10 43

10_42

10_44

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10 43 Quick Notes

10 43 Further Notes and Views

Knot presentations

Planar diagram presentation	X4251 X10,4,11,3 X14,8,15,7 X20,11,1,12 X12,19,13,20 X8,14,9,13 X18,15,19,16 X16,5,17,6 X6,17,7,18 X2,10,3,9
Gauss code	1, -10, 2, -1, 8, -9, 3, -6, 10, -2, 4, -5, 6, -3, 7, -8, 9, -7, 5, -4
Dowker-Thistlethwaite code	4 10 16 14 2 20 8 18 6 12
Conway Notation	[212212]

Three dimensional invariants

Symmetry type Fully amphicheiral Unknotting number 2 3-genus 3 Bridge index 2 Super bridge index Missing Nakanishi index 1 Maximal Thurston-Bennequin number [-6][-6] Hyperbolic Volume 12.6026 A-Polynomial See Data:10 43/A-polynomial

[edit Notes for 10 43's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus $1$ Topological 4 genus $1$ Concordance genus $3$ Rasmussen s-Invariant 0

[edit Notes for 10 43's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{3}+7t^{2}-17t+23-17t^{-1}+7t^{-2}-t^{-3}$
Conway polynomial	$-z^{6}+z^{4}+2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 73, 0 }
Jones polynomial	$-q^{5}+3q^{4}-6q^{3}+9q^{2}-11q+13-11q^{-1}+9q^{-2}-6q^{-3}+3q^{-4}-q^{-5}$
HOMFLY-PT polynomial (db, data sources)	$-z^{6}+2a^{2}z^{4}+2z^{4}a^{-2}-3z^{4}-a^{4}z^{2}+4a^{2}z^{2}+4z^{2}a^{-2}-z^{2}a^{-4}-4z^{2}-a^{4}+2a^{2}+2a^{-2}-a^{-4}-1$
Kauffman polynomial (db, data sources)	$az^{9}+z^{9}a^{-1}+3a^{2}z^{8}+3z^{8}a^{-2}+6z^{8}+4a^{3}z^{7}+7az^{7}+7z^{7}a^{-1}+4z^{7}a^{-3}+3a^{4}z^{6}+3z^{6}a^{-4}-6z^{6}+a^{5}z^{5}-6a^{3}z^{5}-16az^{5}-16z^{5}a^{-1}-6z^{5}a^{-3}+z^{5}a^{-5}-6a^{4}z^{4}-8a^{2}z^{4}-8z^{4}a^{-2}-6z^{4}a^{-4}-4z^{4}-2a^{5}z^{3}+a^{3}z^{3}+12az^{3}+12z^{3}a^{-1}+z^{3}a^{-3}-2z^{3}a^{-5}+3a^{4}z^{2}+7a^{2}z^{2}+7z^{2}a^{-2}+3z^{2}a^{-4}+8z^{2}+a^{5}z-3az-3za^{-1}+za^{-5}-a^{4}-2a^{2}-2a^{-2}-a^{-4}-1$
The A2 invariant	$-q^{16}+q^{12}-2q^{10}+2q^{8}-q^{4}+3q^{2}-1+3q^{-2}-q^{-4}+2q^{-8}-2q^{-10}+q^{-12}-q^{-16}$
The G2 invariant	$q^{80}-2q^{78}+5q^{76}-8q^{74}+8q^{72}-6q^{70}-3q^{68}+16q^{66}-30q^{64}+41q^{62}-46q^{60}+29q^{58}+2q^{56}-45q^{54}+88q^{52}-109q^{50}+105q^{48}-63q^{46}-10q^{44}+86q^{42}-144q^{40}+156q^{38}-115q^{36}+37q^{34}+48q^{32}-108q^{30}+123q^{28}-79q^{26}+5q^{24}+68q^{22}-104q^{20}+78q^{18}-6q^{16}-87q^{14}+161q^{12}-175q^{10}+130q^{8}-25q^{6}-99q^{4}+198q^{2}-235+198q^{-2}-99q^{-4}-25q^{-6}+130q^{-8}-175q^{-10}+161q^{-12}-87q^{-14}-6q^{-16}+78q^{-18}-104q^{-20}+68q^{-22}+5q^{-24}-79q^{-26}+123q^{-28}-108q^{-30}+48q^{-32}+37q^{-34}-115q^{-36}+156q^{-38}-144q^{-40}+86q^{-42}-10q^{-44}-63q^{-46}+105q^{-48}-109q^{-50}+88q^{-52}-45q^{-54}+2q^{-56}+29q^{-58}-46q^{-60}+41q^{-62}-30q^{-64}+16q^{-66}-3q^{-68}-6q^{-70}+8q^{-72}-8q^{-74}+5q^{-76}-2q^{-78}+q^{-80}$

Further Quantum Invariants

Further quantum knot invariants for 10_43.

A1 Invariants.

Weight	Invariant
1	$-q^{11}+2q^{9}-3q^{7}+3q^{5}-2q^{3}+2q+2q^{-1}-2q^{-3}+3q^{-5}-3q^{-7}+2q^{-9}-q^{-11}$
2	$q^{32}-2q^{30}-q^{28}+7q^{26}-6q^{24}-7q^{22}+18q^{20}-7q^{18}-19q^{16}+24q^{14}-25q^{10}+18q^{8}+9q^{6}-17q^{4}+2q^{2}+13+2q^{-2}-17q^{-4}+9q^{-6}+18q^{-8}-25q^{-10}+24q^{-14}-19q^{-16}-7q^{-18}+18q^{-20}-7q^{-22}-6q^{-24}+7q^{-26}-q^{-28}-2q^{-30}+q^{-32}$
3	$-q^{63}+2q^{61}+q^{59}-3q^{57}-4q^{55}+6q^{53}+10q^{51}-12q^{49}-19q^{47}+16q^{45}+36q^{43}-15q^{41}-61q^{39}+8q^{37}+87q^{35}+11q^{33}-108q^{31}-42q^{29}+121q^{27}+75q^{25}-118q^{23}-108q^{21}+101q^{19}+125q^{17}-70q^{15}-134q^{13}+37q^{11}+122q^{9}+5q^{7}-104q^{5}-36q^{3}+74q+74q^{-1}-36q^{-3}-104q^{-5}+5q^{-7}+122q^{-9}+37q^{-11}-134q^{-13}-70q^{-15}+125q^{-17}+101q^{-19}-108q^{-21}-118q^{-23}+75q^{-25}+121q^{-27}-42q^{-29}-108q^{-31}+11q^{-33}+87q^{-35}+8q^{-37}-61q^{-39}-15q^{-41}+36q^{-43}+16q^{-45}-19q^{-47}-12q^{-49}+10q^{-51}+6q^{-53}-4q^{-55}-3q^{-57}+q^{-59}+2q^{-61}-q^{-63}$
4	$q^{104}-2q^{102}-q^{100}+3q^{98}+4q^{94}-9q^{92}-5q^{90}+13q^{88}+4q^{86}+12q^{84}-36q^{82}-28q^{80}+39q^{78}+42q^{76}+52q^{74}-95q^{72}-123q^{70}+30q^{68}+134q^{66}+214q^{64}-114q^{62}-324q^{60}-141q^{58}+184q^{56}+529q^{54}+76q^{52}-480q^{50}-514q^{48}-11q^{46}+803q^{44}+498q^{42}-361q^{40}-852q^{38}-439q^{36}+762q^{34}+866q^{32}+26q^{30}-866q^{28}-803q^{26}+412q^{24}+904q^{22}+402q^{20}-557q^{18}-859q^{16}-14q^{14}+639q^{12}+585q^{10}-142q^{8}-662q^{6}-364q^{4}+261q^{2}+619+261q^{-2}-364q^{-4}-662q^{-6}-142q^{-8}+585q^{-10}+639q^{-12}-14q^{-14}-859q^{-16}-557q^{-18}+402q^{-20}+904q^{-22}+412q^{-24}-803q^{-26}-866q^{-28}+26q^{-30}+866q^{-32}+762q^{-34}-439q^{-36}-852q^{-38}-361q^{-40}+498q^{-42}+803q^{-44}-11q^{-46}-514q^{-48}-480q^{-50}+76q^{-52}+529q^{-54}+184q^{-56}-141q^{-58}-324q^{-60}-114q^{-62}+214q^{-64}+134q^{-66}+30q^{-68}-123q^{-70}-95q^{-72}+52q^{-74}+42q^{-76}+39q^{-78}-28q^{-80}-36q^{-82}+12q^{-84}+4q^{-86}+13q^{-88}-5q^{-90}-9q^{-92}+4q^{-94}+3q^{-98}-q^{-100}-2q^{-102}+q^{-104}$
5	$-q^{155}+2q^{153}+q^{151}-3q^{149}-q^{143}+4q^{141}+4q^{139}-9q^{137}-7q^{135}+6q^{133}+12q^{131}+16q^{129}-q^{127}-36q^{125}-53q^{123}-3q^{121}+79q^{119}+110q^{117}+45q^{115}-113q^{113}-234q^{111}-161q^{109}+133q^{107}+415q^{105}+377q^{103}-50q^{101}-606q^{99}-767q^{97}-218q^{95}+751q^{93}+1294q^{91}+739q^{89}-678q^{87}-1864q^{85}-1592q^{83}+260q^{81}+2330q^{79}+2681q^{77}+596q^{75}-2449q^{73}-3818q^{71}-1917q^{69}+2069q^{67}+4773q^{65}+3487q^{63}-1137q^{61}-5240q^{59}-5042q^{57}-286q^{55}+5120q^{53}+6284q^{51}+1888q^{49}-4364q^{47}-6945q^{45}-3430q^{43}+3166q^{41}+6948q^{39}+4586q^{37}-1734q^{35}-6354q^{33}-5255q^{31}+370q^{29}+5315q^{27}+5370q^{25}+839q^{23}-4086q^{21}-5123q^{19}-1709q^{17}+2816q^{15}+4583q^{13}+2406q^{11}-1615q^{9}-4030q^{7}-2923q^{5}+534q^{3}+3448q+3448q^{-1}+534q^{-3}-2923q^{-5}-4030q^{-7}-1615q^{-9}+2406q^{-11}+4583q^{-13}+2816q^{-15}-1709q^{-17}-5123q^{-19}-4086q^{-21}+839q^{-23}+5370q^{-25}+5315q^{-27}+370q^{-29}-5255q^{-31}-6354q^{-33}-1734q^{-35}+4586q^{-37}+6948q^{-39}+3166q^{-41}-3430q^{-43}-6945q^{-45}-4364q^{-47}+1888q^{-49}+6284q^{-51}+5120q^{-53}-286q^{-55}-5042q^{-57}-5240q^{-59}-1137q^{-61}+3487q^{-63}+4773q^{-65}+2069q^{-67}-1917q^{-69}-3818q^{-71}-2449q^{-73}+596q^{-75}+2681q^{-77}+2330q^{-79}+260q^{-81}-1592q^{-83}-1864q^{-85}-678q^{-87}+739q^{-89}+1294q^{-91}+751q^{-93}-218q^{-95}-767q^{-97}-606q^{-99}-50q^{-101}+377q^{-103}+415q^{-105}+133q^{-107}-161q^{-109}-234q^{-111}-113q^{-113}+45q^{-115}+110q^{-117}+79q^{-119}-3q^{-121}-53q^{-123}-36q^{-125}-q^{-127}+16q^{-129}+12q^{-131}+6q^{-133}-7q^{-135}-9q^{-137}+4q^{-139}+4q^{-141}-q^{-143}-3q^{-149}+q^{-151}+2q^{-153}-q^{-155}$

A2 Invariants.

Weight	Invariant
1,0	$-q^{16}+q^{12}-2q^{10}+2q^{8}-q^{4}+3q^{2}-1+3q^{-2}-q^{-4}+2q^{-8}-2q^{-10}+q^{-12}-q^{-16}$
1,1	$q^{44}-4q^{42}+12q^{40}-28q^{38}+56q^{36}-98q^{34}+162q^{32}-252q^{30}+353q^{28}-462q^{26}+576q^{24}-666q^{22}+697q^{20}-664q^{18}+554q^{16}-362q^{14}+80q^{12}+252q^{10}-592q^{8}+920q^{6}-1186q^{4}+1366q^{2}-1422+1366q^{-2}-1186q^{-4}+920q^{-6}-592q^{-8}+252q^{-10}+80q^{-12}-362q^{-14}+554q^{-16}-664q^{-18}+697q^{-20}-666q^{-22}+576q^{-24}-462q^{-26}+353q^{-28}-252q^{-30}+162q^{-32}-98q^{-34}+56q^{-36}-28q^{-38}+12q^{-40}-4q^{-42}+q^{-44}$
2,0	$q^{42}-2q^{38}-q^{36}+3q^{34}+3q^{32}-5q^{30}-3q^{28}+8q^{26}+2q^{24}-11q^{22}-5q^{20}+11q^{18}+2q^{16}-14q^{14}+2q^{12}+13q^{10}-4q^{8}-7q^{6}+8q^{4}+4q^{2}-4+4q^{-2}+8q^{-4}-7q^{-6}-4q^{-8}+13q^{-10}+2q^{-12}-14q^{-14}+2q^{-16}+11q^{-18}-5q^{-20}-11q^{-22}+2q^{-24}+8q^{-26}-3q^{-28}-5q^{-30}+3q^{-32}+3q^{-34}-q^{-36}-2q^{-38}+q^{-42}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{34}-2q^{32}+q^{30}+4q^{28}-8q^{26}+2q^{24}+9q^{22}-17q^{20}+3q^{18}+17q^{16}-20q^{14}+q^{12}+19q^{10}-14q^{8}-4q^{6}+13q^{4}-4+13q^{-4}-4q^{-6}-14q^{-8}+19q^{-10}+q^{-12}-20q^{-14}+17q^{-16}+3q^{-18}-17q^{-20}+9q^{-22}+2q^{-24}-8q^{-26}+4q^{-28}+q^{-30}-2q^{-32}+q^{-34}$
1,0,0	$-q^{21}-q^{17}+q^{15}-2q^{13}+3q^{11}-q^{9}+2q^{7}-q^{5}+2q^{3}+2q^{-3}-q^{-5}+2q^{-7}-q^{-9}+3q^{-11}-2q^{-13}+q^{-15}-q^{-17}-q^{-21}$

B2 Invariants.

Weight	Invariant
0,1	$-q^{34}+2q^{32}-5q^{30}+8q^{28}-12q^{26}+16q^{24}-21q^{22}+23q^{20}-23q^{18}+21q^{16}-14q^{14}+7q^{12}+5q^{10}-16q^{8}+28q^{6}-37q^{4}+44q^{2}-46+44q^{-2}-37q^{-4}+28q^{-6}-16q^{-8}+5q^{-10}+7q^{-12}-14q^{-14}+21q^{-16}-23q^{-18}+23q^{-20}-21q^{-22}+16q^{-24}-12q^{-26}+8q^{-28}-5q^{-30}+2q^{-32}-q^{-34}$
1,0	$q^{56}-2q^{52}-2q^{50}+3q^{48}+6q^{46}-q^{44}-10q^{42}-6q^{40}+10q^{38}+15q^{36}-5q^{34}-22q^{32}-8q^{30}+20q^{28}+20q^{26}-11q^{24}-25q^{22}-2q^{20}+23q^{18}+11q^{16}-16q^{14}-15q^{12}+10q^{10}+16q^{8}-4q^{6}-15q^{4}+3q^{2}+17+3q^{-2}-15q^{-4}-4q^{-6}+16q^{-8}+10q^{-10}-15q^{-12}-16q^{-14}+11q^{-16}+23q^{-18}-2q^{-20}-25q^{-22}-11q^{-24}+20q^{-26}+20q^{-28}-8q^{-30}-22q^{-32}-5q^{-34}+15q^{-36}+10q^{-38}-6q^{-40}-10q^{-42}-q^{-44}+6q^{-46}+3q^{-48}-2q^{-50}-2q^{-52}+q^{-56}$

G2 Invariants.

Weight

Invariant

1,0

$q^{80}-2q^{78}+5q^{76}-8q^{74}+8q^{72}-6q^{70}-3q^{68}+16q^{66}-30q^{64}+41q^{62}-46q^{60}+29q^{58}+2q^{56}-45q^{54}+88q^{52}-109q^{50}+105q^{48}-63q^{46}-10q^{44}+86q^{42}-144q^{40}+156q^{38}-115q^{36}+37q^{34}+48q^{32}-108q^{30}+123q^{28}-79q^{26}+5q^{24}+68q^{22}-104q^{20}+78q^{18}-6q^{16}-87q^{14}+161q^{12}-175q^{10}+130q^{8}-25q^{6}-99q^{4}+198q^{2}-235+198q^{-2}-99q^{-4}-25q^{-6}+130q^{-8}-175q^{-10}+161q^{-12}-87q^{-14}-6q^{-16}+78q^{-18}-104q^{-20}+68q^{-22}+5q^{-24}-79q^{-26}+123q^{-28}-108q^{-30}+48q^{-32}+37q^{-34}-115q^{-36}+156q^{-38}-144q^{-40}+86q^{-42}-10q^{-44}-63q^{-46}+105q^{-48}-109q^{-50}+88q^{-52}-45q^{-54}+2q^{-56}+29q^{-58}-46q^{-60}+41q^{-62}-30q^{-64}+16q^{-66}-3q^{-68}-6q^{-70}+8q^{-72}-8q^{-74}+5q^{-76}-2q^{-78}+q^{-80}$

.

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 43"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t^{3}+7t^{2}-17t+23-17t^{-1}+7t^{-2}-t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 73, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{5}+3q^{4}-6q^{3}+9q^{2}-11q+13-11q^{-1}+9q^{-2}-6q^{-3}+3q^{-4}-q^{-5}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$az^{9}+z^{9}a^{-1}+3a^{2}z^{8}+3z^{8}a^{-2}+6z^{8}+4a^{3}z^{7}+7az^{7}+7z^{7}a^{-1}+4z^{7}a^{-3}+3a^{4}z^{6}+3z^{6}a^{-4}-6z^{6}+a^{5}z^{5}-6a^{3}z^{5}-16az^{5}-16z^{5}a^{-1}-6z^{5}a^{-3}+z^{5}a^{-5}-6a^{4}z^{4}-8a^{2}z^{4}-8z^{4}a^{-2}-6z^{4}a^{-4}-4z^{4}-2a^{5}z^{3}+a^{3}z^{3}+12az^{3}+12z^{3}a^{-1}+z^{3}a^{-3}-2z^{3}a^{-5}+3a^{4}z^{2}+7a^{2}z^{2}+7z^{2}a^{-2}+3z^{2}a^{-4}+8z^{2}+a^{5}z-3az-3za^{-1}+za^{-5}-a^{4}-2a^{2}-2a^{-2}-a^{-4}-1$

Vassiliev invariants

V2 and V3:

(2, 0)

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 $8$ $0$ $32$ ${\frac {172}{3}}$ ${\frac {20}{3}}$ $0$ $0$ $0$ $0$ ${\frac {256}{3}}$ $0$ ${\frac {1376}{3}}$ ${\frac {160}{3}}$ ${\frac {8551}{15}}$ ${\frac {676}{15}}$ ${\frac {8284}{45}}$ ${\frac {89}{9}}$ $-{\frac {89}{15}}$

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=$0 is the signature of 10 43. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \	-5	-4	-3	-2	-1	0	1	2	3	4	5	χ
11											1	-1
9										2		2
7									4	1		-3
5								5	2			3
3							6	4				-2
1						7	5					2
-1					5	7						2
-3				4	6							-2
-5			2	5								3
-7		1	4									-3
-9		2										2
-11	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, 43]]
Out[2]=	10
In[3]:=	PD[Knot[10, 43]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 4, 11, 3], X[14, 8, 15, 7], X[20, 11, 1, 12], X[12, 19, 13, 20], X[8, 14, 9, 13], X[18, 15, 19, 16], X[16, 5, 17, 6], X[6, 17, 7, 18], X[2, 10, 3, 9]]
In[4]:=	GaussCode[Knot[10, 43]]
Out[4]=	GaussCode[1, -10, 2, -1, 8, -9, 3, -6, 10, -2, 4, -5, 6, -3, 7, -8, 9, -7, 5, -4]
In[5]:=	BR[Knot[10, 43]]
Out[5]=	BR[5, {-1, -1, 2, -1, -3, 2, 4, -3, 4, 4}]
In[6]:=	alex = Alexander[Knot[10, 43]][t]
Out[6]=	-3 7 17 2 3 23 - t + -- - -- - 17 t + 7 t - t 2 t t
In[7]:=	Conway[Knot[10, 43]][z]
Out[7]=	2 4 6 1 + 2 z + z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 43]}
In[9]:=	{KnotDet[Knot[10, 43]], KnotSignature[Knot[10, 43]]}
Out[9]=	{73, 0}
In[10]:=	J=Jones[Knot[10, 43]][q]
Out[10]=	-5 3 6 9 11 2 3 4 5 13 - q + -- - -- + -- - -- - 11 q + 9 q - 6 q + 3 q - q 4 3 2 q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[10, 43], Knot[10, 91]}
In[12]:=	A2Invariant[Knot[10, 43]][q]
Out[12]=	-16 -12 2 2 -4 3 2 4 8 10 -1 - q + q - --- + -- - q + -- + 3 q - q + 2 q - 2 q + 10 8 2 q q q 12 16 q - q
In[13]:=	Kauffman[Knot[10, 43]][a, z]
Out[13]=	2 -4 2 2 4 z 3 z 5 2 3 z -1 - a - -- - 2 a - a + -- - --- - 3 a z + a z + 8 z + ---- + 2 5 a 4 a a a 2 3 3 3 7 z 2 2 4 2 2 z z 12 z 3 3 3 ---- + 7 a z + 3 a z - ---- + -- + ----- + 12 a z + a z - 2 5 3 a a a a 4 4 5 5 5 3 4 6 z 8 z 2 4 4 4 z 6 z 2 a z - 4 z - ---- - ---- - 8 a z - 6 a z + -- - ---- - 4 2 5 3 a a a a 5 6 7 16 z 5 3 5 5 5 6 3 z 4 6 4 z ----- - 16 a z - 6 a z + a z - 6 z + ---- + 3 a z + ---- + a 4 3 a a 7 8 9 7 z 7 3 7 8 3 z 2 8 z 9 ---- + 7 a z + 4 a z + 6 z + ---- + 3 a z + -- + a z a 2 a a
In[14]:=	{Vassiliev[2][Knot[10, 43]], Vassiliev[3][Knot[10, 43]]}
Out[14]=	{0, 0}
In[15]:=	Kh[Knot[10, 43]][q, t]
Out[15]=	7 1 2 1 4 2 5 4 - + 7 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + q 11 5 9 4 7 4 7 3 5 3 5 2 3 2 q t q t q t q t q t q t q t 6 5 3 3 2 5 2 5 3 7 3 ---- + --- + 5 q t + 6 q t + 4 q t + 5 q t + 2 q t + 4 q t + 3 q t q t 7 4 9 4 11 5 q t + 2 q t + q t