10 71

10_70

10_72

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10 71 Further Notes and Views

Knot presentations

Planar diagram presentation	X1425 X3849 X11,15,12,14 X5,13,6,12 X13,7,14,6 X9,19,10,18 X15,20,16,1 X19,16,20,17 X17,11,18,10 X7283
Gauss code	-1, 10, -2, 1, -4, 5, -10, 2, -6, 9, -3, 4, -5, 3, -7, 8, -9, 6, -8, 7
Dowker-Thistlethwaite code	4 8 12 2 18 14 6 20 10 16
Conway Notation	[22,21,2+]

Minimum Braid Representative: Length is 10, width is 5. Braid index is 5.

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 [-6][-6] Hyperbolic Volume 13.3852 A-Polynomial See Data:10 71/A-polynomial

[edit Notes for 10 71'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 71's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{3}+7t^{2}-18t+25-18t^{-1}+7t^{-2}-t^{-3}$
Conway polynomial	$-z^{6}+z^{4}+z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 77, 0 }
Jones polynomial	$-q^{5}+3q^{4}-6q^{3}+10q^{2}-12q+13-12q^{-1}+10q^{-2}-6q^{-3}+3q^{-4}-q^{-5}$
HOMFLY-PT polynomial (db, data sources)	$-z^{6}+2a^{2}z^{4}+2z^{4}a^{-2}-3z^{4}-a^{4}z^{2}+4a^{2}z^{2}+4z^{2}a^{-2}-z^{2}a^{-4}-5z^{2}-a^{4}+3a^{2}+3a^{-2}-a^{-4}-3$
Kauffman polynomial (db, data sources)	$az^{9}+z^{9}a^{-1}+3a^{2}z^{8}+3z^{8}a^{-2}+6z^{8}+4a^{3}z^{7}+8az^{7}+8z^{7}a^{-1}+4z^{7}a^{-3}+3a^{4}z^{6}+2a^{2}z^{6}+2z^{6}a^{-2}+3z^{6}a^{-4}-2z^{6}+a^{5}z^{5}-5a^{3}z^{5}-15az^{5}-15z^{5}a^{-1}-5z^{5}a^{-3}+z^{5}a^{-5}-6a^{4}z^{4}-12a^{2}z^{4}-12z^{4}a^{-2}-6z^{4}a^{-4}-12z^{4}-2a^{5}z^{3}+7az^{3}+7z^{3}a^{-1}-2z^{3}a^{-5}+4a^{4}z^{2}+10a^{2}z^{2}+10z^{2}a^{-2}+4z^{2}a^{-4}+12z^{2}+a^{5}z+a^{3}z-az-za^{-1}+za^{-3}+za^{-5}-a^{4}-3a^{2}-3a^{-2}-a^{-4}-3$
The A2 invariant	$-q^{16}+q^{12}-2q^{10}+3q^{8}+q^{6}-q^{4}+2q^{2}-3+2q^{-2}-q^{-4}+q^{-6}+3q^{-8}-2q^{-10}+q^{-12}-q^{-16}$
The G2 invariant	$q^{80}-2q^{78}+5q^{76}-8q^{74}+8q^{72}-7q^{70}-2q^{68}+16q^{66}-32q^{64}+47q^{62}-52q^{60}+36q^{58}-3q^{56}-48q^{54}+102q^{52}-137q^{50}+132q^{48}-84q^{46}-6q^{44}+105q^{42}-179q^{40}+206q^{38}-155q^{36}+56q^{34}+61q^{32}-147q^{30}+164q^{28}-107q^{26}+10q^{24}+90q^{22}-135q^{20}+110q^{18}-14q^{16}-110q^{14}+208q^{12}-235q^{10}+166q^{8}-34q^{6}-128q^{4}+254q^{2}-299+253q^{-2}-128q^{-4}-32q^{-6}+166q^{-8}-233q^{-10}+206q^{-12}-108q^{-14}-12q^{-16}+109q^{-18}-135q^{-20}+90q^{-22}+10q^{-24}-107q^{-26}+164q^{-28}-150q^{-30}+63q^{-32}+55q^{-34}-156q^{-36}+206q^{-38}-180q^{-40}+106q^{-42}-6q^{-44}-84q^{-46}+132q^{-48}-136q^{-50}+102q^{-52}-47q^{-54}-4q^{-56}+36q^{-58}-51q^{-60}+46q^{-62}-32q^{-64}+16q^{-66}-2q^{-68}-7q^{-70}+8q^{-72}-8q^{-74}+5q^{-76}-2q^{-78}+q^{-80}$

Further Quantum Invariants

Further quantum knot invariants for 10_71.

A1 Invariants.

Weight	Invariant
1	$-q^{11}+2q^{9}-3q^{7}+4q^{5}-2q^{3}+q+q^{-1}-2q^{-3}+4q^{-5}-3q^{-7}+2q^{-9}-q^{-11}$
2	$q^{32}-2q^{30}-q^{28}+7q^{26}-7q^{24}-7q^{22}+21q^{20}-9q^{18}-22q^{16}+30q^{14}-30q^{10}+21q^{8}+11q^{6}-20q^{4}+q^{2}+15+q^{-2}-20q^{-4}+11q^{-6}+21q^{-8}-30q^{-10}+30q^{-14}-22q^{-16}-9q^{-18}+21q^{-20}-7q^{-22}-7q^{-24}+7q^{-26}-q^{-28}-2q^{-30}+q^{-32}$
3	$-q^{63}+2q^{61}+q^{59}-3q^{57}-4q^{55}+7q^{53}+10q^{51}-15q^{49}-21q^{47}+21q^{45}+44q^{43}-23q^{41}-77q^{39}+16q^{37}+115q^{35}+8q^{33}-148q^{31}-54q^{29}+171q^{27}+99q^{25}-165q^{23}-144q^{21}+138q^{19}+175q^{17}-96q^{15}-182q^{13}+46q^{11}+168q^{9}+7q^{7}-138q^{5}-56q^{3}+101q+101q^{-1}-56q^{-3}-138q^{-5}+7q^{-7}+167q^{-9}+46q^{-11}-180q^{-13}-95q^{-15}+174q^{-17}+137q^{-19}-144q^{-21}-166q^{-23}+99q^{-25}+172q^{-27}-54q^{-29}-150q^{-31}+8q^{-33}+119q^{-35}+17q^{-37}-79q^{-39}-25q^{-41}+44q^{-43}+22q^{-45}-21q^{-47}-15q^{-49}+10q^{-51}+7q^{-53}-4q^{-55}-3q^{-57}+q^{-59}+2q^{-61}-q^{-63}$
4	$q^{104}-2q^{102}-q^{100}+3q^{98}+4q^{94}-10q^{92}-5q^{90}+16q^{88}+6q^{86}+11q^{84}-45q^{82}-35q^{80}+50q^{78}+61q^{76}+63q^{74}-128q^{72}-170q^{70}+41q^{68}+205q^{66}+292q^{64}-168q^{62}-477q^{60}-208q^{58}+306q^{56}+779q^{54}+104q^{52}-753q^{50}-794q^{48}+37q^{46}+1242q^{44}+765q^{42}-599q^{40}-1349q^{38}-638q^{36}+1194q^{34}+1370q^{32}+24q^{30}-1377q^{28}-1238q^{26}+621q^{24}+1436q^{22}+642q^{20}-871q^{18}-1332q^{16}-66q^{14}+999q^{12}+932q^{10}-205q^{8}-1027q^{6}-598q^{4}+403q^{2}+976+409q^{-2}-588q^{-4}-1021q^{-6}-209q^{-8}+915q^{-10}+983q^{-12}-62q^{-14}-1311q^{-16}-851q^{-18}+637q^{-20}+1410q^{-22}+603q^{-24}-1234q^{-26}-1355q^{-28}+44q^{-30}+1375q^{-32}+1184q^{-34}-661q^{-36}-1371q^{-38}-596q^{-40}+798q^{-42}+1277q^{-44}+35q^{-46}-841q^{-48}-795q^{-50}+113q^{-52}+829q^{-54}+344q^{-56}-223q^{-58}-520q^{-60}-192q^{-62}+306q^{-64}+235q^{-66}+53q^{-68}-179q^{-70}-143q^{-72}+57q^{-74}+65q^{-76}+55q^{-78}-33q^{-80}-46q^{-82}+10q^{-84}+6q^{-86}+16q^{-88}-5q^{-90}-10q^{-92}+4q^{-94}+3q^{-98}-q^{-100}-2q^{-102}+q^{-104}$
5	$-q^{155}+2q^{153}+q^{151}-3q^{149}-q^{143}+5q^{141}+4q^{139}-12q^{137}-9q^{135}+7q^{133}+17q^{131}+24q^{129}+2q^{127}-49q^{125}-75q^{123}-12q^{121}+101q^{119}+158q^{117}+78q^{115}-145q^{113}-337q^{111}-254q^{109}+171q^{107}+604q^{105}+589q^{103}-39q^{101}-912q^{99}-1202q^{97}-373q^{95}+1165q^{93}+2062q^{91}+1214q^{89}-1099q^{87}-3064q^{85}-2613q^{83}+473q^{81}+3942q^{79}+4493q^{77}+902q^{75}-4300q^{73}-6533q^{71}-3114q^{69}+3789q^{67}+8352q^{65}+5868q^{63}-2305q^{61}-9333q^{59}-8700q^{57}-148q^{55}+9257q^{53}+11041q^{51}+3042q^{49}-8010q^{47}-12339q^{45}-5899q^{43}+5830q^{41}+12446q^{39}+8135q^{37}-3201q^{35}-11407q^{33}-9408q^{31}+579q^{29}+9517q^{27}+9738q^{25}+1643q^{23}-7232q^{21}-9264q^{19}-3313q^{17}+4893q^{15}+8320q^{13}+4522q^{11}-2747q^{9}-7229q^{7}-5426q^{5}+827q^{3}+6187q+6247q^{-1}+985q^{-3}-5254q^{-5}-7142q^{-7}-2811q^{-9}+4318q^{-11}+8093q^{-13}+4802q^{-15}-3177q^{-17}-8965q^{-19}-6968q^{-21}+1664q^{-23}+9472q^{-25}+9135q^{-27}+357q^{-29}-9294q^{-31}-11025q^{-33}-2813q^{-35}+8255q^{-37}+12209q^{-39}+5389q^{-41}-6250q^{-43}-12367q^{-45}-7686q^{-47}+3530q^{-49}+11371q^{-51}+9196q^{-53}-591q^{-55}-9250q^{-57}-9606q^{-59}-2079q^{-61}+6469q^{-63}+8899q^{-65}+3873q^{-67}-3574q^{-69}-7197q^{-71}-4670q^{-73}+1100q^{-75}+5092q^{-77}+4462q^{-79}+541q^{-81}-3006q^{-83}-3577q^{-85}-1341q^{-87}+1376q^{-89}+2443q^{-91}+1448q^{-93}-362q^{-95}-1411q^{-97}-1141q^{-99}-124q^{-101}+664q^{-103}+741q^{-105}+256q^{-107}-257q^{-109}-396q^{-111}-201q^{-113}+61q^{-115}+174q^{-117}+125q^{-119}+q^{-121}-75q^{-123}-56q^{-125}-3q^{-127}+23q^{-129}+18q^{-131}+8q^{-133}-9q^{-135}-12q^{-137}+4q^{-139}+5q^{-141}-q^{-143}-3q^{-149}+q^{-151}+2q^{-153}-q^{-155}$

A2 Invariants.

Weight	Invariant
1,0	$-q^{16}+q^{12}-2q^{10}+3q^{8}+q^{6}-q^{4}+2q^{2}-3+2q^{-2}-q^{-4}+q^{-6}+3q^{-8}-2q^{-10}+q^{-12}-q^{-16}$
2,0	$q^{42}-2q^{38}-q^{36}+3q^{34}+2q^{32}-6q^{30}-3q^{28}+10q^{26}+2q^{24}-13q^{22}-3q^{20}+15q^{18}+5q^{16}-17q^{14}+2q^{12}+14q^{10}-6q^{8}-10q^{6}+6q^{4}+4q^{2}-6+6q^{-2}+9q^{-4}-6q^{-6}-4q^{-8}+14q^{-10}-19q^{-14}+3q^{-16}+14q^{-18}-4q^{-20}-12q^{-22}+4q^{-24}+11q^{-26}-3q^{-28}-7q^{-30}+2q^{-32}+3q^{-34}-q^{-36}-2q^{-38}+q^{-42}$

A3 Invariants.

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

B2 Invariants.

Weight	Invariant
0,1	$-q^{34}+2q^{32}-5q^{30}+8q^{28}-13q^{26}+18q^{24}-23q^{22}+27q^{20}-27q^{18}+25q^{16}-17q^{14}+9q^{12}+5q^{10}-18q^{8}+32q^{6}-43q^{4}+50q^{2}-54+50q^{-2}-43q^{-4}+32q^{-6}-18q^{-8}+5q^{-10}+9q^{-12}-17q^{-14}+25q^{-16}-27q^{-18}+27q^{-20}-23q^{-22}+18q^{-24}-13q^{-26}+8q^{-28}-5q^{-30}+2q^{-32}-q^{-34}$
1,0	$q^{56}-2q^{52}-2q^{50}+3q^{48}+6q^{46}-q^{44}-11q^{42}-7q^{40}+11q^{38}+17q^{36}-4q^{34}-25q^{32}-10q^{30}+23q^{28}+24q^{26}-12q^{24}-29q^{22}-2q^{20}+28q^{18}+13q^{16}-19q^{14}-17q^{12}+12q^{10}+18q^{8}-6q^{6}-18q^{4}+2q^{2}+19+2q^{-2}-18q^{-4}-6q^{-6}+18q^{-8}+12q^{-10}-17q^{-12}-19q^{-14}+13q^{-16}+28q^{-18}-2q^{-20}-29q^{-22}-12q^{-24}+24q^{-26}+23q^{-28}-10q^{-30}-25q^{-32}-4q^{-34}+17q^{-36}+11q^{-38}-7q^{-40}-11q^{-42}-q^{-44}+6q^{-46}+3q^{-48}-2q^{-50}-2q^{-52}+q^{-56}$

G2 Invariants.

Weight

Invariant

1,0

$q^{80}-2q^{78}+5q^{76}-8q^{74}+8q^{72}-7q^{70}-2q^{68}+16q^{66}-32q^{64}+47q^{62}-52q^{60}+36q^{58}-3q^{56}-48q^{54}+102q^{52}-137q^{50}+132q^{48}-84q^{46}-6q^{44}+105q^{42}-179q^{40}+206q^{38}-155q^{36}+56q^{34}+61q^{32}-147q^{30}+164q^{28}-107q^{26}+10q^{24}+90q^{22}-135q^{20}+110q^{18}-14q^{16}-110q^{14}+208q^{12}-235q^{10}+166q^{8}-34q^{6}-128q^{4}+254q^{2}-299+253q^{-2}-128q^{-4}-32q^{-6}+166q^{-8}-233q^{-10}+206q^{-12}-108q^{-14}-12q^{-16}+109q^{-18}-135q^{-20}+90q^{-22}+10q^{-24}-107q^{-26}+164q^{-28}-150q^{-30}+63q^{-32}+55q^{-34}-156q^{-36}+206q^{-38}-180q^{-40}+106q^{-42}-6q^{-44}-84q^{-46}+132q^{-48}-136q^{-50}+102q^{-52}-47q^{-54}-4q^{-56}+36q^{-58}-51q^{-60}+46q^{-62}-32q^{-64}+16q^{-66}-2q^{-68}-7q^{-70}+8q^{-72}-8q^{-74}+5q^{-76}-2q^{-78}+q^{-80}$

.

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 71"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t^{3}+7t^{2}-18t+25-18t^{-1}+7t^{-2}-t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 77, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$az^{9}+z^{9}a^{-1}+3a^{2}z^{8}+3z^{8}a^{-2}+6z^{8}+4a^{3}z^{7}+8az^{7}+8z^{7}a^{-1}+4z^{7}a^{-3}+3a^{4}z^{6}+2a^{2}z^{6}+2z^{6}a^{-2}+3z^{6}a^{-4}-2z^{6}+a^{5}z^{5}-5a^{3}z^{5}-15az^{5}-15z^{5}a^{-1}-5z^{5}a^{-3}+z^{5}a^{-5}-6a^{4}z^{4}-12a^{2}z^{4}-12z^{4}a^{-2}-6z^{4}a^{-4}-12z^{4}-2a^{5}z^{3}+7az^{3}+7z^{3}a^{-1}-2z^{3}a^{-5}+4a^{4}z^{2}+10a^{2}z^{2}+10z^{2}a^{-2}+4z^{2}a^{-4}+12z^{2}+a^{5}z+a^{3}z-az-za^{-1}+za^{-3}+za^{-5}-a^{4}-3a^{2}-3a^{-2}-a^{-4}-3$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11n156, K11n179, ...}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {10_104, ...}

Vassiliev invariants

V2 and V3:

(1, 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 $4$ $0$ $8$ ${\frac {62}{3}}$ $-{\frac {14}{3}}$ $0$ $0$ $0$ $0$ ${\frac {32}{3}}$ $0$ ${\frac {248}{3}}$ $-{\frac {56}{3}}$ ${\frac {3631}{30}}$ ${\frac {326}{5}}$ $-{\frac {2578}{45}}$ ${\frac {209}{18}}$ $-{\frac {1169}{30}}$

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=$0 is the signature of 10 71. 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               6 2     4 3             6 4       -2 1           7 6         1 -1         6 7           1 -3       4 6             -2 -5     2 6               4 -7   1 4                 -3 -9   2                   2 -11 1                     -1

Integral Khovanov Homology (db, data source)

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

The Coloured Jones Polynomials

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

$n$	$J_{n}$
2	$q^{15}-3q^{14}+q^{13}+9q^{12}-17q^{11}+q^{10}+37q^{9}-47q^{8}-12q^{7}+89q^{6}-77q^{5}-42q^{4}+140q^{3}-87q^{2}-73q+161-73q^{-1}-87q^{-2}+140q^{-3}-42q^{-4}-77q^{-5}+89q^{-6}-12q^{-7}-47q^{-8}+37q^{-9}+q^{-10}-17q^{-11}+9q^{-12}+q^{-13}-3q^{-14}+q^{-15}$
3	$-q^{30}+3q^{29}-q^{28}-4q^{27}-2q^{26}+14q^{25}+2q^{24}-29q^{23}-8q^{22}+57q^{21}+24q^{20}-98q^{19}-62q^{18}+153q^{17}+126q^{16}-209q^{15}-220q^{14}+249q^{13}+352q^{12}-282q^{11}-485q^{10}+271q^{9}+633q^{8}-245q^{7}-754q^{6}+186q^{5}+859q^{4}-124q^{3}-914q^{2}+41q+941+33q^{-1}-914q^{-2}-116q^{-3}+859q^{-4}+178q^{-5}-753q^{-6}-238q^{-7}+631q^{-8}+264q^{-9}-482q^{-10}-275q^{-11}+349q^{-12}+243q^{-13}-218q^{-14}-203q^{-15}+124q^{-16}+149q^{-17}-62q^{-18}-96q^{-19}+25q^{-20}+56q^{-21}-8q^{-22}-29q^{-23}+2q^{-24}+14q^{-25}-2q^{-26}-4q^{-27}-q^{-28}+3q^{-29}-q^{-30}$
4	$q^{50}-3q^{49}+q^{48}+4q^{47}-3q^{46}+5q^{45}-17q^{44}+6q^{43}+25q^{42}-13q^{41}+9q^{40}-73q^{39}+19q^{38}+113q^{37}-3q^{36}+q^{35}-273q^{34}-17q^{33}+345q^{32}+179q^{31}+72q^{30}-771q^{29}-345q^{28}+642q^{27}+746q^{26}+557q^{25}-1487q^{24}-1253q^{23}+596q^{22}+1622q^{21}+1799q^{20}-1966q^{19}-2647q^{18}-179q^{17}+2332q^{16}+3644q^{15}-1775q^{14}-3978q^{13}-1578q^{12}+2453q^{11}+5481q^{10}-968q^{9}-4751q^{8}-3066q^{7}+1993q^{6}+6730q^{5}+77q^{4}-4819q^{3}-4190q^{2}+1181q+7163+1074q^{-1}-4252q^{-2}-4763q^{-3}+180q^{-4}+6734q^{-5}+1896q^{-6}-3115q^{-7}-4696q^{-8}-885q^{-9}+5468q^{-10}+2357q^{-11}-1602q^{-12}-3902q^{-13}-1700q^{-14}+3609q^{-15}+2218q^{-16}-201q^{-17}-2556q^{-18}-1876q^{-19}+1777q^{-20}+1507q^{-21}+549q^{-22}-1192q^{-23}-1399q^{-24}+572q^{-25}+676q^{-26}+590q^{-27}-335q^{-28}-724q^{-29}+99q^{-30}+162q^{-31}+321q^{-32}-26q^{-33}-264q^{-34}+12q^{-35}-2q^{-36}+110q^{-37}+16q^{-38}-73q^{-39}+10q^{-40}-13q^{-41}+25q^{-42}+6q^{-43}-17q^{-44}+5q^{-45}-3q^{-46}+4q^{-47}+q^{-48}-3q^{-49}+q^{-50}$
5	$-q^{75}+3q^{74}-q^{73}-4q^{72}+3q^{71}-2q^{69}+9q^{68}-2q^{67}-20q^{66}+6q^{65}+17q^{64}+8q^{63}+14q^{62}-28q^{61}-73q^{60}-13q^{59}+93q^{58}+132q^{57}+63q^{56}-141q^{55}-335q^{54}-208q^{53}+232q^{52}+645q^{51}+548q^{50}-218q^{49}-1123q^{48}-1225q^{47}-38q^{46}+1694q^{45}+2358q^{44}+777q^{43}-2190q^{42}-3942q^{41}-2274q^{40}+2265q^{39}+5905q^{38}+4698q^{37}-1560q^{36}-7934q^{35}-8044q^{34}-262q^{33}+9528q^{32}+12145q^{31}+3466q^{30}-10364q^{29}-16592q^{28}-7789q^{27}+9884q^{26}+20804q^{25}+13253q^{24}-8189q^{23}-24433q^{22}-19005q^{21}+5203q^{20}+26921q^{19}+24892q^{18}-1369q^{17}-28387q^{16}-30073q^{15}-3009q^{14}+28652q^{13}+34543q^{12}+7409q^{11}-28050q^{10}-37881q^{9}-11641q^{8}+26655q^{7}+40331q^{6}+15388q^{5}-24759q^{4}-41656q^{3}-18770q^{2}+22324q+42219+21627q^{-1}-19497q^{-2}-41716q^{-3}-24130q^{-4}+16071q^{-5}+40416q^{-6}+26109q^{-7}-12228q^{-8}-37918q^{-9}-27557q^{-10}+7865q^{-11}+34465q^{-12}+28141q^{-13}-3353q^{-14}-29823q^{-15}-27778q^{-16}-1073q^{-17}+24478q^{-18}+26142q^{-19}+4853q^{-20}-18487q^{-21}-23467q^{-22}-7689q^{-23}+12749q^{-24}+19702q^{-25}+9182q^{-26}-7435q^{-27}-15468q^{-28}-9473q^{-29}+3344q^{-30}+11150q^{-31}+8549q^{-32}-407q^{-33}-7295q^{-34}-6989q^{-35}-1219q^{-36}+4247q^{-37}+5130q^{-38}+1826q^{-39}-2093q^{-40}-3398q^{-41}-1770q^{-42}+778q^{-43}+2044q^{-44}+1375q^{-45}-128q^{-46}-1085q^{-47}-922q^{-48}-119q^{-49}+506q^{-50}+546q^{-51}+162q^{-52}-212q^{-53}-294q^{-54}-104q^{-55}+73q^{-56}+121q^{-57}+79q^{-58}-20q^{-59}-71q^{-60}-24q^{-61}+16q^{-62}+8q^{-63}+16q^{-64}+6q^{-65}-20q^{-66}-2q^{-67}+9q^{-68}-2q^{-69}+3q^{-71}-4q^{-72}-q^{-73}+3q^{-74}-q^{-75}$
6	$q^{105}-3q^{104}+q^{103}+4q^{102}-3q^{101}-3q^{99}+10q^{98}-13q^{97}-3q^{96}+27q^{95}-16q^{94}-11q^{93}-18q^{92}+44q^{91}-17q^{90}-4q^{89}+93q^{88}-59q^{87}-94q^{86}-120q^{85}+122q^{84}+43q^{83}+129q^{82}+374q^{81}-127q^{80}-445q^{79}-704q^{78}-23q^{77}+163q^{76}+820q^{75}+1679q^{74}+417q^{73}-1100q^{72}-2754q^{71}-1860q^{70}-890q^{69}+2140q^{68}+5766q^{67}+4320q^{66}+105q^{65}-6429q^{64}-8228q^{63}-7743q^{62}+486q^{61}+12671q^{60}+16071q^{59}+10267q^{58}-6146q^{57}-18829q^{56}-26781q^{55}-14288q^{54}+14126q^{53}+34903q^{52}+37649q^{51}+11689q^{50}-22346q^{49}-56318q^{48}-52069q^{47}-6912q^{46}+46334q^{45}+79069q^{44}+58082q^{43}+1062q^{42}-78635q^{41}-107682q^{40}-61642q^{39}+28514q^{38}+113841q^{37}+125521q^{36}+61506q^{35}-70706q^{34}-157830q^{33}-139732q^{32}-27071q^{31}+118633q^{30}+188775q^{29}+145658q^{28}-25843q^{27}-179362q^{26}-214439q^{25}-105180q^{24}+88395q^{23}+225359q^{22}+226146q^{21}+40067q^{20}-168064q^{19}-264145q^{18}-179823q^{17}+38263q^{16}+231417q^{15}+283013q^{14}+103990q^{13}-136883q^{12}-285072q^{11}-234321q^{10}-12878q^{9}+217193q^{8}+313167q^{7}+153420q^{6}-99859q^{5}-284929q^{4}-267400q^{3}-56989q^{2}+192187q+322667+189245q^{-1}-60743q^{-2}-269454q^{-3}-284321q^{-4}-96924q^{-5}+156541q^{-6}+314357q^{-7}+215957q^{-8}-15153q^{-9}-235908q^{-10}-285368q^{-11}-135952q^{-12}+104928q^{-13}+283046q^{-14}+230811q^{-15}+38316q^{-16}-178507q^{-17}-262541q^{-18}-167495q^{-19}+38345q^{-20}+222561q^{-21}+222168q^{-22}+88504q^{-23}-100992q^{-24}-208570q^{-25}-175561q^{-26}-27261q^{-27}+139141q^{-28}+180916q^{-29}+114899q^{-30}-23730q^{-31}-131093q^{-32}-149140q^{-33}-67729q^{-34}+56486q^{-35}+115371q^{-36}+105182q^{-37}+26906q^{-38}-55236q^{-39}-97296q^{-40}-70861q^{-41}+1388q^{-42}+50932q^{-43}+69344q^{-44}+40128q^{-45}-6076q^{-46}-45190q^{-47}-48017q^{-48}-17786q^{-49}+10095q^{-50}+31626q^{-51}+28508q^{-52}+11165q^{-53}-12553q^{-54}-22005q^{-55}-14284q^{-56}-4461q^{-57}+8682q^{-58}+12688q^{-59}+9763q^{-60}-403q^{-61}-6499q^{-62}-6038q^{-63}-4750q^{-64}+488q^{-65}+3493q^{-66}+4503q^{-67}+1237q^{-68}-1008q^{-69}-1368q^{-70}-2102q^{-71}-677q^{-72}+440q^{-73}+1427q^{-74}+509q^{-75}-6q^{-76}-17q^{-77}-589q^{-78}-320q^{-79}-67q^{-80}+366q^{-81}+81q^{-82}+3q^{-83}+104q^{-84}-114q^{-85}-79q^{-86}-49q^{-87}+97q^{-88}-7q^{-89}-20q^{-90}+42q^{-91}-18q^{-92}-10q^{-93}-16q^{-94}+27q^{-95}-3q^{-96}-13q^{-97}+10q^{-98}-3q^{-99}-3q^{-101}+4q^{-102}+q^{-103}-3q^{-104}+q^{-105}$
7	$-q^{140}+3q^{139}-q^{138}-4q^{137}+3q^{136}+3q^{134}-5q^{133}-6q^{132}+18q^{131}-4q^{130}-17q^{129}+10q^{128}+5q^{127}+19q^{126}-22q^{125}-49q^{124}+46q^{123}-q^{122}-25q^{121}+50q^{120}+35q^{119}+95q^{118}-67q^{117}-237q^{116}-40q^{115}-72q^{114}+13q^{113}+305q^{112}+327q^{111}+499q^{110}-6q^{109}-834q^{108}-835q^{107}-981q^{106}-329q^{105}+1035q^{104}+1848q^{103}+2739q^{102}+1559q^{101}-1414q^{100}-3589q^{99}-5583q^{98}-4419q^{97}+249q^{96}+5324q^{95}+10843q^{94}+10858q^{93}+3756q^{92}-6224q^{91}-18002q^{90}-22048q^{89}-13892q^{88}+2682q^{87}+25611q^{86}+39508q^{85}+33822q^{84}+10089q^{83}-29584q^{82}-61860q^{81}-66529q^{80}-38785q^{79}+22063q^{78}+84042q^{77}+112851q^{76}+90294q^{75}+7196q^{74}-96800q^{73}-168206q^{72}-167942q^{71}-69313q^{70}+85394q^{69}+221631q^{68}+270081q^{67}+172974q^{66}-34028q^{65}-256579q^{64}-386292q^{63}-319674q^{62}-71005q^{61}+252161q^{60}+498162q^{59}+502190q^{58}+237245q^{57}-189446q^{56}-583385q^{55}-703245q^{54}-460322q^{53}+56550q^{52}+617416q^{51}+897896q^{50}+726787q^{49}+149273q^{48}-583820q^{47}-1061619q^{46}-1012012q^{45}-415748q^{44}+474654q^{43}+1169976q^{42}+1289213q^{41}+724519q^{40}-294998q^{39}-1212370q^{38}-1533060q^{37}-1046266q^{36}+60784q^{35}+1184229q^{34}+1724267q^{33}+1357122q^{32}+206118q^{31}-1096577q^{30}-1854829q^{29}-1633207q^{28}-480434q^{27}+963299q^{26}+1925041q^{25}+1862512q^{24}+741680q^{23}-804916q^{22}-1944029q^{21}-2038995q^{20}-974677q^{19}+638143q^{18}+1924202q^{17}+2166570q^{16}+1172133q^{15}-477017q^{14}-1878611q^{13}-2252020q^{12}-1334234q^{11}+327516q^{10}+1818177q^{9}+2306060q^{8}+1465819q^{7}-191775q^{6}-1748494q^{5}-2335919q^{4}-1575157q^{3}+63986q^{2}+1670691q+2348802+1669944q^{-1}+61475q^{-2}-1580758q^{-3}-2343857q^{-4}-1755851q^{-5}-194546q^{-6}+1471175q^{-7}+2319234q^{-8}+1834551q^{-9}+339718q^{-10}-1333529q^{-11}-2265344q^{-12}-1901929q^{-13}-500698q^{-14}+1159908q^{-15}+2173423q^{-16}+1949453q^{-17}+672252q^{-18}-947289q^{-19}-2032365q^{-20}-1963836q^{-21}-844916q^{-22}+698683q^{-23}+1836854q^{-24}+1930889q^{-25}+1001339q^{-26}-425443q^{-27}-1586172q^{-28}-1839147q^{-29}-1122928q^{-30}+146973q^{-31}+1291084q^{-32}+1682678q^{-33}+1189349q^{-34}+113341q^{-35}-968596q^{-36}-1466308q^{-37}-1189743q^{-38}-329709q^{-39}+646595q^{-40}+1203823q^{-41}+1118353q^{-42}+482769q^{-43}-350263q^{-44}-919307q^{-45}-986055q^{-46}-561821q^{-47}+106912q^{-48}+640217q^{-49}+808886q^{-50}+568347q^{-51}+70019q^{-52}-391940q^{-53}-613950q^{-54}-516053q^{-55}-174862q^{-56}+194438q^{-57}+425070q^{-58}+424638q^{-59}+216201q^{-60}-55025q^{-61}-263003q^{-62}-318303q^{-63}-209546q^{-64}-27811q^{-65}+139438q^{-66}+216128q^{-67}+173949q^{-68}+65113q^{-69}-56221q^{-70}-131497q^{-71}-127792q^{-72}-71267q^{-73}+8490q^{-74}+70271q^{-75}+83625q^{-76}+60143q^{-77}+13260q^{-78}-31096q^{-79}-48559q^{-80}-43446q^{-81}-18945q^{-82}+9772q^{-83}+24774q^{-84}+27496q^{-85}+16498q^{-86}-164q^{-87}-10549q^{-88}-15397q^{-89}-11692q^{-90}-2922q^{-91}+3329q^{-92}+7818q^{-93}+7170q^{-94}+2757q^{-95}-364q^{-96}-3350q^{-97}-3768q^{-98}-2015q^{-99}-706q^{-100}+1365q^{-101}+1990q^{-102}+1035q^{-103}+541q^{-104}-395q^{-105}-735q^{-106}-483q^{-107}-569q^{-108}+88q^{-109}+442q^{-110}+196q^{-111}+192q^{-112}-44q^{-113}-70q^{-114}-q^{-115}-195q^{-116}-42q^{-117}+98q^{-118}+25q^{-119}+39q^{-120}-32q^{-121}-4q^{-122}+48q^{-123}-46q^{-124}-20q^{-125}+19q^{-126}+4q^{-127}+10q^{-128}-17q^{-129}-4q^{-130}+18q^{-131}-6q^{-132}-5q^{-133}+3q^{-134}+3q^{-136}-4q^{-137}-q^{-138}+3q^{-139}-q^{-140}$

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, 71]]
Out[2]=	PD[X[1, 4, 2, 5], X[3, 8, 4, 9], X[11, 15, 12, 14], X[5, 13, 6, 12], X[13, 7, 14, 6], X[9, 19, 10, 18], X[15, 20, 16, 1], X[19, 16, 20, 17], X[17, 11, 18, 10], X[7, 2, 8, 3]]
In[3]:=	GaussCode[Knot[10, 71]]
Out[3]=	GaussCode[-1, 10, -2, 1, -4, 5, -10, 2, -6, 9, -3, 4, -5, 3, -7, 8, -9, 6, -8, 7]
In[4]:=	DTCode[Knot[10, 71]]
Out[4]=	DTCode[4, 8, 12, 2, 18, 14, 6, 20, 10, 16]
In[5]:=	br = BR[Knot[10, 71]]
Out[5]=	BR[5, {-1, -1, 2, -1, -3, 2, 2, 4, -3, 4}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{5, 10}
In[7]:=	BraidIndex[Knot[10, 71]]
Out[7]=	5
In[8]:=	Show[DrawMorseLink[Knot[10, 71]]]
Out[8]=	-Graphics-
In[9]:=	(#[Knot[10, 71]]&) /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 1, 3, 3, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 71]][t]
Out[10]=	-3 7 18 2 3 25 - t + -- - -- - 18 t + 7 t - t 2 t t
In[11]:=	Conway[Knot[10, 71]][z]
Out[11]=	2 4 6 1 + z + z - z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 71], Knot[11, NonAlternating, 156], Knot[11, NonAlternating, 179]}
In[13]:=	{KnotDet[Knot[10, 71]], KnotSignature[Knot[10, 71]]}
Out[13]=	{77, 0}
In[14]:=	Jones[Knot[10, 71]][q]
Out[14]=	-5 3 6 10 12 2 3 4 5 13 - q + -- - -- + -- - -- - 12 q + 10 q - 6 q + 3 q - q 4 3 2 q q q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[10, 71], Knot[10, 104]}
In[16]:=	A2Invariant[Knot[10, 71]][q]
Out[16]=	-16 -12 2 3 -6 -4 2 2 4 6 8 -3 - q + q - --- + -- + q - q + -- + 2 q - q + q + 3 q - 10 8 2 q q q 10 12 16 2 q + q - q
In[17]:=	HOMFLYPT[Knot[10, 71]][a, z]
Out[17]=	2 2 -4 3 2 4 2 z 4 z 2 2 4 2 4 -3 - a + -- + 3 a - a - 5 z - -- + ---- + 4 a z - a z - 3 z + 2 4 2 a a a 4 2 z 2 4 6 ---- + 2 a z - z 2 a
In[18]:=	Kauffman[Knot[10, 71]][a, z]
Out[18]=	-4 3 2 4 z z z 3 5 2 -3 - a - -- - 3 a - a + -- + -- - - - a z + a z + a z + 12 z + 2 5 3 a a a a 2 2 3 3 4 z 10 z 2 2 4 2 2 z 7 z 3 5 3 ---- + ----- + 10 a z + 4 a z - ---- + ---- + 7 a z - 2 a z - 4 2 5 a a a a 4 4 5 5 5 4 6 z 12 z 2 4 4 4 z 5 z 15 z 12 z - ---- - ----- - 12 a z - 6 a z + -- - ---- - ----- - 4 2 5 3 a a a a a 6 6 5 3 5 5 5 6 3 z 2 z 2 6 4 6 15 a z - 5 a z + a z - 2 z + ---- + ---- + 2 a z + 3 a z + 4 2 a a 7 7 8 9 4 z 8 z 7 3 7 8 3 z 2 8 z 9 ---- + ---- + 8 a z + 4 a z + 6 z + ---- + 3 a z + -- + a z 3 a 2 a a a
In[19]:=	{Vassiliev[2][Knot[10, 71]], Vassiliev[3][Knot[10, 71]]}
Out[19]=	{1, 0}
In[20]:=	Kh[Knot[10, 71]][q, t]
Out[20]=	7 1 2 1 4 2 6 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 6 3 3 2 5 2 5 3 7 3 ---- + --- + 6 q t + 6 q t + 4 q t + 6 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
In[21]:=	ColouredJones[Knot[10, 71], 2][q]
Out[21]=	-15 3 -13 9 17 -10 37 47 12 89 77 161 + q - --- + q + --- - --- + q + -- - -- - -- + -- - -- - 14 12 11 9 8 7 6 5 q q q q q q q q 42 140 87 73 2 3 4 5 6 -- + --- - -- - -- - 73 q - 87 q + 140 q - 42 q - 77 q + 89 q - 4 3 2 q q q q 7 8 9 10 11 12 13 14 15 12 q - 47 q + 37 q + q - 17 q + 9 q + q - 3 q + q

See/edit the Rolfsen_Splice_Template.