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]

Minimum Braid Representative: Length is 12, width is 5. Braid index is 5.

A Morse Link Presentation:

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]=

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

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {...}

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

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)

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

The Coloured Jones Polynomials

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

$n$	$J_{n}$
2	$q^{7}-5q^{6}+4q^{5}+17q^{4}-36q^{3}+q^{2}+76q-86-35q^{-1}+167q^{-2}-119q^{-3}-99q^{-4}+241q^{-5}-114q^{-6}-155q^{-7}+258q^{-8}-77q^{-9}-171q^{-10}+207q^{-11}-26q^{-12}-136q^{-13}+116q^{-14}+7q^{-15}-71q^{-16}+40q^{-17}+9q^{-18}-21q^{-19}+8q^{-20}+2q^{-21}-3q^{-22}+q^{-23}$
3	$-q^{15}+5q^{14}-4q^{13}-12q^{12}+6q^{11}+36q^{10}+3q^{9}-97q^{8}-18q^{7}+167q^{6}+101q^{5}-277q^{4}-236q^{3}+368q^{2}+461q-432-736q^{-1}+411q^{-2}+1062q^{-3}-325q^{-4}-1368q^{-5}+159q^{-6}+1633q^{-7}+57q^{-8}-1827q^{-9}-294q^{-10}+1939q^{-11}+524q^{-12}-1957q^{-13}-741q^{-14}+1896q^{-15}+910q^{-16}-1730q^{-17}-1053q^{-18}+1507q^{-19}+1114q^{-20}-1204q^{-21}-1117q^{-22}+885q^{-23}+1026q^{-24}-562q^{-25}-881q^{-26}+307q^{-27}+673q^{-28}-114q^{-29}-467q^{-30}+6q^{-31}+292q^{-32}+28q^{-33}-153q^{-34}-35q^{-35}+75q^{-36}+20q^{-37}-31q^{-38}-10q^{-39}+14q^{-40}+q^{-41}-3q^{-42}-2q^{-43}+3q^{-44}-q^{-45}$
4	$q^{26}-5q^{25}+4q^{24}+12q^{23}-11q^{22}-6q^{21}-40q^{20}+33q^{19}+104q^{18}-21q^{17}-56q^{16}-278q^{15}+34q^{14}+480q^{13}+224q^{12}-53q^{11}-1109q^{10}-484q^{9}+1097q^{8}+1336q^{7}+809q^{6}-2544q^{5}-2490q^{4}+888q^{3}+3370q^{2}+3825q-3238-6045q^{-1}-1766q^{-2}+4781q^{-3}+9022q^{-4}-1392q^{-5}-9359q^{-6}-6894q^{-7}+3737q^{-8}+14355q^{-9}+2985q^{-10}-10439q^{-11}-12452q^{-12}+285q^{-13}+17662q^{-14}+7990q^{-15}-9072q^{-16}-16384q^{-17}-3972q^{-18}+18345q^{-19}+11911q^{-20}-6245q^{-21}-18027q^{-22}-7773q^{-23}+16767q^{-24}+14189q^{-25}-2636q^{-26}-17371q^{-27}-10699q^{-28}+13103q^{-29}+14582q^{-30}+1434q^{-31}-14273q^{-32}-12208q^{-33}+7749q^{-34}+12568q^{-35}+4913q^{-36}-9075q^{-37}-11306q^{-38}+2290q^{-39}+8297q^{-40}+6165q^{-41}-3589q^{-42}-7924q^{-43}-1046q^{-44}+3577q^{-45}+4756q^{-46}-128q^{-47}-3859q^{-48}-1584q^{-49}+594q^{-50}+2326q^{-51}+751q^{-52}-1186q^{-53}-774q^{-54}-268q^{-55}+695q^{-56}+419q^{-57}-218q^{-58}-161q^{-59}-181q^{-60}+130q^{-61}+106q^{-62}-40q^{-63}-45q^{-65}+20q^{-66}+16q^{-67}-13q^{-68}+6q^{-69}-6q^{-70}+3q^{-71}+2q^{-72}-3q^{-73}+q^{-74}$
5	$-q^{40}+5q^{39}-4q^{38}-12q^{37}+11q^{36}+11q^{35}+10q^{34}+4q^{33}-40q^{32}-80q^{31}+7q^{30}+140q^{29}+171q^{28}+54q^{27}-254q^{26}-490q^{25}-319q^{24}+449q^{23}+1152q^{22}+895q^{21}-429q^{20}-2084q^{19}-2458q^{18}-222q^{17}+3485q^{16}+5070q^{15}+2143q^{14}-4223q^{13}-9204q^{12}-6649q^{11}+3800q^{10}+14187q^{9}+14066q^{8}+18q^{7}-18809q^{6}-25002q^{5}-8346q^{4}+21019q^{3}+37988q^{2}+22598q-18483-51260q^{-1}-42095q^{-2}+9165q^{-3}+61828q^{-4}+65515q^{-5}+7408q^{-6}-67235q^{-7}-89671q^{-8}-30559q^{-9}+65579q^{-10}+111902q^{-11}+57841q^{-12}-56827q^{-13}-129342q^{-14}-86346q^{-15}+41963q^{-16}+140724q^{-17}+113285q^{-18}-23266q^{-19}-145828q^{-20}-136469q^{-21}+2968q^{-22}+145474q^{-23}+154852q^{-24}+17071q^{-25}-140939q^{-26}-168406q^{-27}-35656q^{-28}+133463q^{-29}+177419q^{-30}+52498q^{-31}-123496q^{-32}-182731q^{-33}-67920q^{-34}+111567q^{-35}+184355q^{-36}+82072q^{-37}-96779q^{-38}-182393q^{-39}-95395q^{-40}+79292q^{-41}+176022q^{-42}+106998q^{-43}-58440q^{-44}-164552q^{-45}-116191q^{-46}+35395q^{-47}+147300q^{-48}+121050q^{-49}-11230q^{-50}-124675q^{-51}-120244q^{-52}-11282q^{-53}+97755q^{-54}+112623q^{-55}+30011q^{-56}-69237q^{-57}-98701q^{-58}-42121q^{-59}+41861q^{-60}+79782q^{-61}+47105q^{-62}-18807q^{-63}-58903q^{-64}-44958q^{-65}+2139q^{-66}+38711q^{-67}+37774q^{-68}+7668q^{-69}-21982q^{-70}-28159q^{-71}-11294q^{-72}+10043q^{-73}+18409q^{-74}+10815q^{-75}-2801q^{-76}-10599q^{-77}-8233q^{-78}-497q^{-79}+5144q^{-80}+5264q^{-81}+1500q^{-82}-2082q^{-83}-2871q^{-84}-1305q^{-85}+601q^{-86}+1359q^{-87}+827q^{-88}-107q^{-89}-522q^{-90}-404q^{-91}-67q^{-92}+202q^{-93}+183q^{-94}+12q^{-95}-49q^{-96}-40q^{-97}-38q^{-98}+18q^{-99}+32q^{-100}-10q^{-101}-5q^{-102}+7q^{-103}-7q^{-104}-q^{-105}+6q^{-106}-3q^{-107}-2q^{-108}+3q^{-109}-q^{-110}$
6	$q^{57}-5q^{56}+4q^{55}+12q^{54}-11q^{53}-11q^{52}-15q^{51}+26q^{50}+3q^{49}+16q^{48}+94q^{47}-76q^{46}-130q^{45}-166q^{44}+70q^{43}+159q^{42}+285q^{41}+592q^{40}-162q^{39}-786q^{38}-1315q^{37}-492q^{36}+358q^{35}+1823q^{34}+3589q^{33}+1467q^{32}-1862q^{31}-6117q^{30}-5872q^{29}-3275q^{28}+4221q^{27}+14196q^{26}+13763q^{25}+4612q^{24}-13711q^{23}-25068q^{22}-27520q^{21}-7712q^{20}+29398q^{19}+52006q^{18}+47384q^{17}+2707q^{16}-50689q^{15}-95331q^{14}-79513q^{13}+4575q^{12}+101533q^{11}+157541q^{10}+110029q^{9}-14670q^{8}-179840q^{7}-249195q^{6}-151498q^{5}+65923q^{4}+290494q^{3}+350261q^{2}+197545q-153211-449328q^{-1}-478721q^{-2}-188657q^{-3}+287272q^{-4}+629512q^{-5}+615513q^{-6}+130178q^{-7}-493832q^{-8}-853697q^{-9}-674173q^{-10}+85q^{-11}+741503q^{-12}+1085958q^{-13}+658679q^{-14}-241986q^{-15}-1056845q^{-16}-1211702q^{-17}-533358q^{-18}+559209q^{-19}+1383990q^{-20}+1234571q^{-21}+245365q^{-22}-976920q^{-23}-1580502q^{-24}-1106663q^{-25}+159076q^{-26}+1414442q^{-27}+1649785q^{-28}+766439q^{-29}-695670q^{-30}-1702203q^{-31}-1533185q^{-32}-274757q^{-33}+1256130q^{-34}+1843110q^{-35}+1164970q^{-36}-369544q^{-37}-1649075q^{-38}-1770803q^{-39}-622190q^{-40}+1033995q^{-41}+1879600q^{-42}+1420734q^{-43}-80945q^{-44}-1516518q^{-45}-1879548q^{-46}-885422q^{-47}+795906q^{-48}+1828892q^{-49}+1591422q^{-50}+192914q^{-51}-1322990q^{-52}-1907246q^{-53}-1120225q^{-54}+502100q^{-55}+1683461q^{-56}+1704725q^{-57}+504094q^{-58}-1014325q^{-59}-1821200q^{-60}-1332607q^{-61}+107102q^{-62}+1374897q^{-63}+1700374q^{-64}+832731q^{-65}-553067q^{-66}-1536799q^{-67}-1430264q^{-68}-339098q^{-69}+873701q^{-70}+1474204q^{-71}+1049771q^{-72}-20239q^{-73}-1031393q^{-74}-1287386q^{-75}-669820q^{-76}+290363q^{-77}+1009106q^{-78}+1008702q^{-79}+385642q^{-80}-441190q^{-81}-893256q^{-82}-724847q^{-83}-154155q^{-84}+461572q^{-85}+708363q^{-86}+503434q^{-87}+1901q^{-88}-419792q^{-89}-518360q^{-90}-309994q^{-91}+61894q^{-92}+330955q^{-93}+368560q^{-94}+169279q^{-95}-85271q^{-96}-237747q^{-97}-234071q^{-98}-89280q^{-99}+74382q^{-100}+167164q^{-101}+135206q^{-102}+38741q^{-103}-54925q^{-104}-101709q^{-105}-76580q^{-106}-16054q^{-107}+41676q^{-108}+55570q^{-109}+37906q^{-110}+5490q^{-111}-23995q^{-112}-30296q^{-113}-18510q^{-114}+1825q^{-115}+11931q^{-116}+13858q^{-117}+8319q^{-118}-1409q^{-119}-6551q^{-120}-6486q^{-121}-1947q^{-122}+633q^{-123}+2561q^{-124}+2778q^{-125}+757q^{-126}-723q^{-127}-1264q^{-128}-474q^{-129}-299q^{-130}+163q^{-131}+538q^{-132}+222q^{-133}-39q^{-134}-183q^{-135}+6q^{-136}-80q^{-137}-39q^{-138}+84q^{-139}+29q^{-140}-2q^{-141}-35q^{-142}+23q^{-143}-5q^{-144}-18q^{-145}+13q^{-146}+2q^{-147}+q^{-148}-6q^{-149}+3q^{-150}+2q^{-151}-3q^{-152}+q^{-153}$
7	Not Available

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, 89]]
Out[2]=	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[3]:=	GaussCode[Knot[10, 89]]
Out[3]=	GaussCode[1, -4, 3, -1, 6, -7, 2, -3, 4, -10, 9, -2, 5, -6, 8, -9, 10, -8, 7, -5]
In[4]:=	DTCode[Knot[10, 89]]
Out[4]=	DTCode[4, 8, 14, 12, 2, 16, 20, 18, 10, 6]
In[5]:=	br = BR[Knot[10, 89]]
Out[5]=	BR[5, {-1, 2, -1, 2, 3, -2, -1, -4, -3, 2, -3, -4}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{5, 12}
In[7]:=	BraidIndex[Knot[10, 89]]
Out[7]=	5
In[8]:=	Show[DrawMorseLink[Knot[10, 89]]]
Out[8]=	-Graphics-
In[9]:=	(#[Knot[10, 89]]&) /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 2, 3, 3, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 89]][t]
Out[10]=	-3 8 24 2 3 -33 + t - -- + -- + 24 t - 8 t + t 2 t t
In[11]:=	Conway[Knot[10, 89]][z]
Out[11]=	2 4 6 1 + z - 2 z + z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 89]}
In[13]:=	{KnotDet[Knot[10, 89]], KnotSignature[Knot[10, 89]]}
Out[13]=	{99, -2}
In[14]:=	Jones[Knot[10, 89]][q]
Out[14]=	-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[15]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[10, 89]}
In[16]:=	A2Invariant[Knot[10, 89]][q]
Out[16]=	-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[17]:=	HOMFLYPT[Knot[10, 89]][a, z]
Out[17]=	4 6 8 2 2 4 2 6 2 4 2 4 1 - a + 2 a - a + 2 a z - 4 a z + 3 a z - z + 2 a z - 4 4 2 6 3 a z + a z
In[18]:=	Kauffman[Knot[10, 89]][a, z]
Out[18]=	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[19]:=	{Vassiliev[2][Knot[10, 89]], Vassiliev[3][Knot[10, 89]]}
Out[19]=	{1, -3}
In[20]:=	Kh[Knot[10, 89]][q, t]
Out[20]=	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
In[21]:=	ColouredJones[Knot[10, 89], 2][q]
Out[21]=	-23 3 2 8 21 9 40 71 7 116 -86 + q - --- + --- + --- - --- + --- + --- - --- + --- + --- - 22 21 20 19 18 17 16 15 14 q q q q q q q q q 136 26 207 171 77 258 155 114 241 99 119 167 --- - --- + --- - --- - -- + --- - --- - --- + --- - -- - --- + --- - 13 12 11 10 9 8 7 6 5 4 3 2 q q q q q q q q q q q q 35 2 3 4 5 6 7 -- + 76 q + q - 36 q + 17 q + 4 q - 5 q + q q

See/edit the Rolfsen_Splice_Template.