10 106

10_105

10_107

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10 106 Further Notes and Views

Knot presentations

Planar diagram presentation	X6271 X16,8,17,7 X10,3,11,4 X2,15,3,16 X14,5,15,6 X4,11,5,12 X18,10,19,9 X20,14,1,13 X8,18,9,17 X12,20,13,19
Gauss code	1, -4, 3, -6, 5, -1, 2, -9, 7, -3, 6, -10, 8, -5, 4, -2, 9, -7, 10, -8
Dowker-Thistlethwaite code	6 10 14 16 18 4 20 2 8 12
Conway Notation	[30:2:20]

Three dimensional invariants

Symmetry type Chiral Unknotting number 2 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index 1 Maximal Thurston-Bennequin number [-3][-9] Hyperbolic Volume 13.933 A-Polynomial See Data:10 106/A-polynomial

[edit Notes for 10 106's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus $1$ Topological 4 genus $1$ Concordance genus $1$ Rasmussen s-Invariant -2

[edit Notes for 10 106's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{4}+4t^{3}-9t^{2}+15t-17+15t^{-1}-9t^{-2}+4t^{-3}-t^{-4}$
Conway polynomial	$-z^{8}-4z^{6}-5z^{4}-z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 75, 2 }
Jones polynomial	$q^{7}-3q^{6}+6q^{5}-10q^{4}+12q^{3}-12q^{2}+12q-9+6q^{-1}-3q^{-2}+q^{-3}$
HOMFLY-PT polynomial (db, data sources)	$-z^{8}a^{-2}-6z^{6}a^{-2}+z^{6}a^{-4}+z^{6}-13z^{4}a^{-2}+4z^{4}a^{-4}+4z^{4}-11z^{2}a^{-2}+5z^{2}a^{-4}+5z^{2}-2a^{-2}+a^{-4}+2$
Kauffman polynomial (db, data sources)	$2z^{9}a^{-1}+2z^{9}a^{-3}+9z^{8}a^{-2}+5z^{8}a^{-4}+4z^{8}+3az^{7}+z^{7}a^{-1}+4z^{7}a^{-3}+6z^{7}a^{-5}+a^{2}z^{6}-23z^{6}a^{-2}-6z^{6}a^{-4}+5z^{6}a^{-6}-11z^{6}-9az^{5}-12z^{5}a^{-1}-13z^{5}a^{-3}-7z^{5}a^{-5}+3z^{5}a^{-7}-3a^{2}z^{4}+22z^{4}a^{-2}+4z^{4}a^{-4}-5z^{4}a^{-6}+z^{4}a^{-8}+9z^{4}+7az^{3}+9z^{3}a^{-1}+8z^{3}a^{-3}+3z^{3}a^{-5}-3z^{3}a^{-7}+2a^{2}z^{2}-13z^{2}a^{-2}-3z^{2}a^{-4}+2z^{2}a^{-6}-z^{2}a^{-8}-5z^{2}-az-2za^{-1}-za^{-3}+za^{-5}+za^{-7}+2a^{-2}+a^{-4}+2$
The A2 invariant	$q^{8}-q^{6}+2q^{4}-q^{2}+2q^{-2}-2q^{-4}+4q^{-6}-2q^{-8}+q^{-10}-q^{-12}-2q^{-14}+2q^{-16}-q^{-18}+q^{-20}$
The G2 invariant	$q^{46}-2q^{44}+5q^{42}-9q^{40}+10q^{38}-10q^{36}+2q^{34}+16q^{32}-37q^{30}+60q^{28}-66q^{26}+44q^{24}+4q^{22}-72q^{20}+133q^{18}-156q^{16}+126q^{14}-39q^{12}-72q^{10}+165q^{8}-193q^{6}+153q^{4}-51q^{2}-64+139q^{-2}-149q^{-4}+84q^{-6}+21q^{-8}-110q^{-10}+151q^{-12}-114q^{-14}+21q^{-16}+89q^{-18}-181q^{-20}+214q^{-22}-176q^{-24}+69q^{-26}+67q^{-28}-185q^{-30}+250q^{-32}-225q^{-34}+127q^{-36}+7q^{-38}-128q^{-40}+187q^{-42}-171q^{-44}+81q^{-46}+34q^{-48}-113q^{-50}+129q^{-52}-73q^{-54}-27q^{-56}+114q^{-58}-152q^{-60}+124q^{-62}-52q^{-64}-41q^{-66}+115q^{-68}-146q^{-70}+135q^{-72}-79q^{-74}+16q^{-76}+38q^{-78}-74q^{-80}+81q^{-82}-71q^{-84}+50q^{-86}-19q^{-88}-6q^{-90}+24q^{-92}-31q^{-94}+28q^{-96}-20q^{-98}+11q^{-100}-2q^{-102}-4q^{-104}+5q^{-106}-6q^{-108}+4q^{-110}-2q^{-112}+q^{-114}$

Further Quantum Invariants

Further quantum knot invariants for 10_106.

A1 Invariants.

Weight	Invariant
1	$q^{7}-2q^{5}+3q^{3}-3q+3q^{-1}+2q^{-7}-4q^{-9}+3q^{-11}-2q^{-13}+q^{-15}$
2	$q^{22}-2q^{20}-q^{18}+8q^{16}-5q^{14}-12q^{12}+18q^{10}+4q^{8}-26q^{6}+15q^{4}+17q^{2}-27+3q^{-2}+22q^{-4}-13q^{-6}-9q^{-8}+15q^{-10}+6q^{-12}-17q^{-14}-3q^{-16}+24q^{-18}-15q^{-20}-18q^{-22}+29q^{-24}-4q^{-26}-19q^{-28}+15q^{-30}+q^{-32}-8q^{-34}+5q^{-36}-2q^{-40}+q^{-42}$
3	$q^{45}-2q^{43}-q^{41}+4q^{39}+5q^{37}-8q^{35}-17q^{33}+10q^{31}+37q^{29}+3q^{27}-59q^{25}-37q^{23}+71q^{21}+83q^{19}-55q^{17}-130q^{15}+12q^{13}+160q^{11}+45q^{9}-159q^{7}-102q^{5}+131q^{3}+147q-94q^{-1}-164q^{-3}+48q^{-5}+169q^{-7}-6q^{-9}-156q^{-11}-26q^{-13}+138q^{-15}+59q^{-17}-109q^{-19}-92q^{-21}+67q^{-23}+126q^{-25}-21q^{-27}-149q^{-29}-45q^{-31}+157q^{-33}+103q^{-35}-133q^{-37}-151q^{-39}+96q^{-41}+167q^{-43}-46q^{-45}-148q^{-47}-q^{-49}+113q^{-51}+20q^{-53}-68q^{-55}-21q^{-57}+33q^{-59}+13q^{-61}-15q^{-63}-5q^{-65}+9q^{-67}-q^{-69}-4q^{-71}+q^{-73}+2q^{-75}-2q^{-79}+q^{-81}$
4	$q^{76}-2q^{74}-q^{72}+4q^{70}+q^{68}+2q^{66}-14q^{64}-11q^{62}+19q^{60}+26q^{58}+33q^{56}-47q^{54}-96q^{52}-27q^{50}+75q^{48}+210q^{46}+62q^{44}-199q^{42}-302q^{40}-152q^{38}+366q^{36}+495q^{34}+130q^{32}-467q^{30}-778q^{28}-82q^{26}+697q^{24}+903q^{22}+123q^{20}-1048q^{18}-978q^{16}+37q^{14}+1221q^{12}+1121q^{10}-406q^{8}-1359q^{6}-971q^{4}+666q^{2}+1556+524q^{-2}-942q^{-4}-1426q^{-6}-108q^{-8}+1268q^{-10}+983q^{-12}-352q^{-14}-1274q^{-16}-500q^{-18}+798q^{-20}+996q^{-22}+7q^{-24}-974q^{-26}-667q^{-28}+398q^{-30}+979q^{-32}+360q^{-34}-664q^{-36}-944q^{-38}-179q^{-40}+922q^{-42}+953q^{-44}-23q^{-46}-1153q^{-48}-1068q^{-50}+408q^{-52}+1389q^{-54}+972q^{-56}-734q^{-58}-1666q^{-60}-549q^{-62}+1014q^{-64}+1564q^{-66}+197q^{-68}-1288q^{-70}-1082q^{-72}+96q^{-74}+1161q^{-76}+706q^{-78}-397q^{-80}-743q^{-82}-382q^{-84}+385q^{-86}+465q^{-88}+61q^{-90}-191q^{-92}-255q^{-94}+23q^{-96}+120q^{-98}+56q^{-100}+12q^{-102}-68q^{-104}-6q^{-106}+7q^{-108}-q^{-110}+18q^{-112}-10q^{-114}+4q^{-116}-q^{-118}-6q^{-120}+5q^{-122}-2q^{-124}+2q^{-126}-2q^{-130}+q^{-132}$
5	$q^{115}-2q^{113}-q^{111}+4q^{109}+q^{107}-2q^{105}-4q^{103}-8q^{101}-3q^{99}+22q^{97}+33q^{95}+6q^{93}-37q^{91}-81q^{89}-74q^{87}+25q^{85}+178q^{83}+231q^{81}+85q^{79}-213q^{77}-484q^{75}-440q^{73}+36q^{71}+708q^{69}+1018q^{67}+561q^{65}-545q^{63}-1597q^{61}-1657q^{59}-330q^{57}+1698q^{55}+2876q^{53}+2017q^{51}-708q^{49}-3548q^{47}-4170q^{45}-1535q^{43}+2901q^{41}+5852q^{39}+4665q^{37}-481q^{35}-6115q^{33}-7726q^{31}-3350q^{29}+4339q^{27}+9500q^{25}+7639q^{23}-607q^{21}-9216q^{19}-11152q^{17}-4163q^{15}+6760q^{13}+12846q^{11}+8763q^{9}-2714q^{7}-12414q^{5}-12140q^{3}-1758q+10187q^{-1}+13621q^{-3}+5673q^{-5}-6931q^{-7}-13374q^{-9}-8268q^{-11}+3640q^{-13}+11804q^{-15}+9362q^{-17}-934q^{-19}-9663q^{-21}-9256q^{-23}-771q^{-25}+7594q^{-27}+8418q^{-29}+1607q^{-31}-6002q^{-33}-7460q^{-35}-1929q^{-37}+5004q^{-39}+6855q^{-41}+2218q^{-43}-4368q^{-45}-6831q^{-47}-3018q^{-49}+3666q^{-51}+7297q^{-53}+4619q^{-55}-2340q^{-57}-7850q^{-59}-7018q^{-61}+8q^{-63}+7779q^{-65}+9759q^{-67}+3538q^{-69}-6516q^{-71}-12052q^{-73}-7759q^{-75}+3621q^{-77}+12924q^{-79}+11916q^{-81}+612q^{-83}-11842q^{-85}-14738q^{-87}-5337q^{-89}+8656q^{-91}+15445q^{-93}+9364q^{-95}-4217q^{-97}-13764q^{-99}-11559q^{-101}-274q^{-103}+10203q^{-105}+11546q^{-107}+3667q^{-109}-6029q^{-111}-9622q^{-113}-5228q^{-115}+2330q^{-117}+6697q^{-119}+5152q^{-121}+103q^{-123}-3846q^{-125}-4005q^{-127}-1180q^{-129}+1713q^{-131}+2546q^{-133}+1313q^{-135}-473q^{-137}-1354q^{-139}-989q^{-141}-35q^{-143}+586q^{-145}+567q^{-147}+176q^{-149}-189q^{-151}-286q^{-153}-142q^{-155}+45q^{-157}+109q^{-159}+75q^{-161}+13q^{-163}-34q^{-165}-40q^{-167}-14q^{-169}+14q^{-171}+10q^{-173}+4q^{-175}+5q^{-177}-6q^{-179}-6q^{-181}+3q^{-183}+2q^{-185}-2q^{-187}+2q^{-189}-2q^{-193}+q^{-195}$
6	$q^{162}-2q^{160}-q^{158}+4q^{156}+q^{154}-2q^{152}-8q^{150}+2q^{148}+28q^{142}+20q^{140}-7q^{138}-60q^{136}-58q^{134}-47q^{132}-q^{130}+151q^{128}+228q^{126}+184q^{124}-76q^{122}-314q^{120}-558q^{118}-560q^{116}-23q^{114}+716q^{112}+1351q^{110}+1216q^{108}+463q^{106}-1114q^{104}-2678q^{102}-2921q^{100}-1456q^{98}+1537q^{96}+4345q^{94}+5854q^{92}+3881q^{90}-1166q^{88}-6867q^{86}-10071q^{84}-7998q^{82}-978q^{80}+9250q^{78}+16021q^{76}+14916q^{74}+4752q^{72}-10457q^{70}-22829q^{68}-24985q^{66}-11829q^{64}+10246q^{62}+30703q^{60}+36804q^{58}+22735q^{56}-7299q^{54}-38783q^{52}-51318q^{50}-36410q^{48}+2519q^{46}+45216q^{44}+67760q^{42}+52791q^{40}+3889q^{38}-51963q^{36}-84154q^{34}-69232q^{32}-11890q^{30}+59181q^{28}+100583q^{26}+84373q^{24}+17068q^{22}-66349q^{20}-114780q^{18}-97115q^{16}-17973q^{14}+75300q^{12}+126173q^{10}+102446q^{8}+14232q^{6}-84737q^{4}-134038q^{2}-99524-3780q^{-2}+94336q^{-4}+133757q^{-6}+88598q^{-8}-11232q^{-10}-102497q^{-12}-125121q^{-14}-68754q^{-16}+28975q^{-18}+103992q^{-20}+108501q^{-22}+43447q^{-24}-45877q^{-26}-98057q^{-28}-83655q^{-30}-15916q^{-32}+56151q^{-34}+84595q^{-36}+54398q^{-38}-9259q^{-40}-58910q^{-42}-63956q^{-44}-24327q^{-46}+27408q^{-48}+54507q^{-50}+39931q^{-52}-2396q^{-54}-38911q^{-56}-44649q^{-58}-16038q^{-60}+23464q^{-62}+45468q^{-64}+33221q^{-66}-4535q^{-68}-40860q^{-70}-50087q^{-72}-23278q^{-74}+21846q^{-76}+56889q^{-78}+55912q^{-80}+16776q^{-82}-38527q^{-84}-74395q^{-86}-64416q^{-88}-11493q^{-90}+56103q^{-92}+94238q^{-94}+75439q^{-96}+4528q^{-98}-76288q^{-100}-115181q^{-102}-84044q^{-104}+4965q^{-106}+97437q^{-108}+134564q^{-110}+86668q^{-112}-18467q^{-114}-116520q^{-116}-145346q^{-118}-82483q^{-120}+33295q^{-122}+130209q^{-124}+144654q^{-126}+70030q^{-128}-46524q^{-130}-132155q^{-132}-133183q^{-134}-53004q^{-136}+55728q^{-138}+122461q^{-140}+111514q^{-142}+34764q^{-144}-56323q^{-146}-104450q^{-148}-85533q^{-150}-18001q^{-152}+50216q^{-154}+80520q^{-156}+59713q^{-158}+7170q^{-160}-40695q^{-162}-56810q^{-164}-37106q^{-166}-968q^{-168}+29071q^{-170}+36496q^{-172}+21387q^{-174}-2257q^{-176}-18999q^{-178}-20775q^{-180}-11148q^{-182}+2568q^{-184}+11330q^{-186}+11137q^{-188}+4840q^{-190}-2223q^{-192}-5788q^{-194}-5470q^{-196}-2000q^{-198}+1586q^{-200}+2961q^{-202}+2247q^{-204}+578q^{-206}-724q^{-208}-1423q^{-210}-964q^{-212}-43q^{-214}+454q^{-216}+527q^{-218}+298q^{-220}+61q^{-222}-250q^{-224}-254q^{-226}-59q^{-228}+34q^{-230}+75q^{-232}+64q^{-234}+60q^{-236}-36q^{-238}-51q^{-240}-7q^{-242}-q^{-244}+6q^{-246}+2q^{-248}+19q^{-250}-5q^{-252}-11q^{-254}+3q^{-256}+2q^{-260}-2q^{-262}+2q^{-264}-2q^{-268}+q^{-270}$

A2 Invariants.

Weight	Invariant
1,0	$q^{8}-q^{6}+2q^{4}-q^{2}+2q^{-2}-2q^{-4}+4q^{-6}-2q^{-8}+q^{-10}-q^{-12}-2q^{-14}+2q^{-16}-q^{-18}+q^{-20}$
1,1	$q^{28}-4q^{26}+12q^{24}-30q^{22}+66q^{20}-126q^{18}+222q^{16}-344q^{14}+481q^{12}-624q^{10}+730q^{8}-762q^{6}+707q^{4}-554q^{2}+308+22q^{-2}-377q^{-4}+724q^{-6}-1024q^{-8}+1252q^{-10}-1374q^{-12}+1372q^{-14}-1256q^{-16}+1030q^{-18}-728q^{-20}+394q^{-22}-60q^{-24}-226q^{-26}+442q^{-28}-564q^{-30}+600q^{-32}-588q^{-34}+532q^{-36}-446q^{-38}+358q^{-40}-282q^{-42}+213q^{-44}-150q^{-46}+102q^{-48}-66q^{-50}+38q^{-52}-20q^{-54}+10q^{-56}-4q^{-58}+q^{-60}$
2,0	$q^{24}-q^{22}-q^{20}+4q^{18}+q^{16}-6q^{14}+7q^{10}-8q^{6}+2q^{4}+10q^{2}-6-8q^{-2}+10q^{-4}-6q^{-8}+4q^{-10}+7q^{-12}-3q^{-14}-3q^{-16}+10q^{-18}-q^{-20}-12q^{-22}+2q^{-24}+9q^{-26}-9q^{-28}-4q^{-30}+9q^{-32}+2q^{-34}-4q^{-36}-3q^{-38}+4q^{-40}-3q^{-44}+2q^{-46}-q^{-50}+q^{-52}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{20}-2q^{18}+q^{16}+3q^{14}-8q^{12}+6q^{10}+6q^{8}-15q^{6}+14q^{4}+7q^{2}-19+14q^{-2}+8q^{-4}-18q^{-6}+6q^{-8}+9q^{-10}-7q^{-12}-4q^{-14}+2q^{-16}+7q^{-18}-11q^{-20}-6q^{-22}+21q^{-24}-12q^{-26}-9q^{-28}+23q^{-30}-9q^{-32}-10q^{-34}+14q^{-36}-3q^{-38}-7q^{-40}+5q^{-42}-2q^{-46}+q^{-48}$
1,0,0	$q^{9}-q^{7}+3q^{5}-2q^{3}+3q-2q^{-1}+2q^{-3}-q^{-5}+q^{-7}+q^{-9}-q^{-11}+q^{-13}-3q^{-15}+2q^{-17}-3q^{-19}+3q^{-21}-q^{-23}+q^{-25}$
1,0,1	$q^{34}-4q^{32}+10q^{30}-16q^{28}+13q^{26}+12q^{24}-60q^{22}+117q^{20}-130q^{18}+69q^{16}+82q^{14}-280q^{12}+414q^{10}-408q^{8}+212q^{6}+130q^{4}-473q^{2}+684-630q^{-2}+364q^{-4}+17q^{-6}-321q^{-8}+420q^{-10}-333q^{-12}+152q^{-14}-46q^{-16}+88q^{-18}-240q^{-20}+363q^{-22}-309q^{-24}+47q^{-26}+322q^{-28}-600q^{-30}+648q^{-32}-420q^{-34}+39q^{-36}+320q^{-38}-493q^{-40}+411q^{-42}-167q^{-44}-105q^{-46}+261q^{-48}-244q^{-50}+111q^{-52}+38q^{-54}-125q^{-56}+125q^{-58}-63q^{-60}-5q^{-62}+43q^{-64}-45q^{-66}+24q^{-68}-2q^{-70}-8q^{-72}+8q^{-74}-4q^{-76}+q^{-78}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{22}-q^{20}+3q^{16}-4q^{14}-q^{12}+6q^{10}-4q^{8}-4q^{6}+13q^{4}+2q^{2}-9+8q^{-2}+14q^{-4}-8q^{-6}-13q^{-8}+15q^{-10}+4q^{-12}-22q^{-14}+2q^{-16}+19q^{-18}-19q^{-20}-8q^{-22}+18q^{-24}-5q^{-26}-15q^{-28}+11q^{-30}+13q^{-32}-11q^{-34}-2q^{-36}+16q^{-38}+q^{-40}-15q^{-42}+5q^{-44}+7q^{-46}-8q^{-48}-2q^{-50}+4q^{-52}-q^{-54}-q^{-56}+q^{-58}$
1,0,0,0	$q^{10}-q^{8}+3q^{6}-q^{4}+2q^{2}+1-q^{-2}+2q^{-4}-2q^{-6}+2q^{-8}-2q^{-10}+2q^{-12}-2q^{-14}+q^{-16}-2q^{-18}+q^{-22}-2q^{-24}+3q^{-26}-q^{-28}+q^{-30}$

B2 Invariants.

Weight	Invariant
0,1	$q^{20}-2q^{18}+5q^{16}-9q^{14}+14q^{12}-20q^{10}+24q^{8}-25q^{6}+26q^{4}-21q^{2}+15-4q^{-2}-8q^{-4}+22q^{-6}-34q^{-8}+43q^{-10}-49q^{-12}+50q^{-14}-46q^{-16}+37q^{-18}-25q^{-20}+14q^{-22}-q^{-24}-10q^{-26}+19q^{-28}-25q^{-30}+25q^{-32}-24q^{-34}+20q^{-36}-15q^{-38}+11q^{-40}-7q^{-42}+4q^{-44}-2q^{-46}+q^{-48}$
1,0	$q^{34}-2q^{30}-2q^{28}+3q^{26}+6q^{24}-2q^{22}-11q^{20}-4q^{18}+15q^{16}+14q^{14}-12q^{12}-22q^{10}+2q^{8}+28q^{6}+12q^{4}-23q^{2}-22+12q^{-2}+26q^{-4}+q^{-6}-23q^{-8}-8q^{-10}+17q^{-12}+12q^{-14}-12q^{-16}-12q^{-18}+10q^{-20}+15q^{-22}-7q^{-24}-19q^{-26}+2q^{-28}+20q^{-30}+2q^{-32}-22q^{-34}-11q^{-36}+20q^{-38}+19q^{-40}-13q^{-42}-25q^{-44}+3q^{-46}+27q^{-48}+10q^{-50}-17q^{-52}-18q^{-54}+5q^{-56}+17q^{-58}+5q^{-60}-9q^{-62}-9q^{-64}+q^{-66}+6q^{-68}+2q^{-70}-2q^{-72}-2q^{-74}+q^{-78}$

D4 Invariants.

Weight

Invariant

1,0,0,0

$q^{26}-2q^{24}+3q^{22}-5q^{20}+8q^{18}-12q^{16}+14q^{14}-17q^{12}+21q^{10}-20q^{8}+21q^{6}-17q^{4}+20q^{2}-10+6q^{-2}+2q^{-4}-6q^{-6}+14q^{-8}-25q^{-10}+28q^{-12}-33q^{-14}+36q^{-16}-41q^{-18}+36q^{-20}-35q^{-22}+33q^{-24}-27q^{-26}+17q^{-28}-12q^{-30}+8q^{-32}+4q^{-34}-9q^{-36}+12q^{-38}-16q^{-40}+23q^{-42}-19q^{-44}+16q^{-46}-18q^{-48}+17q^{-50}-11q^{-52}+8q^{-54}-9q^{-56}+6q^{-58}-3q^{-60}+2q^{-62}-2q^{-64}+q^{-66}$

G2 Invariants.

Weight

Invariant

1,0

$q^{46}-2q^{44}+5q^{42}-9q^{40}+10q^{38}-10q^{36}+2q^{34}+16q^{32}-37q^{30}+60q^{28}-66q^{26}+44q^{24}+4q^{22}-72q^{20}+133q^{18}-156q^{16}+126q^{14}-39q^{12}-72q^{10}+165q^{8}-193q^{6}+153q^{4}-51q^{2}-64+139q^{-2}-149q^{-4}+84q^{-6}+21q^{-8}-110q^{-10}+151q^{-12}-114q^{-14}+21q^{-16}+89q^{-18}-181q^{-20}+214q^{-22}-176q^{-24}+69q^{-26}+67q^{-28}-185q^{-30}+250q^{-32}-225q^{-34}+127q^{-36}+7q^{-38}-128q^{-40}+187q^{-42}-171q^{-44}+81q^{-46}+34q^{-48}-113q^{-50}+129q^{-52}-73q^{-54}-27q^{-56}+114q^{-58}-152q^{-60}+124q^{-62}-52q^{-64}-41q^{-66}+115q^{-68}-146q^{-70}+135q^{-72}-79q^{-74}+16q^{-76}+38q^{-78}-74q^{-80}+81q^{-82}-71q^{-84}+50q^{-86}-19q^{-88}-6q^{-90}+24q^{-92}-31q^{-94}+28q^{-96}-20q^{-98}+11q^{-100}-2q^{-102}-4q^{-104}+5q^{-106}-6q^{-108}+4q^{-110}-2q^{-112}+q^{-114}$

.

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["10 106"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t^{4}+4t^{3}-9t^{2}+15t-17+15t^{-1}-9t^{-2}+4t^{-3}-t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

$-z^{8}-4z^{6}-5z^{4}-z^{2}+1$

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]=

{ 75, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-z^{8}a^{-2}-6z^{6}a^{-2}+z^{6}a^{-4}+z^{6}-13z^{4}a^{-2}+4z^{4}a^{-4}+4z^{4}-11z^{2}a^{-2}+5z^{2}a^{-4}+5z^{2}-2a^{-2}+a^{-4}+2$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$2z^{9}a^{-1}+2z^{9}a^{-3}+9z^{8}a^{-2}+5z^{8}a^{-4}+4z^{8}+3az^{7}+z^{7}a^{-1}+4z^{7}a^{-3}+6z^{7}a^{-5}+a^{2}z^{6}-23z^{6}a^{-2}-6z^{6}a^{-4}+5z^{6}a^{-6}-11z^{6}-9az^{5}-12z^{5}a^{-1}-13z^{5}a^{-3}-7z^{5}a^{-5}+3z^{5}a^{-7}-3a^{2}z^{4}+22z^{4}a^{-2}+4z^{4}a^{-4}-5z^{4}a^{-6}+z^{4}a^{-8}+9z^{4}+7az^{3}+9z^{3}a^{-1}+8z^{3}a^{-3}+3z^{3}a^{-5}-3z^{3}a^{-7}+2a^{2}z^{2}-13z^{2}a^{-2}-3z^{2}a^{-4}+2z^{2}a^{-6}-z^{2}a^{-8}-5z^{2}-az-2za^{-1}-za^{-3}+za^{-5}+za^{-7}+2a^{-2}+a^{-4}+2$

Vassiliev invariants

V2 and V3:

(-1, -1)

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 $-4$ $-8$ $8$ ${\frac {130}{3}}$ ${\frac {134}{3}}$ $32$ ${\frac {496}{3}}$ ${\frac {160}{3}}$ $88$ $-{\frac {32}{3}}$ $32$ $-{\frac {520}{3}}$ $-{\frac {536}{3}}$ ${\frac {3569}{30}}$ ${\frac {4102}{15}}$ $-{\frac {16742}{45}}$ ${\frac {3151}{18}}$ $-{\frac {4591}{30}}$

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=$2 is the signature of 10 106. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -4 -3 -2 -1 0 1 2 3 4 5 6 χ 15                     1 1 13                   2   -2 11                 4 1   3 9               6 2     -4 7             6 4       2 5           6 6         0 3         6 6           0 1       4 7             3 -1     2 5               -3 -3   1 4                 3 -5   2                   -2 -7 1                     1

Integral Khovanov Homology (db, data source)

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

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, 106]]
Out[2]=	10
In[3]:=	PD[Knot[10, 106]]
Out[3]=	PD[X[6, 2, 7, 1], X[16, 8, 17, 7], X[10, 3, 11, 4], X[2, 15, 3, 16], X[14, 5, 15, 6], X[4, 11, 5, 12], X[18, 10, 19, 9], X[20, 14, 1, 13], X[8, 18, 9, 17], X[12, 20, 13, 19]]
In[4]:=	GaussCode[Knot[10, 106]]
Out[4]=	GaussCode[1, -4, 3, -6, 5, -1, 2, -9, 7, -3, 6, -10, 8, -5, 4, -2, 9, -7, 10, -8]
In[5]:=	BR[Knot[10, 106]]
Out[5]=	BR[3, {1, 1, 1, -2, 1, -2, 1, 1, -2, -2}]
In[6]:=	alex = Alexander[Knot[10, 106]][t]
Out[6]=	-4 4 9 15 2 3 4 -17 - t + -- - -- + -- + 15 t - 9 t + 4 t - t 3 2 t t t
In[7]:=	Conway[Knot[10, 106]][z]
Out[7]=	2 4 6 8 1 - z - 5 z - 4 z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 106]}
In[9]:=	{KnotDet[Knot[10, 106]], KnotSignature[Knot[10, 106]]}
Out[9]=	{75, 2}
In[10]:=	J=Jones[Knot[10, 106]][q]
Out[10]=	-3 3 6 2 3 4 5 6 7 -9 + q - -- + - + 12 q - 12 q + 12 q - 10 q + 6 q - 3 q + q 2 q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[10, 59], Knot[10, 106]}
In[12]:=	A2Invariant[Knot[10, 106]][q]
Out[12]=	-8 -6 2 -2 2 4 6 8 10 12 14 q - q + -- - q + 2 q - 2 q + 4 q - 2 q + q - q - 2 q + 4 q 16 18 20 2 q - q + q
In[13]:=	Kauffman[Knot[10, 106]][a, z]
Out[13]=	2 2 2 -4 2 z z z 2 z 2 z 2 z 3 z 2 + a + -- + -- + -- - -- - --- - a z - 5 z - -- + ---- - ---- - 2 7 5 3 a 8 6 4 a a a a a a a 2 3 3 3 3 4 13 z 2 2 3 z 3 z 8 z 9 z 3 4 z ----- + 2 a z - ---- + ---- + ---- + ---- + 7 a z + 9 z + -- - 2 7 5 3 a 8 a a a a a 4 4 4 5 5 5 5 5 z 4 z 22 z 2 4 3 z 7 z 13 z 12 z ---- + ---- + ----- - 3 a z + ---- - ---- - ----- - ----- - 6 4 2 7 5 3 a a a a a a a 6 6 6 7 7 7 5 6 5 z 6 z 23 z 2 6 6 z 4 z z 9 a z - 11 z + ---- - ---- - ----- + a z + ---- + ---- + -- + 6 4 2 5 3 a a a a a a 8 8 9 9 7 8 5 z 9 z 2 z 2 z 3 a z + 4 z + ---- + ---- + ---- + ---- 4 2 3 a a a a
In[14]:=	{Vassiliev[2][Knot[10, 106]], Vassiliev[3][Knot[10, 106]]}
Out[14]=	{0, -1}
In[15]:=	Kh[Knot[10, 106]][q, t]
Out[15]=	3 1 2 1 4 2 5 4 q 7 q + 6 q + ----- + ----- + ----- + ----- + ---- + --- + --- + 7 4 5 3 3 3 3 2 2 q t t q t q t q t q t q t 3 5 5 2 7 2 7 3 9 3 9 4 6 q t + 6 q t + 6 q t + 6 q t + 4 q t + 6 q t + 2 q t + 11 4 11 5 13 5 15 6 4 q t + q t + 2 q t + q t