+416q^{-225}+592q^{-227}+226q^{-229}-119q^{-231}-217q^{-233}-121q^{-235}+23q^{-237}+83q^{-239}+48q^{-241}-15q^{-243}-24q^{-245}-5q^{-247}+2q^{-251}+6q^{-253}-q^{-255}-6q^{-257}+5q^{-259}+q^{-261}-3q^{-263}+q^{-265}$

A2 Invariants.

Weight	Invariant
1,0	$1-q^{-2}+q^{-4}+2q^{-6}-2q^{-8}+4q^{-10}-q^{-12}+q^{-14}+q^{-16}-3q^{-18}+2q^{-20}-3q^{-22}+q^{-24}+q^{-26}-2q^{-28}+q^{-30}$
1,1	$q^{4}-4q^{2}+14-36q^{-2}+82q^{-4}-162q^{-6}+298q^{-8}-484q^{-10}+726q^{-12}-994q^{-14}+1244q^{-16}-1420q^{-18}+1479q^{-20}-1354q^{-22}+1034q^{-24}-528q^{-26}-104q^{-28}+814q^{-30}-1524q^{-32}+2132q^{-34}-2575q^{-36}+2796q^{-38}-2780q^{-40}+2510q^{-42}-2033q^{-44}+1402q^{-46}-700q^{-48}+16q^{-50}+570q^{-52}-998q^{-54}+1256q^{-56}-1340q^{-58}+1276q^{-60}-1116q^{-62}+916q^{-64}-706q^{-66}+506q^{-68}-344q^{-70}+226q^{-72}-136q^{-74}+73q^{-76}-38q^{-78}+18q^{-80}-6q^{-82}+q^{-84}$
2,0	$q^{4}-q^{2}-2+3q^{-2}+5q^{-4}-2q^{-6}-8q^{-8}+4q^{-10}+14q^{-12}-7q^{-14}-14q^{-16}+10q^{-18}+15q^{-20}-10q^{-22}-9q^{-24}+14q^{-26}+7q^{-28}-11q^{-30}+q^{-32}+10q^{-34}-10q^{-36}-6q^{-38}+10q^{-40}-8q^{-42}-14q^{-44}+8q^{-46}+14q^{-48}-12q^{-50}-10q^{-52}+16q^{-54}+8q^{-56}-13q^{-58}-5q^{-60}+11q^{-62}+2q^{-64}-5q^{-66}-2q^{-68}+3q^{-70}-2q^{-74}+q^{-76}$

A3 Invariants.

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

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{-2}-q^{-4}+4q^{-8}-q^{-10}-4q^{-12}+6q^{-14}+6q^{-16}-8q^{-18}-2q^{-20}+18q^{-22}+4q^{-24}-15q^{-26}+11q^{-28}+26q^{-30}-17q^{-32}-19q^{-34}+20q^{-36}+3q^{-38}-32q^{-40}-3q^{-42}+20q^{-44}-14q^{-46}-15q^{-48}+23q^{-50}+9q^{-52}-24q^{-54}+9q^{-56}+23q^{-58}-17q^{-60}-15q^{-62}+20q^{-64}+7q^{-66}-20q^{-68}-2q^{-70}+16q^{-72}-2q^{-74}-12q^{-76}+4q^{-78}+7q^{-80}-3q^{-82}-2q^{-84}+q^{-86}$
1,0,0,0	$q^{-2}-q^{-4}+2q^{-6}+3q^{-12}-2q^{-14}+4q^{-16}-q^{-18}+2q^{-20}+q^{-22}-q^{-28}-3q^{-32}+2q^{-34}-3q^{-36}+2q^{-38}-q^{-40}-q^{-42}+2q^{-44}-2q^{-46}+q^{-48}$

B2 Invariants.

Weight	Invariant
0,1	$1-2q^{-2}+6q^{-4}-10q^{-6}+17q^{-8}-24q^{-10}+31q^{-12}-35q^{-14}+38q^{-16}-33q^{-18}+26q^{-20}-12q^{-22}-4q^{-24}+23q^{-26}-41q^{-28}+57q^{-30}-68q^{-32}+72q^{-34}-69q^{-36}+59q^{-38}-45q^{-40}+26q^{-42}-7q^{-44}-10q^{-46}+23q^{-48}-33q^{-50}+37q^{-52}-37q^{-54}+32q^{-56}-25q^{-58}+18q^{-60}-11q^{-62}+6q^{-64}-3q^{-66}+q^{-68}$
1,0	$q^{2}-2q^{-2}-2q^{-4}+4q^{-6}+7q^{-8}-2q^{-10}-13q^{-12}-5q^{-14}+18q^{-16}+19q^{-18}-13q^{-20}-30q^{-22}-q^{-24}+38q^{-26}+21q^{-28}-28q^{-30}-34q^{-32}+13q^{-34}+40q^{-36}+6q^{-38}-34q^{-40}-17q^{-42}+22q^{-44}+20q^{-46}-16q^{-48}-23q^{-50}+9q^{-52}+23q^{-54}-6q^{-56}-27q^{-58}+28q^{-62}+9q^{-64}-28q^{-66}-19q^{-68}+25q^{-70}+30q^{-72}-15q^{-74}-38q^{-76}-q^{-78}+38q^{-80}+19q^{-82}-25q^{-84}-30q^{-86}+7q^{-88}+27q^{-90}+8q^{-92}-15q^{-94}-14q^{-96}+3q^{-98}+10q^{-100}+3q^{-102}-3q^{-104}-3q^{-106}+q^{-110}$

D4 Invariants.

Weight

Invariant

1,0,0,0

$q^{-2}-2q^{-4}+4q^{-6}-5q^{-8}+10q^{-10}-13q^{-12}+18q^{-14}-20q^{-16}+27q^{-18}-28q^{-20}+30q^{-22}-26q^{-24}+30q^{-26}-19q^{-28}+14q^{-30}-3q^{-32}-2q^{-34}+14q^{-36}-31q^{-38}+33q^{-40}-44q^{-42}+48q^{-44}-59q^{-46}+53q^{-48}-52q^{-50}+52q^{-52}-42q^{-54}+31q^{-56}-23q^{-58}+17q^{-60}-10q^{-64}+13q^{-66}-21q^{-68}+30q^{-70}-29q^{-72}+25q^{-74}-29q^{-76}+27q^{-78}-18q^{-80}+14q^{-82}-14q^{-84}+10q^{-86}-4q^{-88}+3q^{-90}-3q^{-92}+q^{-94}$

G2 Invariants.

Weight

Invariant

1,0

$q^{-2}-2q^{-4}+6q^{-6}-10q^{-8}+13q^{-10}-12q^{-12}+3q^{-14}+19q^{-16}-45q^{-18}+75q^{-20}-88q^{-22}+65q^{-24}-7q^{-26}-85q^{-28}+181q^{-30}-229q^{-32}+207q^{-34}-97q^{-36}-66q^{-38}+227q^{-40}-319q^{-42}+300q^{-44}-165q^{-46}-29q^{-48}+202q^{-50}-276q^{-52}+226q^{-54}-72q^{-56}-100q^{-58}+223q^{-60}-234q^{-62}+120q^{-64}+62q^{-66}-250q^{-68}+358q^{-70}-329q^{-72}+175q^{-74}+56q^{-76}-283q^{-78}+422q^{-80}-428q^{-82}+287q^{-84}-63q^{-86}-176q^{-88}+333q^{-90}-352q^{-92}+236q^{-94}-38q^{-96}-143q^{-98}+230q^{-100}-197q^{-102}+55q^{-104}+115q^{-106}-235q^{-108}+256q^{-110}-158q^{-112}-6q^{-114}+173q^{-116}-275q^{-118}+278q^{-120}-193q^{-122}+61q^{-124}+66q^{-126}-157q^{-128}+184q^{-130}-156q^{-132}+99q^{-134}-28q^{-136}-26q^{-138}+54q^{-140}-65q^{-142}+54q^{-144}-34q^{-146}+16q^{-148}+q^{-150}-8q^{-152}+10q^{-154}-10q^{-156}+6q^{-158}-3q^{-160}+q^{-162}$

.

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. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["10 92"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-2t^{3}+10t^{2}-20t+25-20t^{-1}+10t^{-2}-2t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

$-2z^{6}-2z^{4}+2z^{2}+1$

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]=

{ 89, 4 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^{10}-4q^{9}+8q^{8}-12q^{7}+14q^{6}-15q^{5}+14q^{4}-10q^{3}+7q^{2}-3q+1$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-z^{6}a^{-4}-z^{6}a^{-6}+z^{4}a^{-2}-2z^{4}a^{-4}-2z^{4}a^{-6}+z^{4}a^{-8}+2z^{2}a^{-2}-z^{2}a^{-6}+z^{2}a^{-8}+a^{-2}+a^{-4}-a^{-6}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$2z^{9}a^{-5}+2z^{9}a^{-7}+4z^{8}a^{-4}+11z^{8}a^{-6}+7z^{8}a^{-8}+3z^{7}a^{-3}+5z^{7}a^{-5}+12z^{7}a^{-7}+10z^{7}a^{-9}+z^{6}a^{-2}-8z^{6}a^{-4}-22z^{6}a^{-6}-5z^{6}a^{-8}+8z^{6}a^{-10}-8z^{5}a^{-3}-22z^{5}a^{-5}-32z^{5}a^{-7}-14z^{5}a^{-9}+4z^{5}a^{-11}-3z^{4}a^{-2}+2z^{4}a^{-4}+10z^{4}a^{-6}-4z^{4}a^{-8}-8z^{4}a^{-10}+z^{4}a^{-12}+6z^{3}a^{-3}+18z^{3}a^{-5}+21z^{3}a^{-7}+7z^{3}a^{-9}-2z^{3}a^{-11}+3z^{2}a^{-2}+z^{2}a^{-4}-2z^{2}a^{-6}+2z^{2}a^{-8}+2z^{2}a^{-10}-za^{-3}-5za^{-5}-5za^{-7}-za^{-9}-a^{-2}+a^{-4}+a^{-6}$

Vassiliev invariants

V2 and V3:

(2, 3)

V2,1 through V6,9:

V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9 $8$ $24$ $32$ ${\frac {316}{3}}$ ${\frac {92}{3}}$ $192$ $464$ $128$ $88$ ${\frac {256}{3}}$ $288$ ${\frac {2528}{3}}$ ${\frac {736}{3}}$ ${\frac {29311}{15}}$ ${\frac {1276}{15}}$ ${\frac {44644}{45}}$ ${\frac {161}{9}}$ ${\frac {2191}{15}}$

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 10 92. 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	1
19										3		-3
17									5	1		4
15								7	3			-4
13							7	5				2
11						8	7					-1
9					6	7						-1
7				4	8							4
5			3	6								-3
3		1	5									4
1		2										-2
-1	1											1

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, 92]]
Out[2]=	10
In[3]:=	PD[Knot[10, 92]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 4, 11, 3], X[14, 6, 15, 5], X[20, 16, 1, 15], X[16, 12, 17, 11], X[18, 7, 19, 8], X[12, 18, 13, 17], X[6, 19, 7, 20], X[8, 14, 9, 13], X[2, 10, 3, 9]]
In[4]:=	GaussCode[Knot[10, 92]]
Out[4]=	GaussCode[1, -10, 2, -1, 3, -8, 6, -9, 10, -2, 5, -7, 9, -3, 4, -5, 7, -6, 8, -4]
In[5]:=	BR[Knot[10, 92]]
Out[5]=	BR[4, {1, 1, 1, 2, 2, -3, 2, -1, 2, -3, 2}]
In[6]:=	alex = Alexander[Knot[10, 92]][t]
Out[6]=	2 10 20 2 3 25 - -- + -- - -- - 20 t + 10 t - 2 t 3 2 t t t
In[7]:=	Conway[Knot[10, 92]][z]
Out[7]=	2 4 6 1 + 2 z - 2 z - 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 92], Knot[11, Alternating, 153], Knot[11, Alternating, 224], Knot[11, NonAlternating, 35], Knot[11, NonAlternating, 43]}
In[9]:=	{KnotDet[Knot[10, 92]], KnotSignature[Knot[10, 92]]}
Out[9]=	{89, 4}
In[10]:=	J=Jones[Knot[10, 92]][q]
Out[10]=	2 3 4 5 6 7 8 9 1 - 3 q + 7 q - 10 q + 14 q - 15 q + 14 q - 12 q + 8 q - 4 q + 10 q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[10, 92]}
In[12]:=	A2Invariant[Knot[10, 92]][q]
Out[12]=	2 4 6 8 10 12 14 16 18 20 1 - q + q + 2 q - 2 q + 4 q - q + q + q - 3 q + 2 q - 22 24 26 28 30 3 q + q + q - 2 q + q
In[13]:=	Kauffman[Knot[10, 92]][a, z]
Out[13]=	2 2 2 2 -6 -4 -2 z 5 z 5 z z 2 z 2 z 2 z z a + a - a - -- - --- - --- - -- + ---- + ---- - ---- + -- + 9 7 5 3 10 8 6 4 a a a a a a a a 2 3 3 3 3 3 4 4 4 3 z 2 z 7 z 21 z 18 z 6 z z 8 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 10 z 2 z 3 z 4 z 14 z 32 z 22 z 8 z 8 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 5 z 22 z 8 z z 10 z 12 z 5 z 3 z 7 z ---- - ----- - ---- + -- + ----- + ----- + ---- + ---- + ---- + 8 6 4 2 9 7 5 3 8 a a a a a a a a a 8 8 9 9 11 z 4 z 2 z 2 z ----- + ---- + ---- + ---- 6 4 7 5 a a a a
In[14]:=	{Vassiliev[2][Knot[10, 92]], Vassiliev[3][Knot[10, 92]]}
Out[14]=	{0, 3}
In[15]:=	Kh[Knot[10, 92]][q, t]
Out[15]=	3 3 5 1 2 q q 5 7 7 2 9 2 5 q + 3 q + ---- + --- + -- + 6 q t + 4 q t + 8 q t + 6 q t + 2 t t q t 9 3 11 3 11 4 13 4 13 5 15 5 7 q t + 8 q t + 7 q t + 7 q t + 5 q t + 7 q t + 15 6 17 6 17 7 19 7 21 8 3 q t + 5 q t + q t + 3 q t + q t