10 89

10_88

10_90

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10 89 Further Notes and Views

Knot presentations

Planar diagram presentation	X4251 X12,8,13,7 X8394 X2,9,3,10 X20,13,1,14 X14,5,15,6 X6,19,7,20 X18,16,19,15 X16,11,17,12 X10,17,11,18
Gauss code	1, -4, 3, -1, 6, -7, 2, -3, 4, -10, 9, -2, 5, -6, 8, -9, 10, -8, 7, -5
Dowker-Thistlethwaite code	4 8 14 12 2 16 20 18 10 6
Conway Notation	[.21.210]

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index 1 Maximal Thurston-Bennequin number [-10][-2] Hyperbolic Volume 15.5661 A-Polynomial See Data:10 89/A-polynomial

[edit Notes for 10 89's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus $1$ Topological 4 genus $1$ Concordance genus $3$ Rasmussen s-Invariant 2

[edit Notes for 10 89's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{3}-8t^{2}+24t-33+24t^{-1}-8t^{-2}+t^{-3}$
Conway polynomial	$z^{6}-2z^{4}+z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 99, -2 }
Jones polynomial	$-q^{2}+5q-9+13q^{-1}-16q^{-2}+17q^{-3}-15q^{-4}+12q^{-5}-7q^{-6}+3q^{-7}-q^{-8}$
HOMFLY-PT polynomial (db, data sources)	$-a^{8}+3z^{2}a^{6}+2a^{6}-3z^{4}a^{4}-4z^{2}a^{4}-a^{4}+z^{6}a^{2}+2z^{4}a^{2}+2z^{2}a^{2}-z^{4}+1$
Kauffman polynomial (db, data sources)	$z^{5}a^{9}-2z^{3}a^{9}+za^{9}+3z^{6}a^{8}-5z^{4}a^{8}+3z^{2}a^{8}-a^{8}+5z^{7}a^{7}-7z^{5}a^{7}+4z^{3}a^{7}-za^{7}+5z^{8}a^{6}-3z^{6}a^{6}-4z^{4}a^{6}+6z^{2}a^{6}-2a^{6}+2z^{9}a^{5}+11z^{7}a^{5}-27z^{5}a^{5}+20z^{3}a^{5}-4za^{5}+12z^{8}a^{4}-15z^{6}a^{4}-2z^{4}a^{4}+6z^{2}a^{4}-a^{4}+2z^{9}a^{3}+15z^{7}a^{3}-35z^{5}a^{3}+19z^{3}a^{3}-2za^{3}+7z^{8}a^{2}-4z^{6}a^{2}-9z^{4}a^{2}+3z^{2}a^{2}+9z^{7}a-15z^{5}a+5z^{3}a+5z^{6}-6z^{4}+1+z^{5}a^{-1}$
The A2 invariant	$-q^{26}-q^{24}+2q^{22}-q^{20}-q^{18}+4q^{16}-2q^{14}+2q^{12}-2q^{8}+2q^{6}-4q^{4}+4q^{2}-q^{-2}+3q^{-4}-q^{-6}$
The G2 invariant	$q^{128}-2q^{126}+5q^{124}-8q^{122}+9q^{120}-9q^{118}+q^{116}+13q^{114}-32q^{112}+52q^{110}-67q^{108}+62q^{106}-34q^{104}-26q^{102}+111q^{100}-190q^{98}+234q^{96}-208q^{94}+89q^{92}+87q^{90}-276q^{88}+405q^{86}-402q^{84}+261q^{82}-14q^{80}-243q^{78}+405q^{76}-399q^{74}+229q^{72}+28q^{70}-252q^{68}+332q^{66}-238q^{64}+7q^{62}+270q^{60}-447q^{58}+447q^{56}-250q^{54}-76q^{52}+406q^{50}-617q^{48}+623q^{46}-423q^{44}+92q^{42}+264q^{40}-515q^{38}+577q^{36}-433q^{34}+148q^{32}+148q^{30}-350q^{28}+361q^{26}-198q^{24}-53q^{22}+282q^{20}-371q^{18}+283q^{16}-53q^{14}-226q^{12}+424q^{10}-463q^{8}+336q^{6}-99q^{4}-147q^{2}+319-360q^{-2}+293q^{-4}-149q^{-6}-q^{-8}+105q^{-10}-152q^{-12}+134q^{-14}-84q^{-16}+37q^{-18}+4q^{-20}-22q^{-22}+25q^{-24}-20q^{-26}+10q^{-28}-4q^{-30}+q^{-32}$

Further Quantum Invariants

Further quantum knot invariants for 10_89.

A1 Invariants.

Weight	Invariant
1	$-q^{17}+2q^{15}-4q^{13}+5q^{11}-3q^{9}+2q^{7}+q^{5}-3q^{3}+4q-4q^{-1}+4q^{-3}-q^{-5}$
2	$q^{48}-2q^{46}+7q^{42}-11q^{40}-4q^{38}+28q^{36}-22q^{34}-24q^{32}+52q^{30}-13q^{28}-46q^{26}+45q^{24}+10q^{22}-41q^{20}+10q^{18}+26q^{16}-11q^{14}-28q^{12}+28q^{10}+23q^{8}-51q^{6}+13q^{4}+46q^{2}-45-9q^{-2}+41q^{-4}-18q^{-6}-15q^{-8}+16q^{-10}-4q^{-14}+q^{-16}$
3	$-q^{93}+2q^{91}-3q^{87}-q^{85}+10q^{83}+2q^{81}-26q^{79}-7q^{77}+54q^{75}+29q^{73}-93q^{71}-85q^{69}+132q^{67}+173q^{65}-141q^{63}-283q^{61}+98q^{59}+399q^{57}-15q^{55}-463q^{53}-110q^{51}+468q^{49}+232q^{47}-410q^{45}-322q^{43}+300q^{41}+364q^{39}-162q^{37}-366q^{35}+23q^{33}+335q^{31}+108q^{29}-278q^{27}-235q^{25}+212q^{23}+342q^{21}-125q^{19}-431q^{17}+22q^{15}+481q^{13}+99q^{11}-472q^{9}-220q^{7}+412q^{5}+305q^{3}-296q-339q^{-1}+161q^{-3}+316q^{-5}-44q^{-7}-245q^{-9}-27q^{-11}+153q^{-13}+55q^{-15}-76q^{-17}-52q^{-19}+33q^{-21}+26q^{-23}-5q^{-25}-12q^{-27}+4q^{-31}-q^{-33}$
4	$q^{152}-2q^{150}+3q^{146}-3q^{144}+2q^{142}-8q^{140}+6q^{138}+23q^{136}-16q^{134}-22q^{132}-49q^{130}+41q^{128}+151q^{126}+15q^{124}-146q^{122}-324q^{120}-11q^{118}+554q^{116}+467q^{114}-146q^{112}-1114q^{110}-782q^{108}+849q^{106}+1711q^{104}+901q^{102}-1772q^{100}-2651q^{98}-221q^{96}+2762q^{94}+3300q^{92}-765q^{90}-4226q^{88}-2817q^{86}+1903q^{84}+5239q^{82}+1857q^{80}-3629q^{78}-4881q^{76}-610q^{74}+4849q^{72}+3947q^{70}-1251q^{68}-4730q^{66}-2716q^{64}+2638q^{62}+4147q^{60}+1049q^{58}-3021q^{56}-3414q^{54}+250q^{52}+3176q^{50}+2520q^{48}-1089q^{46}-3367q^{44}-1789q^{42}+2012q^{40}+3655q^{38}+828q^{36}-3093q^{34}-3776q^{32}+481q^{30}+4413q^{28}+3046q^{26}-1959q^{24}-5266q^{22}-1814q^{20}+3744q^{18}+4824q^{16}+447q^{14}-4886q^{12}-3842q^{10}+1286q^{8}+4600q^{6}+2754q^{4}-2443q^{2}-3854-1200q^{-2}+2355q^{-4}+3049q^{-6}+33q^{-8}-2001q^{-10}-1792q^{-12}+214q^{-14}+1649q^{-16}+787q^{-18}-325q^{-20}-942q^{-22}-424q^{-24}+407q^{-26}+404q^{-28}+159q^{-30}-217q^{-32}-218q^{-34}+20q^{-36}+70q^{-38}+80q^{-40}-12q^{-42}-41q^{-44}-6q^{-46}+q^{-48}+12q^{-50}-4q^{-54}+q^{-56}$
5	$-q^{225}+2q^{223}-3q^{219}+3q^{217}+2q^{215}-4q^{213}-3q^{209}-10q^{207}+16q^{205}+35q^{203}+4q^{201}-43q^{199}-87q^{197}-65q^{195}+86q^{193}+270q^{191}+241q^{189}-123q^{187}-596q^{185}-715q^{183}-71q^{181}+1086q^{179}+1754q^{177}+853q^{175}-1496q^{173}-3471q^{171}-2798q^{169}+1107q^{167}+5650q^{165}+6458q^{163}+1096q^{161}-7421q^{159}-11722q^{157}-6171q^{155}+7094q^{153}+17634q^{151}+14573q^{149}-2987q^{147}-22168q^{145}-25315q^{143}-5950q^{141}+22718q^{139}+36151q^{137}+19352q^{135}-17569q^{133}-44044q^{131}-34713q^{129}+6358q^{127}+46080q^{125}+48917q^{123}+9119q^{121}-41311q^{119}-58405q^{117}-25564q^{115}+30330q^{113}+61169q^{111}+39626q^{109}-15812q^{107}-57053q^{105}-48626q^{103}+919q^{101}+47596q^{99}+51649q^{97}+11772q^{95}-35438q^{93}-49490q^{91}-20768q^{89}+23129q^{87}+43925q^{85}+26084q^{83}-12255q^{81}-37181q^{79}-28848q^{77}+3427q^{75}+30902q^{73}+30564q^{71}+3847q^{69}-25727q^{67}-32663q^{65}-10767q^{63}+21497q^{61}+35822q^{59}+18379q^{57}-17048q^{55}-39615q^{53}-27604q^{51}+11020q^{49}+42957q^{47}+38068q^{45}-2269q^{43}-43836q^{41}-48586q^{39}-9591q^{37}+40532q^{35}+57018q^{33}+23457q^{31}-31987q^{29}-60809q^{27}-37193q^{25}+18594q^{23}+58265q^{21}+47857q^{19}-2576q^{17}-48963q^{15}-52714q^{13}-12990q^{11}+34586q^{9}+50561q^{7}+24670q^{5}-18247q^{3}-42087q-30233q^{-1}+3516q^{-3}+29774q^{-5}+29448q^{-7}+6868q^{-9}-17054q^{-11}-23886q^{-13}-11740q^{-15}+6613q^{-17}+16218q^{-19}+11977q^{-21}+54q^{-23}-9063q^{-25}-9378q^{-27}-2951q^{-29}+3795q^{-31}+5934q^{-33}+3362q^{-35}-813q^{-37}-3146q^{-39}-2475q^{-41}-336q^{-43}+1258q^{-45}+1433q^{-47}+592q^{-49}-389q^{-51}-698q^{-53}-372q^{-55}+38q^{-57}+252q^{-59}+202q^{-61}+41q^{-63}-88q^{-65}-84q^{-67}-16q^{-69}+20q^{-71}+21q^{-73}+10q^{-75}-q^{-77}-12q^{-79}+4q^{-83}-q^{-85}$

A2 Invariants.

Weight	Invariant
1,0	$-q^{26}-q^{24}+2q^{22}-q^{20}-q^{18}+4q^{16}-2q^{14}+2q^{12}-2q^{8}+2q^{6}-4q^{4}+4q^{2}-q^{-2}+3q^{-4}-q^{-6}$
2,0	$q^{66}+q^{64}-q^{62}-3q^{60}+q^{58}+6q^{56}-3q^{54}-12q^{52}+18q^{48}+3q^{46}-22q^{44}-2q^{42}+27q^{40}+2q^{38}-27q^{36}+21q^{32}-4q^{30}-21q^{28}+8q^{26}+9q^{24}-11q^{22}+8q^{20}+9q^{18}-11q^{16}-3q^{14}+23q^{12}-q^{10}-28q^{8}+4q^{6}+28q^{4}-7q^{2}-27+11q^{-2}+19q^{-4}-6q^{-6}-11q^{-8}+2q^{-10}+9q^{-12}-2q^{-14}-3q^{-16}+q^{-18}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{54}-2q^{52}+q^{50}+5q^{48}-10q^{46}+q^{44}+17q^{42}-25q^{40}-q^{38}+34q^{36}-35q^{34}-3q^{32}+39q^{30}-28q^{28}-7q^{26}+26q^{24}-7q^{22}-11q^{20}+16q^{16}-5q^{14}-25q^{12}+30q^{10}+7q^{8}-40q^{6}+30q^{4}+11q^{2}-34+22q^{-2}+7q^{-4}-17q^{-6}+10q^{-8}+2q^{-10}-4q^{-12}+q^{-14}$
1,0,0	$-q^{35}-q^{33}-q^{31}+2q^{29}-q^{27}+2q^{25}-q^{23}+4q^{21}-3q^{19}+3q^{17}-q^{15}-q^{11}-q^{9}+q^{7}-3q^{5}+4q^{3}-q+3q^{-1}-2q^{-3}+3q^{-5}-q^{-7}$

B2 Invariants.

Weight	Invariant
0,1	$-q^{54}+2q^{52}-5q^{50}+9q^{48}-16q^{46}+23q^{44}-33q^{42}+41q^{40}-45q^{38}+46q^{36}-39q^{34}+27q^{32}-7q^{30}-14q^{28}+39q^{26}-60q^{24}+79q^{22}-89q^{20}+90q^{18}-84q^{16}+67q^{14}-47q^{12}+22q^{10}+q^{8}-22q^{6}+38q^{4}-45q^{2}+48-44q^{-2}+39q^{-4}-27q^{-6}+18q^{-8}-10q^{-10}+4q^{-12}-q^{-14}$
1,0	$q^{88}-2q^{84}-2q^{82}+3q^{80}+7q^{78}-13q^{74}-10q^{72}+12q^{70}+25q^{68}-q^{66}-36q^{64}-21q^{62}+30q^{60}+42q^{58}-11q^{56}-51q^{54}-13q^{52}+45q^{50}+31q^{48}-28q^{46}-38q^{44}+14q^{42}+38q^{40}-q^{38}-35q^{36}-6q^{34}+31q^{32}+12q^{30}-28q^{28}-20q^{26}+25q^{24}+27q^{22}-20q^{20}-36q^{18}+12q^{16}+46q^{14}+7q^{12}-47q^{10}-29q^{8}+36q^{6}+45q^{4}-13q^{2}-46-11q^{-2}+33q^{-4}+25q^{-6}-15q^{-8}-22q^{-10}+14q^{-14}+6q^{-16}-4q^{-18}-4q^{-20}+q^{-24}$

G2 Invariants.

Weight

Invariant

1,0

$q^{128}-2q^{126}+5q^{124}-8q^{122}+9q^{120}-9q^{118}+q^{116}+13q^{114}-32q^{112}+52q^{110}-67q^{108}+62q^{106}-34q^{104}-26q^{102}+111q^{100}-190q^{98}+234q^{96}-208q^{94}+89q^{92}+87q^{90}-276q^{88}+405q^{86}-402q^{84}+261q^{82}-14q^{80}-243q^{78}+405q^{76}-399q^{74}+229q^{72}+28q^{70}-252q^{68}+332q^{66}-238q^{64}+7q^{62}+270q^{60}-447q^{58}+447q^{56}-250q^{54}-76q^{52}+406q^{50}-617q^{48}+623q^{46}-423q^{44}+92q^{42}+264q^{40}-515q^{38}+577q^{36}-433q^{34}+148q^{32}+148q^{30}-350q^{28}+361q^{26}-198q^{24}-53q^{22}+282q^{20}-371q^{18}+283q^{16}-53q^{14}-226q^{12}+424q^{10}-463q^{8}+336q^{6}-99q^{4}-147q^{2}+319-360q^{-2}+293q^{-4}-149q^{-6}-q^{-8}+105q^{-10}-152q^{-12}+134q^{-14}-84q^{-16}+37q^{-18}+4q^{-20}-22q^{-22}+25q^{-24}-20q^{-26}+10q^{-28}-4q^{-30}+q^{-32}$

.

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 89"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^{3}-8t^{2}+24t-33+24t^{-1}-8t^{-2}+t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 99, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{2}+5q-9+13q^{-1}-16q^{-2}+17q^{-3}-15q^{-4}+12q^{-5}-7q^{-6}+3q^{-7}-q^{-8}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{5}a^{9}-2z^{3}a^{9}+za^{9}+3z^{6}a^{8}-5z^{4}a^{8}+3z^{2}a^{8}-a^{8}+5z^{7}a^{7}-7z^{5}a^{7}+4z^{3}a^{7}-za^{7}+5z^{8}a^{6}-3z^{6}a^{6}-4z^{4}a^{6}+6z^{2}a^{6}-2a^{6}+2z^{9}a^{5}+11z^{7}a^{5}-27z^{5}a^{5}+20z^{3}a^{5}-4za^{5}+12z^{8}a^{4}-15z^{6}a^{4}-2z^{4}a^{4}+6z^{2}a^{4}-a^{4}+2z^{9}a^{3}+15z^{7}a^{3}-35z^{5}a^{3}+19z^{3}a^{3}-2za^{3}+7z^{8}a^{2}-4z^{6}a^{2}-9z^{4}a^{2}+3z^{2}a^{2}+9z^{7}a-15z^{5}a+5z^{3}a+5z^{6}-6z^{4}+1+z^{5}a^{-1}$

Vassiliev invariants

V2 and V3:

(1, -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 $4$ $-24$ $8$ ${\frac {302}{3}}$ ${\frac {58}{3}}$ $-96$ $-400$ $-64$ $-88$ ${\frac {32}{3}}$ $288$ ${\frac {1208}{3}}$ ${\frac {232}{3}}$ ${\frac {49471}{30}}$ $-{\frac {3062}{15}}$ ${\frac {43862}{45}}$ ${\frac {641}{18}}$ ${\frac {3871}{30}}$

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 89. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 χ 5                     1 -1 3                   4   4 1                 5 1   -4 -1               8 4     4 -3             9 6       -3 -5           8 7         1 -7         7 9           2 -9       5 8             -3 -11     2 7               5 -13   1 5                 -4 -15   2                   2 -17 1                     -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-3$ $i=-1$ $r=-7$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-4$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=-3$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=-2$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=-1$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=0$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{8}$ $r=1$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=2$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=3$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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, 89]]
Out[2]=	10
In[3]:=	PD[Knot[10, 89]]
Out[3]=	PD[X[4, 2, 5, 1], X[12, 8, 13, 7], X[8, 3, 9, 4], X[2, 9, 3, 10], X[20, 13, 1, 14], X[14, 5, 15, 6], X[6, 19, 7, 20], X[18, 16, 19, 15], X[16, 11, 17, 12], X[10, 17, 11, 18]]
In[4]:=	GaussCode[Knot[10, 89]]
Out[4]=	GaussCode[1, -4, 3, -1, 6, -7, 2, -3, 4, -10, 9, -2, 5, -6, 8, -9, 10, -8, 7, -5]
In[5]:=	BR[Knot[10, 89]]
Out[5]=	BR[5, {-1, 2, -1, 2, 3, -2, -1, -4, -3, 2, -3, -4}]
In[6]:=	alex = Alexander[Knot[10, 89]][t]
Out[6]=	-3 8 24 2 3 -33 + t - -- + -- + 24 t - 8 t + t 2 t t
In[7]:=	Conway[Knot[10, 89]][z]
Out[7]=	2 4 6 1 + z - 2 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 89]}
In[9]:=	{KnotDet[Knot[10, 89]], KnotSignature[Knot[10, 89]]}
Out[9]=	{99, -2}
In[10]:=	J=Jones[Knot[10, 89]][q]
Out[10]=	-8 3 7 12 15 17 16 13 2 -9 - q + -- - -- + -- - -- + -- - -- + -- + 5 q - q 7 6 5 4 3 2 q q q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[10, 89]}
In[12]:=	A2Invariant[Knot[10, 89]][q]
Out[12]=	-26 -24 2 -20 -18 4 2 2 2 2 4 -q - q + --- - q - q + --- - --- + --- - -- + -- - -- + 22 16 14 12 8 6 4 q q q q q q q 4 2 4 6 -- - q + 3 q - q 2 q
In[13]:=	Kauffman[Knot[10, 89]][a, z]
Out[13]=	4 6 8 3 5 7 9 2 2 1 - a - 2 a - a - 2 a z - 4 a z - a z + a z + 3 a z + 4 2 6 2 8 2 3 3 3 5 3 6 a z + 6 a z + 3 a z + 5 a z + 19 a z + 20 a z + 7 3 9 3 4 2 4 4 4 6 4 8 4 4 a z - 2 a z - 6 z - 9 a z - 2 a z - 4 a z - 5 a z + 5 z 5 3 5 5 5 7 5 9 5 6 -- - 15 a z - 35 a z - 27 a z - 7 a z + a z + 5 z - a 2 6 4 6 6 6 8 6 7 3 7 4 a z - 15 a z - 3 a z + 3 a z + 9 a z + 15 a z + 5 7 7 7 2 8 4 8 6 8 3 9 5 9 11 a z + 5 a z + 7 a z + 12 a z + 5 a z + 2 a z + 2 a z
In[14]:=	{Vassiliev[2][Knot[10, 89]], Vassiliev[3][Knot[10, 89]]}
Out[14]=	{0, -3}
In[15]:=	Kh[Knot[10, 89]][q, t]
Out[15]=	6 8 1 2 1 5 2 7 5 -- + - + ------ + ------ + ------ + ------ + ------ + ------ + ----- + 3 q 17 7 15 6 13 6 13 5 11 5 11 4 9 4 q q t q t q t q t q t q t q t 8 7 9 8 7 9 4 t 2 ----- + ----- + ----- + ----- + ---- + ---- + --- + 5 q t + q t + 9 3 7 3 7 2 5 2 5 3 q q t q t q t q t q t q t 3 2 5 3 4 q t + q t