10 112

10_111

10_113

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10 112 Further Notes and Views

Knot presentations

Planar diagram presentation	X6271 X8394 X18,11,19,12 X20,13,1,14 X2,16,3,15 X4,17,5,18 X12,19,13,20 X10,6,11,5 X14,7,15,8 X16,10,17,9
Gauss code	1, -5, 2, -6, 8, -1, 9, -2, 10, -8, 3, -7, 4, -9, 5, -10, 6, -3, 7, -4
Dowker-Thistlethwaite code	6 8 10 14 16 18 20 2 4 12
Conway Notation	[8*3]

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index 1 Maximal Thurston-Bennequin number [-9][-3] Hyperbolic Volume 14.7559 A-Polynomial See Data:10 112/A-polynomial

[edit Notes for 10 112's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for 10 112's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{4}+5t^{3}-11t^{2}+17t-19+17t^{-1}-11t^{-2}+5t^{-3}-t^{-4}$
Conway polynomial	$-z^{8}-3z^{6}-z^{4}+2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 87, -2 }
Jones polynomial	$q^{3}-4q^{2}+7q-10+14q^{-1}-14q^{-2}+14q^{-3}-11q^{-4}+7q^{-5}-4q^{-6}+q^{-7}$
HOMFLY-PT polynomial (db, data sources)	$-a^{2}z^{8}+a^{4}z^{6}-5a^{2}z^{6}+z^{6}+3a^{4}z^{4}-7a^{2}z^{4}+3z^{4}+a^{4}z^{2}+z^{2}-2a^{4}+4a^{2}-1$
Kauffman polynomial (db, data sources)	$3a^{3}z^{9}+3az^{9}+7a^{4}z^{8}+13a^{2}z^{8}+6z^{8}+8a^{5}z^{7}+4a^{3}z^{7}+4z^{7}a^{-1}+7a^{6}z^{6}-9a^{4}z^{6}-35a^{2}z^{6}+z^{6}a^{-2}-18z^{6}+4a^{7}z^{5}-8a^{5}z^{5}-17a^{3}z^{5}-16az^{5}-11z^{5}a^{-1}+a^{8}z^{4}-7a^{6}z^{4}+3a^{4}z^{4}+28a^{2}z^{4}-2z^{4}a^{-2}+15z^{4}-3a^{7}z^{3}-a^{5}z^{3}+9a^{3}z^{3}+13az^{3}+6z^{3}a^{-1}+a^{6}z^{2}+a^{4}z^{2}-3a^{2}z^{2}-3z^{2}+2a^{5}z+2a^{3}z-2a^{4}-4a^{2}-1$
The A2 invariant	$q^{20}-2q^{18}+q^{16}-3q^{14}-q^{12}+2q^{10}-q^{8}+6q^{6}-q^{4}+3q^{2}-2q^{-2}+q^{-4}-2q^{-6}+q^{-8}$
The G2 invariant	$q^{114}-3q^{112}+6q^{110}-10q^{108}+9q^{106}-6q^{104}-2q^{102}+18q^{100}-32q^{98}+47q^{96}-50q^{94}+34q^{92}-6q^{90}-34q^{88}+75q^{86}-104q^{84}+117q^{82}-101q^{80}+50q^{78}+25q^{76}-110q^{74}+182q^{72}-205q^{70}+161q^{68}-63q^{66}-68q^{64}+177q^{62}-218q^{60}+166q^{58}-45q^{56}-99q^{54}+184q^{52}-176q^{50}+58q^{48}+108q^{46}-239q^{44}+275q^{42}-192q^{40}+16q^{38}+182q^{36}-322q^{34}+358q^{32}-271q^{30}+102q^{28}+103q^{26}-251q^{24}+316q^{22}-264q^{20}+137q^{18}+29q^{16}-166q^{14}+221q^{12}-170q^{10}+47q^{8}+109q^{6}-214q^{4}+212q^{2}-106-64q^{-2}+214q^{-4}-286q^{-6}+244q^{-8}-112q^{-10}-54q^{-12}+184q^{-14}-238q^{-16}+206q^{-18}-112q^{-20}+3q^{-22}+71q^{-24}-104q^{-26}+94q^{-28}-58q^{-30}+25q^{-32}+5q^{-34}-17q^{-36}+17q^{-38}-14q^{-40}+7q^{-42}-3q^{-44}+q^{-46}$

Further Quantum Invariants

Further quantum knot invariants for 10_112.

A1 Invariants.

Weight	Invariant
1	$q^{15}-3q^{13}+3q^{11}-4q^{9}+3q^{7}+4q-3q^{-1}+3q^{-3}-3q^{-5}+q^{-7}$
2	$q^{42}-3q^{40}+8q^{36}-10q^{34}+q^{32}+19q^{30}-24q^{28}-5q^{26}+34q^{24}-24q^{22}-19q^{20}+32q^{18}-3q^{16}-22q^{14}+9q^{12}+19q^{10}-12q^{8}-17q^{6}+28q^{4}+4q^{2}-33+23q^{-2}+18q^{-4}-34q^{-6}+5q^{-8}+23q^{-10}-17q^{-12}-7q^{-14}+12q^{-16}-q^{-18}-3q^{-20}+q^{-22}$
3	$q^{81}-3q^{79}+5q^{75}+2q^{73}-6q^{71}-6q^{69}+12q^{67}-5q^{65}-17q^{63}+17q^{61}+41q^{59}-34q^{57}-85q^{55}+41q^{53}+149q^{51}-16q^{49}-209q^{47}-40q^{45}+238q^{43}+117q^{41}-225q^{39}-182q^{37}+154q^{35}+224q^{33}-74q^{31}-218q^{29}-21q^{27}+189q^{25}+96q^{23}-142q^{21}-152q^{19}+96q^{17}+191q^{15}-48q^{13}-219q^{11}+q^{9}+241q^{7}+53q^{5}-238q^{3}-119q+221q^{-1}+179q^{-3}-157q^{-5}-228q^{-7}+78q^{-9}+231q^{-11}+10q^{-13}-196q^{-15}-81q^{-17}+129q^{-19}+109q^{-21}-58q^{-23}-95q^{-25}+5q^{-27}+62q^{-29}+19q^{-31}-30q^{-33}-15q^{-35}+7q^{-37}+8q^{-39}-q^{-41}-3q^{-43}+q^{-45}$
4	$q^{132}-3q^{130}+5q^{126}-q^{124}+6q^{122}-13q^{120}-7q^{118}+5q^{116}-8q^{114}+43q^{112}+4q^{110}-5q^{108}-39q^{106}-115q^{104}+56q^{102}+140q^{100}+191q^{98}-44q^{96}-479q^{94}-272q^{92}+239q^{90}+819q^{88}+495q^{86}-795q^{84}-1234q^{82}-420q^{80}+1352q^{78}+1806q^{76}-107q^{74}-1950q^{72}-1927q^{70}+623q^{68}+2626q^{66}+1469q^{64}-1162q^{62}-2728q^{60}-1001q^{58}+1753q^{56}+2269q^{54}+494q^{52}-1872q^{50}-1863q^{48}+88q^{46}+1660q^{44}+1462q^{42}-416q^{40}-1605q^{38}-1000q^{36}+713q^{34}+1600q^{32}+521q^{30}-1187q^{28}-1500q^{26}+174q^{24}+1659q^{22}+1136q^{20}-980q^{18}-2018q^{16}-376q^{14}+1728q^{12}+1941q^{10}-397q^{8}-2381q^{6}-1386q^{4}+1078q^{2}+2541+905q^{-2}-1742q^{-4}-2211q^{-6}-471q^{-8}+1943q^{-10}+1957q^{-12}-71q^{-14}-1709q^{-16}-1670q^{-18}+288q^{-20}+1549q^{-22}+1148q^{-24}-202q^{-26}-1332q^{-28}-801q^{-30}+243q^{-32}+887q^{-34}+646q^{-36}-264q^{-38}-566q^{-40}-377q^{-42}+116q^{-44}+400q^{-46}+164q^{-48}-55q^{-50}-195q^{-52}-101q^{-54}+59q^{-56}+62q^{-58}+41q^{-60}-20q^{-62}-29q^{-64}-q^{-66}+3q^{-68}+8q^{-70}-q^{-72}-3q^{-74}+q^{-76}$
5	$q^{195}-3q^{193}+5q^{189}-q^{187}+3q^{185}-q^{183}-14q^{181}-14q^{179}+8q^{177}+26q^{175}+39q^{173}+27q^{171}-44q^{169}-119q^{167}-111q^{165}+46q^{163}+233q^{161}+297q^{159}+89q^{157}-396q^{155}-732q^{153}-406q^{151}+522q^{149}+1377q^{147}+1244q^{145}-325q^{143}-2358q^{141}-2807q^{139}-538q^{137}+3226q^{135}+5261q^{133}+2790q^{131}-3383q^{129}-8363q^{127}-6776q^{125}+1819q^{123}+11098q^{121}+12328q^{119}+2363q^{117}-12050q^{115}-18353q^{113}-9116q^{111}+9844q^{109}+22819q^{107}+17308q^{105}-3982q^{103}-23854q^{101}-24722q^{99}-4508q^{97}+20466q^{95}+29011q^{93}+13444q^{91}-13220q^{89}-28708q^{87}-20448q^{85}+4095q^{83}+24112q^{81}+23630q^{79}+4488q^{77}-16577q^{75}-22887q^{73}-10738q^{71}+8556q^{69}+19145q^{67}+13802q^{65}-1675q^{63}-14248q^{61}-14346q^{59}-2957q^{57}+9853q^{55}+13501q^{53}+5484q^{51}-6920q^{49}-12671q^{47}-6700q^{45}+5592q^{43}+12784q^{41}+7707q^{39}-5350q^{37}-14066q^{35}-9442q^{33}+5142q^{31}+16192q^{29}+12420q^{27}-3966q^{25}-18267q^{23}-16515q^{21}+1026q^{19}+19110q^{17}+21012q^{15}+3968q^{13}-17673q^{11}-24717q^{9}-10372q^{7}+13293q^{5}+26080q^{3}+17093q-6169q^{-1}-24159q^{-3}-22158q^{-5}-2447q^{-7}+18441q^{-9}+24012q^{-11}+10687q^{-13}-10037q^{-15}-21692q^{-17}-16237q^{-19}+756q^{-21}+15621q^{-23}+17721q^{-25}+6925q^{-27}-7576q^{-29}-15018q^{-31}-11105q^{-33}-52q^{-35}+9478q^{-37}+11239q^{-39}+5213q^{-41}-3369q^{-43}-8312q^{-45}-6957q^{-47}-1275q^{-49}+4158q^{-51}+5888q^{-53}+3506q^{-55}-666q^{-57}-3504q^{-59}-3462q^{-61}-1239q^{-63}+1192q^{-65}+2281q^{-67}+1653q^{-69}+133q^{-71}-1007q^{-73}-1162q^{-75}-552q^{-77}+181q^{-79}+572q^{-81}+454q^{-83}+82q^{-85}-172q^{-87}-215q^{-89}-108q^{-91}+16q^{-93}+82q^{-95}+55q^{-97}+q^{-99}-19q^{-101}-15q^{-103}-5q^{-105}+3q^{-107}+8q^{-109}-q^{-111}-3q^{-113}+q^{-115}$
6	$q^{270}-3q^{268}+5q^{264}-q^{262}+3q^{260}-4q^{258}-2q^{256}-21q^{254}-11q^{252}+42q^{250}+28q^{248}+36q^{246}-3q^{244}-55q^{242}-157q^{240}-127q^{238}+107q^{236}+225q^{234}+326q^{232}+190q^{230}-160q^{228}-734q^{226}-821q^{224}-93q^{222}+758q^{220}+1583q^{218}+1449q^{216}+33q^{214}-2441q^{212}-3745q^{210}-2161q^{208}+1426q^{206}+5785q^{204}+7359q^{202}+3530q^{200}-5400q^{198}-13415q^{196}-13431q^{194}-3589q^{192}+13408q^{190}+26609q^{188}+24074q^{186}+1478q^{184}-29679q^{182}-48096q^{180}-37772q^{178}+4186q^{176}+55610q^{174}+81157q^{172}+54349q^{170}-17776q^{168}-93662q^{166}-122736q^{164}-71743q^{162}+39507q^{160}+145090q^{158}+169711q^{156}+83551q^{154}-71025q^{152}-202254q^{150}-215345q^{148}-87509q^{146}+112381q^{144}+257605q^{142}+248375q^{140}+79265q^{138}-154324q^{136}-301417q^{134}-263819q^{132}-57139q^{130}+189978q^{128}+322701q^{126}+256764q^{124}+29559q^{122}-212564q^{120}-320019q^{118}-227969q^{116}-1781q^{114}+216152q^{112}+293743q^{110}+188732q^{108}-21342q^{106}-204151q^{104}-250290q^{102}-145830q^{100}+36003q^{98}+180150q^{96}+202898q^{94}+104568q^{92}-45655q^{90}-151559q^{88}-157267q^{86}-67675q^{84}+52872q^{82}+127501q^{80}+115629q^{78}+31904q^{76}-62677q^{74}-108267q^{72}-76318q^{70}+5389q^{68}+78347q^{66}+90646q^{64}+34071q^{62}-48065q^{60}-95505q^{58}-68472q^{56}+14805q^{54}+95245q^{52}+107948q^{50}+36161q^{48}-72932q^{46}-140329q^{44}-108342q^{42}+8598q^{40}+135384q^{38}+176663q^{36}+93826q^{34}-65353q^{32}-193987q^{30}-197546q^{28}-65014q^{26}+124891q^{24}+241972q^{22}+202838q^{20}+25026q^{18}-178074q^{16}-272696q^{14}-194117q^{12}+14895q^{10}+218361q^{8}+286782q^{6}+174267q^{4}-47186q^{2}-239543-284538q^{-2}-153010q^{-4}+67717q^{-6}+243003q^{-8}+267322q^{-10}+134083q^{-12}-72869q^{-14}-229033q^{-16}-243101q^{-18}-118385q^{-20}+66287q^{-22}+199949q^{-24}+213825q^{-26}+107454q^{-28}-49879q^{-30}-164671q^{-32}-180650q^{-34}-97700q^{-36}+27443q^{-38}+126436q^{-40}+147194q^{-42}+88688q^{-44}-7493q^{-46}-88419q^{-48}-113621q^{-50}-79365q^{-52}-8326q^{-54}+55890q^{-56}+83122q^{-58}+66022q^{-60}+19031q^{-62}-29776q^{-64}-57423q^{-66}-51503q^{-68}-22737q^{-70}+12115q^{-72}+35303q^{-74}+37728q^{-76}+21839q^{-78}-2141q^{-80}-19462q^{-82}-24867q^{-84}-17354q^{-86}-3304q^{-88}+9548q^{-90}+15028q^{-92}+11770q^{-94}+4397q^{-96}-3622q^{-98}-7855q^{-100}-7459q^{-102}-3406q^{-104}+1041q^{-106}+3498q^{-108}+3919q^{-110}+2199q^{-112}+41q^{-114}-1561q^{-116}-1718q^{-118}-1042q^{-120}-227q^{-122}+523q^{-124}+687q^{-126}+471q^{-128}+48q^{-130}-147q^{-132}-200q^{-134}-164q^{-136}-25q^{-138}+53q^{-140}+74q^{-142}+15q^{-144}+2q^{-146}-5q^{-148}-19q^{-150}-5q^{-152}+3q^{-154}+8q^{-156}-q^{-158}-3q^{-160}+q^{-162}$

A2 Invariants.

Weight	Invariant
1,0	$q^{20}-2q^{18}+q^{16}-3q^{14}-q^{12}+2q^{10}-q^{8}+6q^{6}-q^{4}+3q^{2}-2q^{-2}+q^{-4}-2q^{-6}+q^{-8}$
1,1	$q^{60}-6q^{58}+18q^{56}-38q^{54}+69q^{52}-120q^{50}+190q^{48}-268q^{46}+357q^{44}-460q^{42}+576q^{40}-686q^{38}+790q^{36}-878q^{34}+906q^{32}-846q^{30}+671q^{28}-372q^{26}-56q^{24}+558q^{22}-1086q^{20}+1554q^{18}-1926q^{16}+2140q^{14}-2158q^{12}+2014q^{10}-1680q^{8}+1236q^{6}-688q^{4}+132q^{2}+388-822q^{-2}+1101q^{-4}-1220q^{-6}+1182q^{-8}-1040q^{-10}+827q^{-12}-590q^{-14}+384q^{-16}-226q^{-18}+117q^{-20}-52q^{-22}+20q^{-24}-6q^{-26}+q^{-28}$
2,0	$q^{52}-2q^{50}-q^{48}+3q^{46}-3q^{44}+2q^{42}+6q^{40}-q^{38}-3q^{36}+2q^{34}+8q^{32}-11q^{30}-15q^{28}+7q^{26}+q^{24}-15q^{22}+2q^{20}+17q^{18}-q^{16}-2q^{14}+9q^{12}+6q^{10}-7q^{8}+q^{6}+14q^{4}-9q^{2}-6+11q^{-2}-13q^{-6}-q^{-8}+9q^{-10}-q^{-12}-7q^{-14}+2q^{-16}+6q^{-18}-2q^{-20}-2q^{-22}+q^{-24}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{48}-3q^{46}+8q^{42}-9q^{40}-3q^{38}+18q^{36}-13q^{34}-11q^{32}+28q^{30}-12q^{28}-18q^{26}+24q^{24}-11q^{22}-18q^{20}+7q^{18}+3q^{16}-q^{14}-3q^{12}+19q^{10}+14q^{8}-19q^{6}+11q^{4}+14q^{2}-28+4q^{-2}+15q^{-4}-21q^{-6}+7q^{-8}+9q^{-10}-10q^{-12}+5q^{-14}+q^{-16}-3q^{-18}+q^{-20}$
1,0,0	$q^{25}-2q^{23}+2q^{21}-5q^{19}+q^{17}-4q^{15}+2q^{13}+q^{11}+4q^{9}+4q^{7}+q^{5}+3q^{3}-3q+2q^{-1}-4q^{-3}+2q^{-5}-2q^{-7}+q^{-9}$
1,0,1	$q^{78}-6q^{76}+15q^{74}-17q^{72}-3q^{70}+45q^{68}-86q^{66}+84q^{64}-10q^{62}-112q^{60}+210q^{58}-204q^{56}+69q^{54}+154q^{52}-343q^{50}+362q^{48}-147q^{46}-237q^{44}+591q^{42}-689q^{40}+425q^{38}+131q^{36}-695q^{34}+991q^{32}-892q^{30}+417q^{28}+108q^{26}-527q^{24}+596q^{22}-431q^{20}+217q^{18}-87q^{16}+239q^{14}-424q^{12}+582q^{10}-413q^{8}-31q^{6}+577q^{4}-987q^{2}+1018-706q^{-2}+167q^{-4}+339q^{-6}-639q^{-8}+659q^{-10}-445q^{-12}+140q^{-14}+113q^{-16}-223q^{-18}+197q^{-20}-104q^{-22}+17q^{-24}+26q^{-26}-29q^{-28}+17q^{-30}-6q^{-32}+q^{-34}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{58}-2q^{56}-2q^{54}+6q^{52}-2q^{50}-9q^{48}+11q^{46}+10q^{44}-17q^{42}+q^{40}+20q^{38}-5q^{36}-19q^{34}+9q^{32}+7q^{30}-28q^{28}-15q^{26}+17q^{24}-12q^{22}-20q^{20}+34q^{18}+19q^{16}-11q^{14}+20q^{12}+30q^{10}-13q^{8}-14q^{6}+8q^{4}-21-4q^{-2}+13q^{-4}-5q^{-6}-3q^{-8}+9q^{-10}-5q^{-14}+4q^{-16}-q^{-18}-2q^{-20}+q^{-22}$
1,0,0,0	$q^{30}-2q^{28}+2q^{26}-4q^{24}-q^{22}-2q^{20}-3q^{18}+2q^{16}+6q^{12}+2q^{10}+6q^{8}+3q^{4}-2q^{2}-1-3q^{-4}+2q^{-6}-2q^{-8}+q^{-10}$

B2 Invariants.

Weight	Invariant
0,1	$q^{48}-3q^{46}+6q^{44}-10q^{42}+15q^{40}-21q^{38}+26q^{36}-31q^{34}+33q^{32}-32q^{30}+24q^{28}-14q^{26}-2q^{24}+17q^{22}-34q^{20}+49q^{18}-59q^{16}+67q^{14}-63q^{12}+59q^{10}-44q^{8}+31q^{6}-11q^{4}-4q^{2}+18-28q^{-2}+33q^{-4}-35q^{-6}+31q^{-8}-27q^{-10}+20q^{-12}-13q^{-14}+7q^{-16}-3q^{-18}+q^{-20}$
1,0	$q^{78}-3q^{74}-3q^{72}+3q^{70}+9q^{68}+2q^{66}-12q^{64}-12q^{62}+8q^{60}+22q^{58}+6q^{56}-23q^{54}-22q^{52}+13q^{50}+34q^{48}+3q^{46}-33q^{44}-19q^{42}+23q^{40}+25q^{38}-15q^{36}-31q^{34}+q^{32}+26q^{30}+3q^{28}-24q^{26}-9q^{24}+22q^{22}+15q^{20}-13q^{18}-13q^{16}+19q^{14}+24q^{12}-8q^{10}-29q^{8}+q^{6}+33q^{4}+15q^{2}-30-32q^{-2}+13q^{-4}+35q^{-6}+3q^{-8}-31q^{-10}-17q^{-12}+18q^{-14}+21q^{-16}-5q^{-18}-15q^{-20}-2q^{-22}+9q^{-24}+4q^{-26}-3q^{-28}-3q^{-30}+q^{-34}$

D4 Invariants.

Weight

Invariant

1,0,0,0

$q^{66}-3q^{64}+3q^{62}-4q^{60}+9q^{58}-12q^{56}+12q^{54}-15q^{52}+22q^{50}-23q^{48}+21q^{46}-25q^{44}+29q^{42}-21q^{40}+15q^{38}-13q^{36}+3q^{34}+6q^{32}-20q^{30}+18q^{28}-39q^{26}+41q^{24}-46q^{22}+51q^{20}-48q^{18}+56q^{16}-35q^{14}+45q^{12}-27q^{10}+23q^{8}-9q^{6}+q^{4}+3q^{2}-18+20q^{-2}-26q^{-4}+25q^{-6}-28q^{-8}+28q^{-10}-23q^{-12}+20q^{-14}-16q^{-16}+12q^{-18}-7q^{-20}+4q^{-22}-3q^{-24}+q^{-26}$

G2 Invariants.

Weight

Invariant

1,0

$q^{114}-3q^{112}+6q^{110}-10q^{108}+9q^{106}-6q^{104}-2q^{102}+18q^{100}-32q^{98}+47q^{96}-50q^{94}+34q^{92}-6q^{90}-34q^{88}+75q^{86}-104q^{84}+117q^{82}-101q^{80}+50q^{78}+25q^{76}-110q^{74}+182q^{72}-205q^{70}+161q^{68}-63q^{66}-68q^{64}+177q^{62}-218q^{60}+166q^{58}-45q^{56}-99q^{54}+184q^{52}-176q^{50}+58q^{48}+108q^{46}-239q^{44}+275q^{42}-192q^{40}+16q^{38}+182q^{36}-322q^{34}+358q^{32}-271q^{30}+102q^{28}+103q^{26}-251q^{24}+316q^{22}-264q^{20}+137q^{18}+29q^{16}-166q^{14}+221q^{12}-170q^{10}+47q^{8}+109q^{6}-214q^{4}+212q^{2}-106-64q^{-2}+214q^{-4}-286q^{-6}+244q^{-8}-112q^{-10}-54q^{-12}+184q^{-14}-238q^{-16}+206q^{-18}-112q^{-20}+3q^{-22}+71q^{-24}-104q^{-26}+94q^{-28}-58q^{-30}+25q^{-32}+5q^{-34}-17q^{-36}+17q^{-38}-14q^{-40}+7q^{-42}-3q^{-44}+q^{-46}$

.

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["10 112"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t^{4}+5t^{3}-11t^{2}+17t-19+17t^{-1}-11t^{-2}+5t^{-3}-t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

$-z^{8}-3z^{6}-z^{4}+2z^{2}+1$

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]=

{ 87, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^{3}-4q^{2}+7q-10+14q^{-1}-14q^{-2}+14q^{-3}-11q^{-4}+7q^{-5}-4q^{-6}+q^{-7}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-a^{2}z^{8}+a^{4}z^{6}-5a^{2}z^{6}+z^{6}+3a^{4}z^{4}-7a^{2}z^{4}+3z^{4}+a^{4}z^{2}+z^{2}-2a^{4}+4a^{2}-1$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$3a^{3}z^{9}+3az^{9}+7a^{4}z^{8}+13a^{2}z^{8}+6z^{8}+8a^{5}z^{7}+4a^{3}z^{7}+4z^{7}a^{-1}+7a^{6}z^{6}-9a^{4}z^{6}-35a^{2}z^{6}+z^{6}a^{-2}-18z^{6}+4a^{7}z^{5}-8a^{5}z^{5}-17a^{3}z^{5}-16az^{5}-11z^{5}a^{-1}+a^{8}z^{4}-7a^{6}z^{4}+3a^{4}z^{4}+28a^{2}z^{4}-2z^{4}a^{-2}+15z^{4}-3a^{7}z^{3}-a^{5}z^{3}+9a^{3}z^{3}+13az^{3}+6z^{3}a^{-1}+a^{6}z^{2}+a^{4}z^{2}-3a^{2}z^{2}-3z^{2}+2a^{5}z+2a^{3}z-2a^{4}-4a^{2}-1$

Vassiliev invariants

V2 and V3:

(2, -2)

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 $8$ $-16$ $32$ ${\frac {220}{3}}$ ${\frac {68}{3}}$ $-128$ $-{\frac {736}{3}}$ $-{\frac {160}{3}}$ $-48$ ${\frac {256}{3}}$ $128$ ${\frac {1760}{3}}$ ${\frac {544}{3}}$ ${\frac {14431}{15}}$ $-{\frac {604}{15}}$ ${\frac {21724}{45}}$ ${\frac {545}{9}}$ ${\frac {1471}{15}}$

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 $s=$-2 is the signature of 10 112. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 7                     1 1 5                   3   -3 3                 4 1   3 1               6 3     -3 -1             8 4       4 -3           7 7         0 -5         7 7           0 -7       4 7             3 -9     3 7               -4 -11   1 4                 3 -13   3                   -3 -15 1                     1

Integral Khovanov Homology (db, data source)

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

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, 112]]
Out[2]=	10
In[3]:=	PD[Knot[10, 112]]
Out[3]=	PD[X[6, 2, 7, 1], X[8, 3, 9, 4], X[18, 11, 19, 12], X[20, 13, 1, 14], X[2, 16, 3, 15], X[4, 17, 5, 18], X[12, 19, 13, 20], X[10, 6, 11, 5], X[14, 7, 15, 8], X[16, 10, 17, 9]]
In[4]:=	GaussCode[Knot[10, 112]]
Out[4]=	GaussCode[1, -5, 2, -6, 8, -1, 9, -2, 10, -8, 3, -7, 4, -9, 5, -10, 6, -3, 7, -4]
In[5]:=	BR[Knot[10, 112]]
Out[5]=	BR[3, {-1, -1, -1, 2, -1, 2, -1, 2, -1, 2}]
In[6]:=	alex = Alexander[Knot[10, 112]][t]
Out[6]=	-4 5 11 17 2 3 4 -19 - t + -- - -- + -- + 17 t - 11 t + 5 t - t 3 2 t t t
In[7]:=	Conway[Knot[10, 112]][z]
Out[7]=	2 4 6 8 1 + 2 z - z - 3 z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 112], Knot[11, Alternating, 184]}
In[9]:=	{KnotDet[Knot[10, 112]], KnotSignature[Knot[10, 112]]}
Out[9]=	{87, -2}
In[10]:=	J=Jones[Knot[10, 112]][q]
Out[10]=	-7 4 7 11 14 14 14 2 3 -10 + q - -- + -- - -- + -- - -- + -- + 7 q - 4 q + q 6 5 4 3 2 q q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[10, 112]}
In[12]:=	A2Invariant[Knot[10, 112]][q]
Out[12]=	-20 2 -16 3 -12 2 -8 6 -4 3 2 q - --- + q - --- - q + --- - q + -- - q + -- - 2 q + 18 14 10 6 2 q q q q q 4 6 8 q - 2 q + q
In[13]:=	Kauffman[Knot[10, 112]][a, z]
Out[13]=	2 4 3 5 2 2 2 4 2 6 2 -1 - 4 a - 2 a + 2 a z + 2 a z - 3 z - 3 a z + a z + a z + 3 4 6 z 3 3 3 5 3 7 3 4 2 z ---- + 13 a z + 9 a z - a z - 3 a z + 15 z - ---- + a 2 a 5 2 4 4 4 6 4 8 4 11 z 5 3 5 28 a z + 3 a z - 7 a z + a z - ----- - 16 a z - 17 a z - a 6 5 5 7 5 6 z 2 6 4 6 6 6 8 a z + 4 a z - 18 z + -- - 35 a z - 9 a z + 7 a z + 2 a 7 4 z 3 7 5 7 8 2 8 4 8 9 ---- + 4 a z + 8 a z + 6 z + 13 a z + 7 a z + 3 a z + a 3 9 3 a z
In[14]:=	{Vassiliev[2][Knot[10, 112]], Vassiliev[3][Knot[10, 112]]}
Out[14]=	{0, -2}
In[15]:=	Kh[Knot[10, 112]][q, t]
Out[15]=	7 8 1 3 1 4 3 7 4 -- + - + ------ + ------ + ------ + ------ + ----- + ----- + ----- + 3 q 15 6 13 5 11 5 11 4 9 4 9 3 7 3 q q t q t q t q t q t q t q t 7 7 7 7 4 t 2 3 2 ----- + ----- + ---- + ---- + --- + 6 q t + 3 q t + 4 q t + 7 2 5 2 5 3 q q t q t q t q t 3 3 5 3 7 4 q t + 3 q t + q t