10 90

10_89

10_91

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10 90 Quick Notes

10 90 Further Notes and Views

Knot presentations

Planar diagram presentation	X6271 X2837 X18,14,19,13 X14,5,15,6 X20,12,1,11 X12,20,13,19 X8,15,9,16 X10,4,11,3 X16,9,17,10 X4,17,5,18
Gauss code	1, -2, 8, -10, 4, -1, 2, -7, 9, -8, 5, -6, 3, -4, 7, -9, 10, -3, 6, -5
Dowker-Thistlethwaite code	6 10 14 2 16 20 18 8 4 12
Conway Notation	[.3.2.2]

Three dimensional invariants

Symmetry type Chiral Unknotting number 2 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index 1 Maximal Thurston-Bennequin number [-5][-7] Hyperbolic Volume 13.8661 A-Polynomial See Data:10 90/A-polynomial

[edit Notes for 10 90'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 90's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-2t^{3}+8t^{2}-17t+23-17t^{-1}+8t^{-2}-2t^{-3}$
Conway polynomial	$-2z^{6}-4z^{4}-3z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 77, 0 }
Jones polynomial	$q^{6}-3q^{5}+6q^{4}-9q^{3}+12q^{2}-13q+12-10q^{-1}+7q^{-2}-3q^{-3}+q^{-4}$
HOMFLY-PT polynomial (db, data sources)	$-z^{6}a^{-2}-z^{6}+a^{2}z^{4}-3z^{4}a^{-2}+z^{4}a^{-4}-3z^{4}+2a^{2}z^{2}-3z^{2}a^{-2}+2z^{2}a^{-4}-4z^{2}+2a^{2}+a^{-4}-2$
Kauffman polynomial (db, data sources)	$2z^{9}a^{-1}+2z^{9}a^{-3}+9z^{8}a^{-2}+4z^{8}a^{-4}+5z^{8}+7az^{7}+5z^{7}a^{-1}+z^{7}a^{-3}+3z^{7}a^{-5}+6a^{2}z^{6}-21z^{6}a^{-2}-11z^{6}a^{-4}+z^{6}a^{-6}-3z^{6}+3a^{3}z^{5}-10az^{5}-15z^{5}a^{-1}-11z^{5}a^{-3}-9z^{5}a^{-5}+a^{4}z^{4}-8a^{2}z^{4}+15z^{4}a^{-2}+9z^{4}a^{-4}-3z^{4}a^{-6}-6z^{4}-2a^{3}z^{3}+7az^{3}+9z^{3}a^{-1}+7z^{3}a^{-3}+7z^{3}a^{-5}-a^{4}z^{2}+6a^{2}z^{2}-5z^{2}a^{-2}-4z^{2}a^{-4}+2z^{2}a^{-6}+8z^{2}-2az-2za^{-1}-za^{-3}-za^{-5}-2a^{2}+a^{-4}-2$
The A2 invariant	$q^{12}-q^{10}+2q^{8}+2q^{6}-2q^{4}+2q^{2}-3-q^{-6}+3q^{-8}-2q^{-10}+q^{-12}+q^{-14}-q^{-16}+q^{-18}$
The G2 invariant	$q^{66}-2q^{64}+4q^{62}-6q^{60}+6q^{58}-5q^{56}+11q^{52}-22q^{50}+33q^{48}-39q^{46}+30q^{44}-11q^{42}-20q^{40}+62q^{38}-90q^{36}+104q^{34}-85q^{32}+36q^{30}+31q^{28}-103q^{26}+157q^{24}-161q^{22}+114q^{20}-23q^{18}-74q^{16}+146q^{14}-151q^{12}+96q^{10}-2q^{8}-96q^{6}+137q^{4}-108q^{2}+9+113q^{-2}-199q^{-4}+206q^{-6}-126q^{-8}-18q^{-10}+157q^{-12}-255q^{-14}+265q^{-16}-189q^{-18}+54q^{-20}+93q^{-22}-195q^{-24}+229q^{-26}-176q^{-28}+68q^{-30}+46q^{-32}-131q^{-34}+148q^{-36}-93q^{-38}-4q^{-40}+110q^{-42}-162q^{-44}+139q^{-46}-48q^{-48}-75q^{-50}+172q^{-52}-204q^{-54}+161q^{-56}-64q^{-58}-48q^{-60}+133q^{-62}-159q^{-64}+134q^{-66}-69q^{-68}+q^{-70}+45q^{-72}-66q^{-74}+59q^{-76}-37q^{-78}+16q^{-80}+2q^{-82}-10q^{-84}+10q^{-86}-9q^{-88}+5q^{-90}-2q^{-92}+q^{-94}$

Further Quantum Invariants

Further quantum knot invariants for 10_90.

A1 Invariants.

Weight	Invariant
1	$q^{9}-2q^{7}+4q^{5}-3q^{3}+2q-q^{-1}-q^{-3}+3q^{-5}-3q^{-7}+3q^{-9}-2q^{-11}+q^{-13}$
2	$q^{26}-2q^{24}+q^{22}+6q^{20}-10q^{18}+18q^{14}-20q^{12}-7q^{10}+30q^{8}-17q^{6}-16q^{4}+26q^{2}-17q^{-2}+5q^{-4}+16q^{-6}-9q^{-8}-17q^{-10}+22q^{-12}+5q^{-14}-29q^{-16}+16q^{-18}+17q^{-20}-26q^{-22}+3q^{-24}+18q^{-26}-12q^{-28}-5q^{-30}+8q^{-32}-q^{-34}-2q^{-36}+q^{-38}$
3	$q^{51}-2q^{49}+q^{47}+3q^{45}-q^{43}-7q^{41}+q^{39}+15q^{37}-5q^{35}-28q^{33}+11q^{31}+48q^{29}-7q^{27}-86q^{25}-3q^{23}+124q^{21}+31q^{19}-151q^{17}-78q^{15}+160q^{13}+125q^{11}-133q^{9}-160q^{7}+85q^{5}+169q^{3}-21q-155q^{-1}-40q^{-3}+123q^{-5}+91q^{-7}-83q^{-9}-127q^{-11}+42q^{-13}+152q^{-15}-170q^{-19}-37q^{-21}+171q^{-23}+80q^{-25}-159q^{-27}-123q^{-29}+131q^{-31}+155q^{-33}-82q^{-35}-171q^{-37}+25q^{-39}+162q^{-41}+28q^{-43}-126q^{-45}-64q^{-47}+78q^{-49}+73q^{-51}-34q^{-53}-59q^{-55}+2q^{-57}+37q^{-59}+10q^{-61}-17q^{-63}-8q^{-65}+5q^{-67}+4q^{-69}-q^{-71}-2q^{-73}+q^{-75}$
4	$q^{84}-2q^{82}+q^{80}+3q^{78}-4q^{76}+2q^{74}-6q^{72}+4q^{70}+10q^{68}-16q^{66}+7q^{64}-6q^{62}+22q^{60}+21q^{58}-76q^{56}-29q^{54}+15q^{52}+152q^{50}+122q^{48}-214q^{46}-269q^{44}-102q^{42}+421q^{40}+560q^{38}-170q^{36}-731q^{34}-672q^{32}+451q^{30}+1245q^{28}+442q^{26}-844q^{24}-1467q^{22}-195q^{20}+1414q^{18}+1269q^{16}-178q^{14}-1617q^{12}-1035q^{10}+686q^{8}+1430q^{6}+703q^{4}-882q^{2}-1258-279q^{-2}+843q^{-4}+1071q^{-6}+40q^{-8}-903q^{-10}-852q^{-12}+175q^{-14}+1020q^{-16}+633q^{-18}-525q^{-20}-1116q^{-22}-252q^{-24}+928q^{-26}+1047q^{-28}-235q^{-30}-1333q^{-32}-676q^{-34}+758q^{-36}+1448q^{-38}+259q^{-40}-1309q^{-42}-1202q^{-44}+169q^{-46}+1532q^{-48}+976q^{-50}-687q^{-52}-1398q^{-54}-715q^{-56}+911q^{-58}+1307q^{-60}+290q^{-62}-832q^{-64}-1143q^{-66}-63q^{-68}+817q^{-70}+769q^{-72}+45q^{-74}-733q^{-76}-515q^{-78}+59q^{-80}+471q^{-82}+393q^{-84}-121q^{-86}-292q^{-88}-209q^{-90}+50q^{-92}+208q^{-94}+78q^{-96}-24q^{-98}-96q^{-100}-48q^{-102}+33q^{-104}+26q^{-106}+19q^{-108}-11q^{-110}-14q^{-112}+2q^{-114}+q^{-116}+4q^{-118}-q^{-120}-2q^{-122}+q^{-124}$
5	$q^{125}-2q^{123}+q^{121}+3q^{119}-4q^{117}-q^{115}+3q^{113}-3q^{111}-q^{109}+5q^{107}-q^{105}+15q^{101}-q^{99}-27q^{97}-39q^{95}-16q^{93}+48q^{91}+121q^{89}+114q^{87}-61q^{85}-286q^{83}-338q^{81}-35q^{79}+498q^{77}+802q^{75}+409q^{73}-667q^{71}-1577q^{69}-1237q^{67}+547q^{65}+2514q^{63}+2730q^{61}+284q^{59}-3369q^{57}-4870q^{55}-2063q^{53}+3520q^{51}+7226q^{49}+4996q^{47}-2436q^{45}-9142q^{43}-8627q^{41}-136q^{39}+9702q^{37}+12097q^{35}+4020q^{33}-8385q^{31}-14428q^{29}-8283q^{27}+5247q^{25}+14680q^{23}+11867q^{21}-914q^{19}-12781q^{17}-13779q^{15}-3448q^{13}+9153q^{11}+13615q^{9}+6881q^{7}-4783q^{5}-11677q^{3}-8780q+692q^{-1}+8712q^{-3}+9167q^{-5}+2435q^{-7}-5583q^{-9}-8546q^{-11}-4429q^{-13}+3030q^{-15}+7576q^{-17}+5501q^{-19}-1302q^{-21}-6829q^{-23}-6153q^{-25}+275q^{-27}+6647q^{-29}+6865q^{-31}+376q^{-33}-6928q^{-35}-7942q^{-37}-1212q^{-39}+7358q^{-41}+9501q^{-43}+2569q^{-45}-7451q^{-47}-11203q^{-49}-4686q^{-51}+6652q^{-53}+12622q^{-55}+7432q^{-57}-4693q^{-59}-13095q^{-61}-10270q^{-63}+1496q^{-65}+12104q^{-67}+12524q^{-69}+2473q^{-71}-9478q^{-73}-13383q^{-75}-6370q^{-77}+5424q^{-79}+12375q^{-81}+9281q^{-83}-798q^{-85}-9546q^{-87}-10368q^{-89}-3345q^{-91}+5490q^{-93}+9409q^{-95}+6061q^{-97}-1303q^{-99}-6839q^{-101}-6794q^{-103}-1963q^{-105}+3538q^{-107}+5770q^{-109}+3679q^{-111}-605q^{-113}-3737q^{-115}-3768q^{-117}-1258q^{-119}+1580q^{-121}+2800q^{-123}+1913q^{-125}-51q^{-127}-1513q^{-129}-1629q^{-131}-664q^{-133}+462q^{-135}+993q^{-137}+739q^{-139}+80q^{-141}-418q^{-143}-488q^{-145}-226q^{-147}+76q^{-149}+230q^{-151}+181q^{-153}+28q^{-155}-74q^{-157}-82q^{-159}-37q^{-161}+6q^{-163}+33q^{-165}+22q^{-167}-3q^{-169}-8q^{-171}-4q^{-173}-2q^{-175}+q^{-177}+4q^{-179}-q^{-181}-2q^{-183}+q^{-185}$

A2 Invariants.

Weight	Invariant
1,0	$q^{12}-q^{10}+2q^{8}+2q^{6}-2q^{4}+2q^{2}-3-q^{-6}+3q^{-8}-2q^{-10}+q^{-12}+q^{-14}-q^{-16}+q^{-18}$
1,1	$q^{36}-4q^{34}+10q^{32}-20q^{30}+42q^{28}-74q^{26}+120q^{24}-180q^{22}+264q^{20}-358q^{18}+454q^{16}-556q^{14}+643q^{12}-692q^{10}+672q^{8}-584q^{6}+404q^{4}-130q^{2}-206+588q^{-2}-941q^{-4}+1248q^{-6}-1450q^{-8}+1526q^{-10}-1481q^{-12}+1298q^{-14}-1014q^{-16}+658q^{-18}-275q^{-20}-94q^{-22}+424q^{-24}-660q^{-26}+789q^{-28}-818q^{-30}+760q^{-32}-640q^{-34}+489q^{-36}-344q^{-38}+220q^{-40}-126q^{-42}+66q^{-44}-30q^{-46}+12q^{-48}-4q^{-50}+q^{-52}$
2,0	$q^{32}-q^{30}+5q^{26}-6q^{22}+2q^{20}+9q^{18}-2q^{16}-15q^{14}+2q^{12}+12q^{10}-10q^{8}-10q^{6}+11q^{4}+8q^{2}-4+9q^{-4}-2q^{-6}-6q^{-8}+7q^{-10}-q^{-12}-11q^{-14}+5q^{-16}+11q^{-18}-8q^{-20}-8q^{-22}+7q^{-24}+8q^{-26}-7q^{-28}-8q^{-30}+8q^{-32}+4q^{-34}-5q^{-36}-3q^{-38}+2q^{-40}+3q^{-42}-q^{-44}-q^{-46}+q^{-48}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{28}-2q^{26}+6q^{22}-7q^{20}-3q^{18}+17q^{16}-11q^{14}-8q^{12}+24q^{10}-11q^{8}-13q^{6}+21q^{4}-6q^{2}-11+8q^{-2}+2q^{-4}-4q^{-6}-9q^{-8}+10q^{-10}+7q^{-12}-19q^{-14}+10q^{-16}+14q^{-18}-21q^{-20}+7q^{-22}+13q^{-24}-16q^{-26}+6q^{-28}+6q^{-30}-8q^{-32}+3q^{-34}+q^{-36}-2q^{-38}+q^{-40}$
1,0,0	$q^{15}-q^{13}+3q^{11}+3q^{7}-2q^{5}+2q^{3}-3q-q^{-1}-q^{-3}-q^{-5}+q^{-7}-q^{-9}+3q^{-11}-2q^{-13}+2q^{-15}-q^{-17}+2q^{-19}-q^{-21}+q^{-23}$

A4 Invariants.

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

B2 Invariants.

Weight	Invariant
0,1	$q^{28}-2q^{26}+4q^{24}-8q^{22}+13q^{20}-17q^{18}+23q^{16}-25q^{14}+28q^{12}-24q^{10}+19q^{8}-9q^{6}-3q^{4}+16q^{2}-31+40q^{-2}-50q^{-4}+52q^{-6}-51q^{-8}+44q^{-10}-33q^{-12}+21q^{-14}-6q^{-16}-6q^{-18}+17q^{-20}-23q^{-22}+27q^{-24}-26q^{-26}+24q^{-28}-20q^{-30}+14q^{-32}-9q^{-34}+5q^{-36}-2q^{-38}+q^{-40}$
1,0	$q^{46}-2q^{42}-2q^{40}+2q^{38}+7q^{36}+2q^{34}-10q^{32}-10q^{30}+6q^{28}+20q^{26}+5q^{24}-20q^{22}-18q^{20}+13q^{18}+27q^{16}+q^{14}-27q^{12}-13q^{10}+20q^{8}+20q^{6}-12q^{4}-22q^{2}+5+21q^{-2}+2q^{-4}-20q^{-6}-6q^{-8}+16q^{-10}+9q^{-12}-15q^{-14}-12q^{-16}+14q^{-18}+17q^{-20}-10q^{-22}-24q^{-24}+3q^{-26}+28q^{-28}+11q^{-30}-24q^{-32}-23q^{-34}+13q^{-36}+28q^{-38}+q^{-40}-23q^{-42}-12q^{-44}+14q^{-46}+15q^{-48}-4q^{-50}-11q^{-52}-2q^{-54}+6q^{-56}+3q^{-58}-2q^{-60}-2q^{-62}+q^{-66}$

D4 Invariants.

Weight

Invariant

1,0,0,0

$q^{38}-2q^{36}+2q^{34}-3q^{32}+7q^{30}-10q^{28}+10q^{26}-12q^{24}+20q^{22}-19q^{20}+19q^{18}-19q^{16}+24q^{14}-15q^{12}+11q^{10}-9q^{8}+2q^{6}+9q^{4}-16q^{2}+19-31q^{-2}+35q^{-4}-38q^{-6}+38q^{-8}-43q^{-10}+38q^{-12}-32q^{-14}+29q^{-16}-23q^{-18}+14q^{-20}-4q^{-22}+7q^{-26}-13q^{-28}+20q^{-30}-19q^{-32}+21q^{-34}-21q^{-36}+21q^{-38}-17q^{-40}+14q^{-42}-12q^{-44}+8q^{-46}-5q^{-48}+3q^{-50}-2q^{-52}+q^{-54}$

G2 Invariants.

Weight

Invariant

1,0

$q^{66}-2q^{64}+4q^{62}-6q^{60}+6q^{58}-5q^{56}+11q^{52}-22q^{50}+33q^{48}-39q^{46}+30q^{44}-11q^{42}-20q^{40}+62q^{38}-90q^{36}+104q^{34}-85q^{32}+36q^{30}+31q^{28}-103q^{26}+157q^{24}-161q^{22}+114q^{20}-23q^{18}-74q^{16}+146q^{14}-151q^{12}+96q^{10}-2q^{8}-96q^{6}+137q^{4}-108q^{2}+9+113q^{-2}-199q^{-4}+206q^{-6}-126q^{-8}-18q^{-10}+157q^{-12}-255q^{-14}+265q^{-16}-189q^{-18}+54q^{-20}+93q^{-22}-195q^{-24}+229q^{-26}-176q^{-28}+68q^{-30}+46q^{-32}-131q^{-34}+148q^{-36}-93q^{-38}-4q^{-40}+110q^{-42}-162q^{-44}+139q^{-46}-48q^{-48}-75q^{-50}+172q^{-52}-204q^{-54}+161q^{-56}-64q^{-58}-48q^{-60}+133q^{-62}-159q^{-64}+134q^{-66}-69q^{-68}+q^{-70}+45q^{-72}-66q^{-74}+59q^{-76}-37q^{-78}+16q^{-80}+2q^{-82}-10q^{-84}+10q^{-86}-9q^{-88}+5q^{-90}-2q^{-92}+q^{-94}$

.

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 90"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-2t^{3}+8t^{2}-17t+23-17t^{-1}+8t^{-2}-2t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 77, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^{6}-3q^{5}+6q^{4}-9q^{3}+12q^{2}-13q+12-10q^{-1}+7q^{-2}-3q^{-3}+q^{-4}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$2z^{9}a^{-1}+2z^{9}a^{-3}+9z^{8}a^{-2}+4z^{8}a^{-4}+5z^{8}+7az^{7}+5z^{7}a^{-1}+z^{7}a^{-3}+3z^{7}a^{-5}+6a^{2}z^{6}-21z^{6}a^{-2}-11z^{6}a^{-4}+z^{6}a^{-6}-3z^{6}+3a^{3}z^{5}-10az^{5}-15z^{5}a^{-1}-11z^{5}a^{-3}-9z^{5}a^{-5}+a^{4}z^{4}-8a^{2}z^{4}+15z^{4}a^{-2}+9z^{4}a^{-4}-3z^{4}a^{-6}-6z^{4}-2a^{3}z^{3}+7az^{3}+9z^{3}a^{-1}+7z^{3}a^{-3}+7z^{3}a^{-5}-a^{4}z^{2}+6a^{2}z^{2}-5z^{2}a^{-2}-4z^{2}a^{-4}+2z^{2}a^{-6}+8z^{2}-2az-2za^{-1}-za^{-3}-za^{-5}-2a^{2}+a^{-4}-2$

Vassiliev invariants

V2 and V3:

(-3, -1)

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 $-12$ $-8$ $72$ $130$ $70$ $96$ ${\frac {592}{3}}$ ${\frac {160}{3}}$ $56$ $-288$ $32$ $-1560$ $-840$ $-{\frac {13391}{10}}$ ${\frac {9826}{15}}$ $-{\frac {9474}{5}}$ ${\frac {613}{2}}$ $-{\frac {4751}{10}}$

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

\   r    \    j   \	-4	-3	-2	-1	0	1	2	3	4	5	6	χ
13											1	1
11										2		-2
9									4	1		3
7								5	2			-3
5							7	4				3
3						6	5					-1
1					6	7						-1
-1				5	7							2
-3			2	5								-3
-5		1	5									4
-7		2										-2
-9	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, 90]]
Out[2]=	10
In[3]:=	PD[Knot[10, 90]]
Out[3]=	PD[X[6, 2, 7, 1], X[2, 8, 3, 7], X[18, 14, 19, 13], X[14, 5, 15, 6], X[20, 12, 1, 11], X[12, 20, 13, 19], X[8, 15, 9, 16], X[10, 4, 11, 3], X[16, 9, 17, 10], X[4, 17, 5, 18]]
In[4]:=	GaussCode[Knot[10, 90]]
Out[4]=	GaussCode[1, -2, 8, -10, 4, -1, 2, -7, 9, -8, 5, -6, 3, -4, 7, -9, 10, -3, 6, -5]
In[5]:=	BR[Knot[10, 90]]
Out[5]=	BR[4, {-1, -1, 2, -1, 2, 3, -2, -1, 3, 2, 2}]
In[6]:=	alex = Alexander[Knot[10, 90]][t]
Out[6]=	2 8 17 2 3 23 - -- + -- - -- - 17 t + 8 t - 2 t 3 2 t t t
In[7]:=	Conway[Knot[10, 90]][z]
Out[7]=	2 4 6 1 - 3 z - 4 z - 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 90]}
In[9]:=	{KnotDet[Knot[10, 90]], KnotSignature[Knot[10, 90]]}
Out[9]=	{77, 0}
In[10]:=	J=Jones[Knot[10, 90]][q]
Out[10]=	-4 3 7 10 2 3 4 5 6 12 + q - -- + -- - -- - 13 q + 12 q - 9 q + 6 q - 3 q + q 3 2 q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[10, 90]}
In[12]:=	A2Invariant[Knot[10, 90]][q]
Out[12]=	-12 -10 2 2 2 2 6 8 10 12 14 -3 + q - q + -- + -- - -- + -- - q + 3 q - 2 q + q + q - 8 6 4 2 q q q q 16 18 q + q
In[13]:=	Kauffman[Knot[10, 90]][a, z]
Out[13]=	2 2 2 -4 2 z z 2 z 2 2 z 4 z 5 z -2 + a - 2 a - -- - -- - --- - 2 a z + 8 z + ---- - ---- - ---- + 5 3 a 6 4 2 a a a a a 3 3 3 2 2 4 2 7 z 7 z 9 z 3 3 3 4 6 a z - a z + ---- + ---- + ---- + 7 a z - 2 a z - 6 z - 5 3 a a a 4 4 4 5 5 5 3 z 9 z 15 z 2 4 4 4 9 z 11 z 15 z ---- + ---- + ----- - 8 a z + a z - ---- - ----- - ----- - 6 4 2 5 3 a a a a a a 6 6 6 7 7 5 3 5 6 z 11 z 21 z 2 6 3 z z 10 a z + 3 a z - 3 z + -- - ----- - ----- + 6 a z + ---- + -- + 6 4 2 5 3 a a a a a 7 8 8 9 9 5 z 7 8 4 z 9 z 2 z 2 z ---- + 7 a z + 5 z + ---- + ---- + ---- + ---- a 4 2 3 a a a a
In[14]:=	{Vassiliev[2][Knot[10, 90]], Vassiliev[3][Knot[10, 90]]}
Out[14]=	{0, -1}
In[15]:=	Kh[Knot[10, 90]][q, t]
Out[15]=	7 1 2 1 5 2 5 5 - + 6 q + ----- + ----- + ----- + ----- + ----- + ---- + --- + 7 q t + q 9 4 7 3 5 3 5 2 3 2 3 q t q t q t q t q t q t q t 3 3 2 5 2 5 3 7 3 7 4 9 4 6 q t + 5 q t + 7 q t + 4 q t + 5 q t + 2 q t + 4 q t + 9 5 11 5 13 6 q t + 2 q t + q t