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]

Minimum Braid Representative: Length is 10, width is 3. Braid index is 3.

A Morse Link Presentation:

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]=

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

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a184, ...}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {...}

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)

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

The Coloured Jones Polynomials

The Coloured Jones Polynomials (in the $n+1$-dimensional representation of $sl(2)$)

$n$	$J_{n}$
2	$q^{10}-4q^{9}+2q^{8}+14q^{7}-23q^{6}-8q^{5}+54q^{4}-41q^{3}-47q^{2}+106q-36-103q^{-1}+143q^{-2}-12q^{-3}-148q^{-4}+148q^{-5}+19q^{-6}-158q^{-7}+117q^{-8}+38q^{-9}-123q^{-10}+66q^{-11}+33q^{-12}-65q^{-13}+27q^{-14}+14q^{-15}-22q^{-16}+9q^{-17}+3q^{-18}-4q^{-19}+q^{-20}$
3	$q^{21}-4q^{20}+2q^{19}+9q^{18}-26q^{16}-13q^{15}+58q^{14}+43q^{13}-83q^{12}-113q^{11}+95q^{10}+210q^{9}-63q^{8}-323q^{7}-20q^{6}+416q^{5}+158q^{4}-476q^{3}-326q^{2}+487q+494-434q^{-1}-666q^{-2}+368q^{-3}+785q^{-4}-246q^{-5}-906q^{-6}+148q^{-7}+956q^{-8}-7q^{-9}-1001q^{-10}-100q^{-11}+966q^{-12}+231q^{-13}-908q^{-14}-310q^{-15}+769q^{-16}+375q^{-17}-610q^{-18}-380q^{-19}+433q^{-20}+332q^{-21}-268q^{-22}-259q^{-23}+155q^{-24}+163q^{-25}-75q^{-26}-94q^{-27}+47q^{-28}+37q^{-29}-24q^{-30}-19q^{-31}+23q^{-32}+3q^{-33}-12q^{-34}-2q^{-35}+5q^{-36}+3q^{-37}-4q^{-38}+q^{-39}$
4	$q^{36}-4q^{35}+2q^{34}+9q^{33}-5q^{32}-3q^{31}-32q^{30}+11q^{29}+70q^{28}+16q^{27}-6q^{26}-192q^{25}-83q^{24}+210q^{23}+235q^{22}+230q^{21}-476q^{20}-576q^{19}+21q^{18}+537q^{17}+1140q^{16}-235q^{15}-1220q^{14}-1023q^{13}+6q^{12}+2270q^{11}+1115q^{10}-819q^{9}-2284q^{8}-1952q^{7}+2231q^{6}+2753q^{5}+1209q^{4}-2298q^{3}-4366q^{2}+491q+3222+3856q^{-1}-662q^{-2}-5829q^{-3}-1973q^{-4}+2227q^{-5}+5840q^{-6}+1676q^{-7}-6042q^{-8}-4077q^{-9}+585q^{-10}+6878q^{-11}+3792q^{-12}-5519q^{-13}-5562q^{-14}-1089q^{-15}+7191q^{-16}+5500q^{-17}-4440q^{-18}-6449q^{-19}-2802q^{-20}+6586q^{-21}+6689q^{-22}-2562q^{-23}-6251q^{-24}-4374q^{-25}+4635q^{-26}+6680q^{-27}-196q^{-28}-4476q^{-29}-4890q^{-30}+1881q^{-31}+4953q^{-32}+1370q^{-33}-1845q^{-34}-3733q^{-35}-122q^{-36}+2403q^{-37}+1347q^{-38}-2q^{-39}-1820q^{-40}-576q^{-41}+631q^{-42}+533q^{-43}+437q^{-44}-530q^{-45}-252q^{-46}+51q^{-47}+22q^{-48}+230q^{-49}-95q^{-50}-17q^{-51}-62q^{-53}+59q^{-54}-19q^{-55}+17q^{-56}+9q^{-57}-23q^{-58}+8q^{-59}-6q^{-60}+5q^{-61}+3q^{-62}-4q^{-63}+q^{-64}$
5	$q^{55}-4q^{54}+2q^{53}+9q^{52}-5q^{51}-8q^{50}-9q^{49}-8q^{48}+22q^{47}+63q^{46}+22q^{45}-74q^{44}-133q^{43}-115q^{42}+65q^{41}+317q^{40}+394q^{39}+44q^{38}-524q^{37}-848q^{36}-545q^{35}+472q^{34}+1534q^{33}+1564q^{32}+104q^{31}-1937q^{30}-2976q^{29}-1751q^{28}+1492q^{27}+4402q^{26}+4276q^{25}+445q^{24}-4706q^{23}-7184q^{22}-4190q^{21}+3047q^{20}+9219q^{19}+9027q^{18}+1320q^{17}-8945q^{16}-13720q^{15}-8006q^{14}+5306q^{13}+16469q^{12}+15821q^{11}+1851q^{10}-15820q^{9}-22871q^{8}-11687q^{7}+11014q^{6}+27476q^{5}+22575q^{4}-2495q^{3}-28442q^{2}-32575q-8697+25475q^{-1}+40565q^{-2}+20767q^{-3}-19455q^{-4}-45362q^{-5}-32362q^{-6}+11130q^{-7}+47609q^{-8}+42408q^{-9}-2411q^{-10}-47264q^{-11}-50446q^{-12}-6411q^{-13}+45857q^{-14}+56709q^{-15}+13975q^{-16}-43492q^{-17}-61496q^{-18}-20995q^{-19}+41233q^{-20}+65425q^{-21}+27032q^{-22}-38415q^{-23}-68688q^{-24}-33287q^{-25}+35262q^{-26}+71176q^{-27}+39436q^{-28}-30398q^{-29}-72336q^{-30}-46097q^{-31}+23873q^{-32}+71274q^{-33}+52009q^{-34}-14921q^{-35}-66993q^{-36}-56686q^{-37}+4579q^{-38}+59125q^{-39}+58319q^{-40}+6144q^{-41}-47851q^{-42}-56204q^{-43}-15438q^{-44}+34582q^{-45}+50059q^{-46}+21632q^{-47}-21187q^{-48}-40637q^{-49}-23983q^{-50}+9608q^{-51}+29845q^{-52}+22500q^{-53}-1315q^{-54}-19347q^{-55}-18472q^{-56}-3367q^{-57}+10885q^{-58}+13263q^{-59}+4988q^{-60}-4934q^{-61}-8507q^{-62}-4597q^{-63}+1606q^{-64}+4668q^{-65}+3401q^{-66}+46q^{-67}-2334q^{-68}-2126q^{-69}-429q^{-70}+904q^{-71}+1132q^{-72}+495q^{-73}-301q^{-74}-557q^{-75}-296q^{-76}+49q^{-77}+204q^{-78}+169q^{-79}+35q^{-80}-72q^{-81}-88q^{-82}-15q^{-83}+17q^{-84}+12q^{-85}+27q^{-86}+3q^{-87}-17q^{-88}-3q^{-89}+4q^{-90}-6q^{-91}+5q^{-92}+3q^{-93}-4q^{-94}+q^{-95}$
6	$q^{78}-4q^{77}+2q^{76}+9q^{75}-5q^{74}-8q^{73}-14q^{72}+15q^{71}+3q^{70}+15q^{69}+68q^{68}-26q^{67}-86q^{66}-153q^{65}-21q^{64}+56q^{63}+210q^{62}+491q^{61}+190q^{60}-250q^{59}-903q^{58}-836q^{57}-620q^{56}+367q^{55}+2093q^{54}+2348q^{53}+1470q^{52}-1324q^{51}-3293q^{50}-5067q^{49}-3686q^{48}+1697q^{47}+6581q^{46}+9489q^{45}+6049q^{44}-35q^{43}-10547q^{42}-16538q^{41}-12353q^{40}-932q^{39}+14894q^{38}+23370q^{37}+23945q^{36}+5342q^{35}-18963q^{34}-35541q^{33}-35784q^{32}-13872q^{31}+17450q^{30}+51592q^{29}+54149q^{28}+28028q^{27}-18441q^{26}-63016q^{25}-78088q^{24}-53589q^{23}+17336q^{22}+79351q^{21}+108954q^{20}+77740q^{19}-4510q^{18}-98846q^{17}-152582q^{16}-107807q^{15}-3599q^{14}+124933q^{13}+192532q^{12}+152823q^{11}+7525q^{10}-166458q^{9}-241469q^{8}-188271q^{7}+217q^{6}+206600q^{5}+308987q^{4}+214477q^{3}-33219q^{2}-265788q-363557-220510q^{-1}+75072q^{-2}+353982q^{-3}+406834q^{-4}+188234q^{-5}-153273q^{-6}-431978q^{-7}-423976q^{-8}-133940q^{-9}+275403q^{-10}+501456q^{-11}+391334q^{-12}+24539q^{-13}-392844q^{-14}-541057q^{-15}-323845q^{-16}+142871q^{-17}+505015q^{-18}+519968q^{-19}+183718q^{-20}-310007q^{-21}-582336q^{-22}-450631q^{-23}+25931q^{-24}+473028q^{-25}+587365q^{-26}+292811q^{-27}-238220q^{-28}-595039q^{-29}-531071q^{-30}-57346q^{-31}+445995q^{-32}+634805q^{-33}+374947q^{-34}-181645q^{-35}-607338q^{-36}-604029q^{-37}-139053q^{-38}+414046q^{-39}+680395q^{-40}+469528q^{-41}-97920q^{-42}-595466q^{-43}-678658q^{-44}-259600q^{-45}+324454q^{-46}+686103q^{-47}+575432q^{-48}+52303q^{-49}-497136q^{-50}-701406q^{-51}-403747q^{-52}+142621q^{-53}+581643q^{-54}+621571q^{-55}+235112q^{-56}-287062q^{-57}-596395q^{-58}-481338q^{-59}-75312q^{-60}+355455q^{-61}+529521q^{-62}+342567q^{-63}-44939q^{-64}-369190q^{-65}-415401q^{-66}-208035q^{-67}+108338q^{-68}+322841q^{-69}+304969q^{-70}+102154q^{-71}-135601q^{-72}-246291q^{-73}-198805q^{-74}-36886q^{-75}+122951q^{-76}+177133q^{-77}+115245q^{-78}-4372q^{-79}-91715q^{-80}-112645q^{-81}-61507q^{-82}+17368q^{-83}+65883q^{-84}+64252q^{-85}+24702q^{-86}-15829q^{-87}-40520q^{-88}-34699q^{-89}-8179q^{-90}+14459q^{-91}+22294q^{-92}+14378q^{-93}+2588q^{-94}-9363q^{-95}-12103q^{-96}-5644q^{-97}+1258q^{-98}+5297q^{-99}+4536q^{-100}+2604q^{-101}-1348q^{-102}-3173q^{-103}-1815q^{-104}-316q^{-105}+938q^{-106}+949q^{-107}+1020q^{-108}-44q^{-109}-699q^{-110}-399q^{-111}-182q^{-112}+113q^{-113}+98q^{-114}+292q^{-115}+43q^{-116}-125q^{-117}-49q^{-118}-46q^{-119}+12q^{-120}-20q^{-121}+58q^{-122}+13q^{-123}-23q^{-124}+3q^{-125}-7q^{-126}+4q^{-127}-6q^{-128}+5q^{-129}+3q^{-130}-4q^{-131}+q^{-132}$
7	$q^{105}-4q^{104}+2q^{103}+9q^{102}-5q^{101}-8q^{100}-14q^{99}+10q^{98}+26q^{97}-4q^{96}+20q^{95}+20q^{94}-39q^{93}-86q^{92}-130q^{91}-18q^{90}+189q^{89}+218q^{88}+307q^{87}+213q^{86}-170q^{85}-575q^{84}-1114q^{83}-937q^{82}+19q^{81}+1068q^{80}+2323q^{79}+2755q^{78}+1680q^{77}-476q^{76}-4106q^{75}-6650q^{74}-6050q^{73}-2726q^{72}+4089q^{71}+10866q^{70}+14273q^{69}+12544q^{68}+2272q^{67}-12339q^{66}-24453q^{65}-29328q^{64}-19717q^{63}+1745q^{62}+27496q^{61}+49490q^{60}+51247q^{59}+29082q^{58}-11109q^{57}-58147q^{56}-86351q^{55}-81674q^{54}-40221q^{53}+32573q^{52}+103269q^{51}+141541q^{50}+125723q^{49}+45576q^{48}-66572q^{47}-170667q^{46}-221936q^{45}-178291q^{44}-51085q^{43}+122230q^{42}+276101q^{41}+328024q^{40}+248280q^{39}+44859q^{38}-219684q^{37}-424559q^{36}-479667q^{35}-329249q^{34}+1427q^{33}+377537q^{32}+650984q^{31}+674467q^{30}+385742q^{29}-116839q^{28}-651507q^{27}-969702q^{26}-876873q^{25}-371829q^{24}+388842q^{23}+1079864q^{22}+1348069q^{21}+1026443q^{20}+165462q^{19}-892107q^{18}-1646414q^{17}-1716158q^{16}-958013q^{15}+358674q^{14}+1638995q^{13}+2276578q^{12}+1862041q^{11}+484452q^{10}-1256298q^{9}-2561391q^{8}-2712791q^{7}-1522784q^{6}+513794q^{5}+2479981q^{4}+3353864q^{3}+2601296q^{2}+498263q-2022845-3681823q^{-1}-3565159q^{-2}-1639523q^{-3}+1255163q^{-4}+3659966q^{-5}+4297269q^{-6}+2766712q^{-7}-288310q^{-8}-3324924q^{-9}-4744622q^{-10}-3758569q^{-11}-745625q^{-12}+2758223q^{-13}+4907479q^{-14}+4544110q^{-15}+1737138q^{-16}-2065024q^{-17}-4840920q^{-18}-5101061q^{-19}-2600896q^{-20}+1345575q^{-21}+4615074q^{-22}+5448137q^{-23}+3301323q^{-24}-677090q^{-25}-4314371q^{-26}-5632970q^{-27}-3829595q^{-28}+110075q^{-29}+4003226q^{-30}+5712096q^{-31}+4213556q^{-32}+338862q^{-33}-3735338q^{-34}-5745173q^{-35}-4491455q^{-36}-677796q^{-37}+3532924q^{-38}+5780653q^{-39}+4718685q^{-40}+940126q^{-41}-3399370q^{-42}-5851981q^{-43}-4945050q^{-44}-1177198q^{-45}+3304370q^{-46}+5969456q^{-47}+5218300q^{-48}+1449833q^{-49}-3198548q^{-50}-6111765q^{-51}-5558109q^{-52}-1818171q^{-53}+3006066q^{-54}+6225844q^{-55}+5956977q^{-56}+2321108q^{-57}-2654449q^{-58}-6228347q^{-59}-6356639q^{-60}-2958913q^{-61}+2081549q^{-62}+6023339q^{-63}+6666957q^{-64}+3679256q^{-65}-1277096q^{-66}-5532402q^{-67}-6768298q^{-68}-4374923q^{-69}+287316q^{-70}+4722091q^{-71}+6558152q^{-72}+4910362q^{-73}+771904q^{-74}-3634733q^{-75}-5980942q^{-76}-5151849q^{-77}-1743975q^{-78}+2384254q^{-79}+5059882q^{-80}+5018303q^{-81}+2473724q^{-82}-1138645q^{-83}-3903117q^{-84}-4509092q^{-85}-2849932q^{-86}+70836q^{-87}+2672182q^{-88}+3709833q^{-89}+2847577q^{-90}+692290q^{-91}-1543588q^{-92}-2768946q^{-93}-2524928q^{-94}-1100025q^{-95}+650464q^{-96}+1846579q^{-97}+2004985q^{-98}+1188097q^{-99}-55431q^{-100}-1072392q^{-101}-1427253q^{-102}-1050542q^{-103}-255713q^{-104}+512977q^{-105}+906860q^{-106}+801291q^{-107}+350733q^{-108}-169154q^{-109}-508679q^{-110}-538782q^{-111}-318086q^{-112}-1268q^{-113}+247773q^{-114}+321322q^{-115}+232766q^{-116}+60507q^{-117}-99750q^{-118}-171068q^{-119}-147523q^{-120}-62536q^{-121}+30928q^{-122}+81618q^{-123}+81789q^{-124}+44758q^{-125}-4126q^{-126}-35039q^{-127}-41684q^{-128}-26819q^{-129}-2055q^{-130}+14365q^{-131}+19478q^{-132}+13821q^{-133}+2242q^{-134}-5527q^{-135}-8897q^{-136}-6990q^{-137}-1259q^{-138}+2497q^{-139}+4247q^{-140}+3300q^{-141}+370q^{-142}-1068q^{-143}-1866q^{-144}-1680q^{-145}-298q^{-146}+445q^{-147}+1047q^{-148}+895q^{-149}+11q^{-150}-214q^{-151}-358q^{-152}-390q^{-153}-94q^{-154}+q^{-155}+207q^{-156}+237q^{-157}-17q^{-158}-23q^{-159}-45q^{-160}-59q^{-161}-9q^{-162}-25q^{-163}+26q^{-164}+44q^{-165}-13q^{-166}-3q^{-167}-q^{-168}-7q^{-169}+4q^{-170}-6q^{-171}+5q^{-172}+3q^{-173}-4q^{-174}+q^{-175}$

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 29, 2005, 15:27:48)...
In[2]:=	PD[Knot[10, 112]]
Out[2]=	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[3]:=	GaussCode[Knot[10, 112]]
Out[3]=	GaussCode[1, -5, 2, -6, 8, -1, 9, -2, 10, -8, 3, -7, 4, -9, 5, -10, 6, -3, 7, -4]
In[4]:=	DTCode[Knot[10, 112]]
Out[4]=	DTCode[6, 8, 10, 14, 16, 18, 20, 2, 4, 12]
In[5]:=	br = BR[Knot[10, 112]]
Out[5]=	BR[3, {-1, -1, -1, 2, -1, 2, -1, 2, -1, 2}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{3, 10}
In[7]:=	BraidIndex[Knot[10, 112]]
Out[7]=	3
In[8]:=	Show[DrawMorseLink[Knot[10, 112]]]
Out[8]=	-Graphics-
In[9]:=	(#[Knot[10, 112]]&) /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 2, 4, 3, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 112]][t]
Out[10]=	-4 5 11 17 2 3 4 -19 - t + -- - -- + -- + 17 t - 11 t + 5 t - t 3 2 t t t
In[11]:=	Conway[Knot[10, 112]][z]
Out[11]=	2 4 6 8 1 + 2 z - z - 3 z - z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 112], Knot[11, Alternating, 184]}
In[13]:=	{KnotDet[Knot[10, 112]], KnotSignature[Knot[10, 112]]}
Out[13]=	{87, -2}
In[14]:=	Jones[Knot[10, 112]][q]
Out[14]=	-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[15]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[10, 112]}
In[16]:=	A2Invariant[Knot[10, 112]][q]
Out[16]=	-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[17]:=	HOMFLYPT[Knot[10, 112]][a, z]
Out[17]=	2 4 2 4 2 4 2 4 4 4 6 -1 + 4 a - 2 a + z + a z + 3 z - 7 a z + 3 a z + z - 2 6 4 6 2 8 5 a z + a z - a z
In[18]:=	Kauffman[Knot[10, 112]][a, z]
Out[18]=	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[19]:=	{Vassiliev[2][Knot[10, 112]], Vassiliev[3][Knot[10, 112]]}
Out[19]=	{2, -2}
In[20]:=	Kh[Knot[10, 112]][q, t]
Out[20]=	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
In[21]:=	ColouredJones[Knot[10, 112], 2][q]
Out[21]=	-20 4 3 9 22 14 27 65 33 66 -36 + q - --- + --- + --- - --- + --- + --- - --- + --- + --- - 19 18 17 16 15 14 13 12 11 q q q q q q q q q 123 38 117 158 19 148 148 12 143 103 --- + -- + --- - --- + -- + --- - --- - -- + --- - --- + 106 q - 10 9 8 7 6 5 4 3 2 q q q q q q q q q q 2 3 4 5 6 7 8 9 10 47 q - 41 q + 54 q - 8 q - 23 q + 14 q + 2 q - 4 q + q

See/edit the Rolfsen_Splice_Template.