10 143

10_142

10_144

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10 143 Further Notes and Views

Knot presentations

Planar diagram presentation	X4251 X10,4,11,3 X5,14,6,15 X7,16,8,17 X15,6,16,7 X17,20,18,1 X11,18,12,19 X19,12,20,13 X13,8,14,9 X2,10,3,9
Gauss code	1, -10, 2, -1, -3, 5, -4, 9, 10, -2, -7, 8, -9, 3, -5, 4, -6, 7, -8, 6
Dowker-Thistlethwaite code	4 10 -14 -16 2 -18 -8 -6 -20 -12
Conway Notation	[31,3,21-]

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 1 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index 1 Maximal Thurston-Bennequin number [-10][0] Hyperbolic Volume 9.0709 A-Polynomial See Data:10 143/A-polynomial

[edit Notes for 10 143's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for 10 143's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{3}-3t^{2}+6t-7+6t^{-1}-3t^{-2}+t^{-3}$
Conway polynomial	$z^{6}+3z^{4}+3z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 27, -2 }
Jones polynomial	$-1+3q^{-1}-3q^{-2}+5q^{-3}-5q^{-4}+4q^{-5}-3q^{-6}+2q^{-7}-q^{-8}$
HOMFLY-PT polynomial (db, data sources)	$-z^{4}a^{6}-3z^{2}a^{6}-2a^{6}+z^{6}a^{4}+5z^{4}a^{4}+8z^{2}a^{4}+3a^{4}-z^{4}a^{2}-2z^{2}a^{2}$
Kauffman polynomial (db, data sources)	$z^{5}a^{9}-3z^{3}a^{9}+za^{9}+2z^{6}a^{8}-6z^{4}a^{8}+3z^{2}a^{8}+2z^{7}a^{7}-6z^{5}a^{7}+5z^{3}a^{7}-2za^{7}+z^{8}a^{6}-2z^{6}a^{6}+2z^{4}a^{6}-3z^{2}a^{6}+2a^{6}+3z^{7}a^{5}-10z^{5}a^{5}+14z^{3}a^{5}-5za^{5}+z^{8}a^{4}-4z^{6}a^{4}+11z^{4}a^{4}-10z^{2}a^{4}+3a^{4}+z^{7}a^{3}-3z^{5}a^{3}+7z^{3}a^{3}-3za^{3}+3z^{4}a^{2}-4z^{2}a^{2}+z^{3}a-za$
The A2 invariant	$-q^{24}-q^{20}+q^{16}-q^{14}+q^{12}+2q^{8}+2q^{6}+q^{2}-1$
The G2 invariant	$q^{128}-q^{126}+2q^{124}-3q^{122}+2q^{120}-q^{118}-2q^{116}+7q^{114}-8q^{112}+9q^{110}-7q^{108}+5q^{104}-13q^{102}+15q^{100}-11q^{98}+2q^{96}+6q^{94}-11q^{92}+11q^{90}-6q^{88}-4q^{86}+8q^{84}-13q^{82}+7q^{80}+2q^{78}-12q^{76}+18q^{74}-14q^{72}+8q^{70}+2q^{68}-11q^{66}+14q^{64}-19q^{62}+15q^{60}-3q^{58}-4q^{56}+10q^{54}-14q^{52}+14q^{50}-q^{48}-5q^{46}+5q^{44}-9q^{42}+9q^{40}+8q^{38}-13q^{36}+14q^{34}-7q^{32}+3q^{30}+10q^{28}-16q^{26}+11q^{24}-6q^{22}+5q^{20}+2q^{18}-8q^{16}+7q^{14}-3q^{12}+3q^{10}-q^{8}-q^{6}-q^{4}-q^{2}+1-q^{-2}+q^{-4}$

Further Quantum Invariants

Further quantum knot invariants for 10_143.

A1 Invariants.

Weight	Invariant
1	$-q^{17}+q^{15}-q^{13}+q^{11}-q^{9}+2q^{5}+2q-q^{-1}$
2	$q^{48}-q^{46}-q^{44}+3q^{42}-q^{40}-4q^{38}+3q^{36}+2q^{34}-5q^{32}+q^{30}+4q^{28}-3q^{26}+3q^{22}-q^{20}-4q^{18}-q^{16}+4q^{14}-3q^{12}-q^{10}+7q^{8}-2q^{4}+4q^{2}-2q^{-2}$
3	$-q^{93}+q^{91}+q^{89}-q^{87}-2q^{85}+q^{83}+5q^{81}-7q^{77}-3q^{75}+7q^{73}+8q^{71}-5q^{69}-13q^{67}+14q^{63}+7q^{61}-13q^{59}-12q^{57}+10q^{55}+15q^{53}-6q^{51}-15q^{49}+3q^{47}+12q^{45}-q^{43}-10q^{41}-2q^{39}+10q^{37}+4q^{35}-4q^{33}-10q^{31}+q^{29}+11q^{27}+q^{25}-16q^{23}-7q^{21}+15q^{19}+11q^{17}-10q^{15}-12q^{13}+9q^{11}+14q^{9}+q^{7}-9q^{5}-q^{3}+4q+4q^{-1}-2q^{-3}-3q^{-5}-q^{-7}+q^{-11}$
5	$-q^{225}+q^{223}+q^{221}-q^{219}+q^{211}+2q^{209}-2q^{207}-5q^{205}-2q^{203}+q^{201}+6q^{199}+10q^{197}+6q^{195}-10q^{193}-20q^{191}-14q^{189}+q^{187}+24q^{185}+33q^{183}+18q^{181}-18q^{179}-47q^{177}-45q^{175}-12q^{173}+41q^{171}+77q^{169}+61q^{167}-7q^{165}-82q^{163}-110q^{161}-63q^{159}+45q^{157}+139q^{155}+144q^{153}+33q^{151}-125q^{149}-205q^{147}-136q^{145}+56q^{143}+229q^{141}+231q^{139}+40q^{137}-199q^{135}-287q^{133}-141q^{131}+129q^{129}+300q^{127}+217q^{125}-49q^{123}-268q^{121}-250q^{119}-22q^{117}+209q^{115}+246q^{113}+68q^{111}-149q^{109}-210q^{107}-84q^{105}+93q^{103}+162q^{101}+81q^{99}-56q^{97}-118q^{95}-61q^{93}+32q^{91}+81q^{89}+54q^{87}-14q^{85}-71q^{83}-55q^{81}+3q^{79}+54q^{77}+78q^{75}+28q^{73}-62q^{71}-112q^{69}-65q^{67}+55q^{65}+155q^{63}+129q^{61}-27q^{59}-192q^{57}-201q^{55}-24q^{53}+205q^{51}+268q^{49}+97q^{47}-181q^{45}-316q^{43}-182q^{41}+112q^{39}+311q^{37}+246q^{35}-22q^{33}-259q^{31}-272q^{29}-75q^{27}+167q^{25}+257q^{23}+139q^{21}-60q^{19}-179q^{17}-160q^{15}-24q^{13}+106q^{11}+134q^{9}+68q^{7}-23q^{5}-83q^{3}-73q-17q^{-1}+34q^{-3}+46q^{-5}+28q^{-7}-2q^{-9}-22q^{-11}-23q^{-13}-9q^{-15}+5q^{-17}+8q^{-19}+7q^{-21}+2q^{-23}-2q^{-25}-2q^{-27}$
6	$q^{312}-q^{310}-q^{308}+q^{306}-2q^{300}+2q^{298}-2q^{294}+4q^{292}+3q^{290}+q^{288}-6q^{286}-2q^{284}-6q^{282}-8q^{280}+6q^{278}+14q^{276}+19q^{274}+3q^{272}-22q^{268}-40q^{266}-22q^{264}+6q^{262}+44q^{260}+47q^{258}+54q^{256}+10q^{254}-57q^{252}-94q^{250}-87q^{248}-25q^{246}+37q^{244}+136q^{242}+155q^{240}+90q^{238}-35q^{236}-160q^{234}-220q^{232}-207q^{230}-24q^{228}+183q^{226}+330q^{224}+318q^{222}+144q^{220}-155q^{218}-457q^{216}-501q^{214}-287q^{212}+138q^{210}+538q^{208}+719q^{206}+481q^{204}-108q^{202}-668q^{200}-921q^{198}-639q^{196}+44q^{194}+810q^{192}+1127q^{190}+758q^{188}-73q^{186}-927q^{184}-1253q^{182}-834q^{180}+168q^{178}+1064q^{176}+1300q^{174}+741q^{172}-292q^{170}-1132q^{168}-1261q^{166}-539q^{164}+471q^{162}+1125q^{160}+1038q^{158}+294q^{156}-587q^{154}-1034q^{152}-735q^{150}-19q^{148}+621q^{146}+788q^{144}+432q^{142}-157q^{140}-569q^{138}-516q^{136}-156q^{134}+239q^{132}+401q^{130}+284q^{128}-241q^{124}-257q^{122}-110q^{120}+99q^{118}+195q^{116}+161q^{114}+16q^{112}-144q^{110}-208q^{108}-130q^{106}+70q^{104}+220q^{102}+266q^{100}+119q^{98}-146q^{96}-392q^{94}-394q^{92}-81q^{90}+319q^{88}+607q^{86}+505q^{84}+29q^{82}-593q^{80}-880q^{78}-560q^{76}+159q^{74}+893q^{72}+1090q^{70}+575q^{68}-422q^{66}-1191q^{64}-1195q^{62}-443q^{60}+652q^{58}+1350q^{56}+1212q^{54}+254q^{52}-833q^{50}-1366q^{48}-1076q^{46}-128q^{44}+854q^{42}+1280q^{40}+879q^{38}+23q^{36}-753q^{34}-1050q^{32}-717q^{30}-11q^{28}+622q^{26}+795q^{24}+536q^{22}+48q^{20}-409q^{18}-574q^{16}-399q^{14}-46q^{12}+238q^{10}+356q^{8}+290q^{6}+79q^{4}-125q^{2}-210-171q^{-2}-74q^{-4}+32q^{-6}+106q^{-8}+103q^{-10}+49q^{-12}-6q^{-14}-38q^{-16}-48q^{-18}-35q^{-20}-5q^{-22}+12q^{-24}+16q^{-26}+12q^{-28}+7q^{-30}+q^{-32}-5q^{-34}-2q^{-36}-q^{-38}-q^{-40}-q^{-42}+q^{-46}$

A2 Invariants.

Weight	Invariant
1,0	$-q^{24}-q^{20}+q^{16}-q^{14}+q^{12}+2q^{8}+2q^{6}+q^{2}-1$
1,1	$q^{68}-2q^{66}+4q^{64}-8q^{62}+15q^{60}-20q^{58}+26q^{56}-34q^{54}+37q^{52}-34q^{50}+24q^{48}-14q^{46}-5q^{44}+24q^{42}-42q^{40}+58q^{38}-61q^{36}+66q^{34}-62q^{32}+50q^{30}-42q^{28}+18q^{26}-6q^{24}-14q^{22}+24q^{20}-32q^{18}+42q^{16}-28q^{14}+30q^{12}-14q^{10}+16q^{8}-6q^{6}+2q^{4}-4q^{2}-2q^{-2}+q^{-4}$
2,0	$q^{62}+q^{56}+q^{54}-q^{52}-2q^{50}-q^{48}-3q^{44}-2q^{42}+2q^{40}+q^{38}+q^{36}+2q^{34}+3q^{32}-3q^{28}-3q^{26}-4q^{24}-5q^{22}+q^{20}+4q^{18}+3q^{16}+4q^{14}+6q^{12}+3q^{10}-q^{6}+q^{4}-q^{2}-2$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{54}-q^{52}+q^{48}-2q^{46}+2q^{44}+2q^{42}-3q^{40}+2q^{38}-6q^{34}-2q^{32}-q^{30}-3q^{28}+2q^{24}+3q^{22}+2q^{20}+q^{18}+6q^{16}+5q^{10}-q^{8}-3q^{6}+2q^{4}-q^{2}-1+q^{-2}$
1,0,0	$-q^{31}-2q^{27}-q^{23}+q^{21}+q^{17}+q^{15}+q^{13}+2q^{11}+q^{9}+2q^{7}-q^{5}+q^{3}-q$
1,0,1	$q^{88}-2q^{86}+3q^{84}-3q^{82}+q^{80}+6q^{78}-11q^{76}+13q^{74}-10q^{72}+13q^{68}-21q^{66}+26q^{64}-25q^{62}+8q^{60}+4q^{58}-25q^{56}+31q^{54}-23q^{52}+23q^{50}-4q^{48}+9q^{46}-7q^{44}-5q^{42}-4q^{40}-25q^{38}+20q^{36}-38q^{34}+34q^{32}-20q^{30}+4q^{28}+22q^{26}-24q^{24}+32q^{22}-9q^{20}+12q^{18}+11q^{16}+q^{14}+7q^{12}+q^{10}-6q^{8}-q^{6}-2q^{4}-4q^{2}+3-2q^{-2}+q^{-4}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{68}+q^{62}-q^{60}-q^{58}+3q^{56}+2q^{54}-q^{52}+2q^{50}+2q^{48}-4q^{46}-9q^{44}-6q^{42}-6q^{40}-10q^{38}-3q^{36}+5q^{34}+3q^{32}+5q^{30}+12q^{28}+7q^{26}+3q^{24}+5q^{22}+4q^{20}-q^{18}-3q^{16}+2q^{14}-3q^{10}-q^{8}+2q^{6}-q^{4}+1$
1,0,0,0	$-q^{38}-2q^{34}-q^{32}-q^{30}-q^{28}+q^{26}+2q^{22}+q^{20}+2q^{18}+q^{16}+2q^{14}+q^{12}+q^{10}+q^{8}-q^{6}+q^{4}-q^{2}$

B2 Invariants.

Weight	Invariant
0,1	$-q^{54}+q^{52}-2q^{50}+3q^{48}-4q^{46}+4q^{44}-4q^{42}+3q^{40}-2q^{38}+2q^{34}-4q^{32}+5q^{30}-7q^{28}+8q^{26}-8q^{24}+7q^{22}-4q^{20}+3q^{18}+4q^{12}-3q^{10}+5q^{8}-3q^{6}+4q^{4}-3q^{2}+1-q^{-2}$
1,0	$q^{88}-q^{84}-q^{82}+q^{80}+2q^{78}-q^{76}-3q^{74}+4q^{70}+3q^{68}-3q^{66}-4q^{64}+q^{62}+4q^{60}-5q^{56}-3q^{54}+q^{52}+q^{50}-2q^{48}-3q^{46}+q^{44}+3q^{42}-3q^{38}+4q^{34}+2q^{32}-2q^{30}-2q^{28}+4q^{26}+4q^{24}-3q^{20}+2q^{18}+5q^{16}+3q^{14}-3q^{12}-3q^{10}+3q^{6}-2q^{2}-1+q^{-4}$

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{74}-q^{72}+q^{70}-2q^{68}+3q^{66}-3q^{64}+3q^{62}-3q^{60}+4q^{58}-2q^{56}+2q^{54}-q^{52}-q^{50}-q^{48}-6q^{46}-7q^{42}+2q^{40}-8q^{38}+6q^{36}-4q^{34}+9q^{32}-q^{30}+7q^{28}+q^{26}+5q^{24}+3q^{22}+2q^{18}-2q^{16}+4q^{14}-3q^{12}+2q^{10}-4q^{8}+3q^{6}-2q^{4}+q^{2}-1+q^{-2}$

G2 Invariants.

Weight

Invariant

1,0

$q^{128}-q^{126}+2q^{124}-3q^{122}+2q^{120}-q^{118}-2q^{116}+7q^{114}-8q^{112}+9q^{110}-7q^{108}+5q^{104}-13q^{102}+15q^{100}-11q^{98}+2q^{96}+6q^{94}-11q^{92}+11q^{90}-6q^{88}-4q^{86}+8q^{84}-13q^{82}+7q^{80}+2q^{78}-12q^{76}+18q^{74}-14q^{72}+8q^{70}+2q^{68}-11q^{66}+14q^{64}-19q^{62}+15q^{60}-3q^{58}-4q^{56}+10q^{54}-14q^{52}+14q^{50}-q^{48}-5q^{46}+5q^{44}-9q^{42}+9q^{40}+8q^{38}-13q^{36}+14q^{34}-7q^{32}+3q^{30}+10q^{28}-16q^{26}+11q^{24}-6q^{22}+5q^{20}+2q^{18}-8q^{16}+7q^{14}-3q^{12}+3q^{10}-q^{8}-q^{6}-q^{4}-q^{2}+1-q^{-2}+q^{-4}$

.

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["10 143"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^{3}-3t^{2}+6t-7+6t^{-1}-3t^{-2}+t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 27, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-1+3q^{-1}-3q^{-2}+5q^{-3}-5q^{-4}+4q^{-5}-3q^{-6}+2q^{-7}-q^{-8}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-z^{4}a^{6}-3z^{2}a^{6}-2a^{6}+z^{6}a^{4}+5z^{4}a^{4}+8z^{2}a^{4}+3a^{4}-z^{4}a^{2}-2z^{2}a^{2}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{5}a^{9}-3z^{3}a^{9}+za^{9}+2z^{6}a^{8}-6z^{4}a^{8}+3z^{2}a^{8}+2z^{7}a^{7}-6z^{5}a^{7}+5z^{3}a^{7}-2za^{7}+z^{8}a^{6}-2z^{6}a^{6}+2z^{4}a^{6}-3z^{2}a^{6}+2a^{6}+3z^{7}a^{5}-10z^{5}a^{5}+14z^{3}a^{5}-5za^{5}+z^{8}a^{4}-4z^{6}a^{4}+11z^{4}a^{4}-10z^{2}a^{4}+3a^{4}+z^{7}a^{3}-3z^{5}a^{3}+7z^{3}a^{3}-3za^{3}+3z^{4}a^{2}-4z^{2}a^{2}+z^{3}a-za$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {8_10, K11n106, ...}

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

Vassiliev invariants

V2 and V3:

(3, -5)

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 $12$ $-40$ $72$ $158$ $10$ $-480$ $-{\frac {2320}{3}}$ $-{\frac {256}{3}}$ $-104$ $288$ $800$ $1896$ $120$ ${\frac {37871}{10}}$ ${\frac {298}{5}}$ ${\frac {18422}{15}}$ ${\frac {593}{6}}$ ${\frac {1231}{10}}$

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 $s=$-2 is the signature of 10 143. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

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

The Coloured Jones Polynomials

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

$n$	$J_{n}$
2	$-2+2q^{-1}+4q^{-2}-8q^{-3}+4q^{-4}+11q^{-5}-16q^{-6}+2q^{-7}+18q^{-8}-21q^{-9}-q^{-10}+21q^{-11}-17q^{-12}-4q^{-13}+18q^{-14}-10q^{-15}-7q^{-16}+12q^{-17}-3q^{-18}-6q^{-19}+5q^{-20}-2q^{-22}+q^{-23}$
3	$q^{4}-q^{3}-q^{2}-2q+2+5q^{-1}-q^{-2}-7q^{-3}-6q^{-4}+15q^{-5}+12q^{-6}-12q^{-7}-27q^{-8}+17q^{-9}+33q^{-10}-8q^{-11}-49q^{-12}+8q^{-13}+50q^{-14}+2q^{-15}-59q^{-16}-3q^{-17}+56q^{-18}+10q^{-19}-53q^{-20}-15q^{-21}+48q^{-22}+19q^{-23}-40q^{-24}-24q^{-25}+30q^{-26}+28q^{-27}-19q^{-28}-29q^{-29}+8q^{-30}+27q^{-31}+q^{-32}-22q^{-33}-6q^{-34}+14q^{-35}+9q^{-36}-9q^{-37}-7q^{-38}+4q^{-39}+5q^{-40}-2q^{-41}-2q^{-42}+2q^{-44}-q^{-45}$
4	$-q^{8}+q^{7}+3q^{6}-2q^{4}-8q^{3}-4q^{2}+11q+10+10q^{-1}-18q^{-2}-32q^{-3}+5q^{-4}+20q^{-5}+52q^{-6}+3q^{-7}-67q^{-8}-39q^{-9}-8q^{-10}+103q^{-11}+71q^{-12}-70q^{-13}-95q^{-14}-82q^{-15}+125q^{-16}+149q^{-17}-36q^{-18}-123q^{-19}-159q^{-20}+115q^{-21}+197q^{-22}+3q^{-23}-116q^{-24}-208q^{-25}+95q^{-26}+211q^{-27}+24q^{-28}-95q^{-29}-222q^{-30}+72q^{-31}+194q^{-32}+39q^{-33}-60q^{-34}-213q^{-35}+34q^{-36}+156q^{-37}+58q^{-38}-8q^{-39}-181q^{-40}-15q^{-41}+89q^{-42}+63q^{-43}+54q^{-44}-119q^{-45}-45q^{-46}+12q^{-47}+35q^{-48}+87q^{-49}-45q^{-50}-33q^{-51}-33q^{-52}-8q^{-53}+69q^{-54}-q^{-55}-28q^{-57}-28q^{-58}+31q^{-59}+5q^{-60}+13q^{-61}-9q^{-62}-19q^{-63}+10q^{-64}-q^{-65}+7q^{-66}-7q^{-68}+3q^{-69}-q^{-70}+2q^{-71}-2q^{-73}+q^{-74}$
5	$-2q^{11}+4q^{9}+5q^{8}+q^{7}-3q^{6}-16q^{5}-14q^{4}+5q^{3}+25q^{2}+31q+15-28q^{-1}-65q^{-2}-51q^{-3}+15q^{-4}+91q^{-5}+106q^{-6}+38q^{-7}-93q^{-8}-181q^{-9}-121q^{-10}+72q^{-11}+225q^{-12}+237q^{-13}+25q^{-14}-271q^{-15}-363q^{-16}-125q^{-17}+238q^{-18}+474q^{-19}+293q^{-20}-206q^{-21}-562q^{-22}-419q^{-23}+104q^{-24}+609q^{-25}+571q^{-26}-35q^{-27}-625q^{-28}-648q^{-29}-73q^{-30}+618q^{-31}+736q^{-32}+121q^{-33}-599q^{-34}-748q^{-35}-193q^{-36}+571q^{-37}+786q^{-38}+211q^{-39}-549q^{-40}-772q^{-41}-244q^{-42}+513q^{-43}+770q^{-44}+268q^{-45}-481q^{-46}-745q^{-47}-293q^{-48}+420q^{-49}+713q^{-50}+330q^{-51}-344q^{-52}-664q^{-53}-362q^{-54}+243q^{-55}+587q^{-56}+391q^{-57}-127q^{-58}-486q^{-59}-399q^{-60}+12q^{-61}+359q^{-62}+373q^{-63}+92q^{-64}-220q^{-65}-316q^{-66}-159q^{-67}+89q^{-68}+227q^{-69}+180q^{-70}+19q^{-71}-125q^{-72}-161q^{-73}-84q^{-74}+35q^{-75}+111q^{-76}+99q^{-77}+33q^{-78}-50q^{-79}-89q^{-80}-59q^{-81}+3q^{-82}+52q^{-83}+61q^{-84}+25q^{-85}-21q^{-86}-43q^{-87}-33q^{-88}-q^{-89}+28q^{-90}+23q^{-91}+8q^{-92}-7q^{-93}-18q^{-94}-10q^{-95}+5q^{-96}+8q^{-97}+2q^{-98}+3q^{-99}-2q^{-100}-6q^{-101}+q^{-102}+3q^{-103}-q^{-104}+q^{-106}-2q^{-107}+2q^{-109}-q^{-110}$
6	$q^{20}-q^{19}-q^{18}-q^{15}-3q^{14}+7q^{13}+5q^{12}+4q^{11}+4q^{10}-4q^{9}-18q^{8}-33q^{7}-6q^{6}+15q^{5}+36q^{4}+59q^{3}+50q^{2}-15q-107-112q^{-1}-82q^{-2}-3q^{-3}+144q^{-4}+254q^{-5}+196q^{-6}-41q^{-7}-230q^{-8}-366q^{-9}-356q^{-10}-31q^{-11}+419q^{-12}+653q^{-13}+447q^{-14}+29q^{-15}-539q^{-16}-989q^{-17}-737q^{-18}+86q^{-19}+950q^{-20}+1223q^{-21}+885q^{-22}-138q^{-23}-1415q^{-24}-1719q^{-25}-862q^{-26}+660q^{-27}+1756q^{-28}+1972q^{-29}+820q^{-30}-1277q^{-31}-2417q^{-32}-1957q^{-33}-92q^{-34}+1760q^{-35}+2741q^{-36}+1817q^{-37}-762q^{-38}-2614q^{-39}-2691q^{-40}-811q^{-41}+1440q^{-42}+3028q^{-43}+2439q^{-44}-286q^{-45}-2512q^{-46}-2979q^{-47}-1211q^{-48}+1127q^{-49}+3030q^{-50}+2685q^{-51}-21q^{-52}-2365q^{-53}-3025q^{-54}-1361q^{-55}+927q^{-56}+2952q^{-57}+2749q^{-58}+139q^{-59}-2220q^{-60}-2991q^{-61}-1457q^{-62}+718q^{-63}+2805q^{-64}+2765q^{-65}+380q^{-66}-1936q^{-67}-2874q^{-68}-1619q^{-69}+323q^{-70}+2445q^{-71}+2712q^{-72}+792q^{-73}-1347q^{-74}-2518q^{-75}-1786q^{-76}-317q^{-77}+1729q^{-78}+2413q^{-79}+1239q^{-80}-466q^{-81}-1774q^{-82}-1699q^{-83}-971q^{-84}+719q^{-85}+1691q^{-86}+1368q^{-87}+374q^{-88}-741q^{-89}-1140q^{-90}-1207q^{-91}-177q^{-92}+689q^{-93}+949q^{-94}+700q^{-95}+113q^{-96}-309q^{-97}-838q^{-98}-494q^{-99}-77q^{-100}+266q^{-101}+418q^{-102}+366q^{-103}+251q^{-104}-249q^{-105}-256q^{-106}-258q^{-107}-134q^{-108}-7q^{-109}+152q^{-110}+295q^{-111}+53q^{-112}+43q^{-113}-84q^{-114}-122q^{-115}-154q^{-116}-55q^{-117}+112q^{-118}+40q^{-119}+103q^{-120}+41q^{-121}+3q^{-122}-89q^{-123}-74q^{-124}+13q^{-125}-22q^{-126}+41q^{-127}+34q^{-128}+40q^{-129}-22q^{-130}-30q^{-131}+6q^{-132}-25q^{-133}+3q^{-134}+6q^{-135}+22q^{-136}-4q^{-137}-8q^{-138}+9q^{-139}-9q^{-140}-2q^{-141}-2q^{-142}+8q^{-143}-2q^{-144}-4q^{-145}+5q^{-146}-2q^{-147}-q^{-149}+2q^{-150}-2q^{-152}+q^{-153}$
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, 143]]
Out[2]=	PD[X[4, 2, 5, 1], X[10, 4, 11, 3], X[5, 14, 6, 15], X[7, 16, 8, 17], X[15, 6, 16, 7], X[17, 20, 18, 1], X[11, 18, 12, 19], X[19, 12, 20, 13], X[13, 8, 14, 9], X[2, 10, 3, 9]]
In[3]:=	GaussCode[Knot[10, 143]]
Out[3]=	GaussCode[1, -10, 2, -1, -3, 5, -4, 9, 10, -2, -7, 8, -9, 3, -5, 4, -6, 7, -8, 6]
In[4]:=	DTCode[Knot[10, 143]]
Out[4]=	DTCode[4, 10, -14, -16, 2, -18, -8, -6, -20, -12]
In[5]:=	br = BR[Knot[10, 143]]
Out[5]=	BR[3, {-1, -1, -1, -1, -2, 1, 1, 1, -2, -2}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{3, 10}
In[7]:=	BraidIndex[Knot[10, 143]]
Out[7]=	3
In[8]:=	Show[DrawMorseLink[Knot[10, 143]]]
Out[8]=	-Graphics-
In[9]:=	(#[Knot[10, 143]]&) /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 1, 3, 3, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 143]][t]
Out[10]=	-3 3 6 2 3 -7 + t - -- + - + 6 t - 3 t + t 2 t t
In[11]:=	Conway[Knot[10, 143]][z]
Out[11]=	2 4 6 1 + 3 z + 3 z + z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[8, 10], Knot[10, 143], Knot[11, NonAlternating, 106]}
In[13]:=	{KnotDet[Knot[10, 143]], KnotSignature[Knot[10, 143]]}
Out[13]=	{27, -2}
In[14]:=	Jones[Knot[10, 143]][q]
Out[14]=	-8 2 3 4 5 5 3 3 -1 - 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, 143]}
In[16]:=	A2Invariant[Knot[10, 143]][q]
Out[16]=	-24 -20 -16 -14 -12 2 2 -2 -1 - q - q + q - q + q + -- + -- + q 8 6 q q
In[17]:=	HOMFLYPT[Knot[10, 143]][a, z]
Out[17]=	4 6 2 2 4 2 6 2 2 4 4 4 6 4 3 a - 2 a - 2 a z + 8 a z - 3 a z - a z + 5 a z - a z + 4 6 a z
In[18]:=	Kauffman[Knot[10, 143]][a, z]
Out[18]=	4 6 3 5 7 9 2 2 3 a + 2 a - a z - 3 a z - 5 a z - 2 a z + a z - 4 a z - 4 2 6 2 8 2 3 3 3 5 3 7 3 10 a z - 3 a z + 3 a z + a z + 7 a z + 14 a z + 5 a z - 9 3 2 4 4 4 6 4 8 4 3 5 3 a z + 3 a z + 11 a z + 2 a z - 6 a z - 3 a z - 5 5 7 5 9 5 4 6 6 6 8 6 3 7 10 a z - 6 a z + a z - 4 a z - 2 a z + 2 a z + a z + 5 7 7 7 4 8 6 8 3 a z + 2 a z + a z + a z
In[19]:=	{Vassiliev[2][Knot[10, 143]], Vassiliev[3][Knot[10, 143]]}
Out[19]=	{3, -5}
In[20]:=	Kh[Knot[10, 143]][q, t]
Out[20]=	2 2 1 1 1 2 1 2 2 -- + - + ------ + ------ + ------ + ------ + ------ + ------ + ----- + 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 3 2 2 3 1 2 ----- + ----- + ----- + ----- + ---- + ---- + q t 9 3 7 3 7 2 5 2 5 3 q t q t q t q t q t q t
In[21]:=	ColouredJones[Knot[10, 143], 2][q]
Out[21]=	-23 2 5 6 3 12 7 10 18 4 17 -2 + q - --- + --- - --- - --- + --- - --- - --- + --- - --- - --- + 22 20 19 18 17 16 15 14 13 12 q q q q q q q q q q 21 -10 21 18 2 16 11 4 8 4 2 --- - q - -- + -- + -- - -- + -- + -- - -- + -- + - 11 9 8 7 6 5 4 3 2 q q q q q q q q q q

See/edit the Rolfsen_Splice_Template.