10 81

10_80

10_82

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10 81 Further Notes and Views

Knot presentations

Planar diagram presentation	X4251 X8493 X12,6,13,5 X16,9,17,10 X20,17,1,18 X18,13,19,14 X14,19,15,20 X10,15,11,16 X6,12,7,11 X2837
Gauss code	1, -10, 2, -1, 3, -9, 10, -2, 4, -8, 9, -3, 6, -7, 8, -4, 5, -6, 7, -5
Dowker-Thistlethwaite code	4 8 12 2 16 6 18 10 20 14
Conway Notation	[(21,2)(21,2)]

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

A Morse Link Presentation:

Three dimensional invariants

Symmetry type Negative amphicheiral Unknotting number 2 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index 1 Maximal Thurston-Bennequin number [-6][-6] Hyperbolic Volume 14.4927 A-Polynomial See Data:10 81/A-polynomial

[edit Notes for 10 81'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 81's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{3}+8t^{2}-20t+27-20t^{-1}+8t^{-2}-t^{-3}$
Conway polynomial	$-z^{6}+2z^{4}+3z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 85, 0 }
Jones polynomial	$-q^{5}+3q^{4}-7q^{3}+11q^{2}-13q+15-13q^{-1}+11q^{-2}-7q^{-3}+3q^{-4}-q^{-5}$
HOMFLY-PT polynomial (db, data sources)	$-z^{6}+2a^{2}z^{4}+2z^{4}a^{-2}-2z^{4}-a^{4}z^{2}+3a^{2}z^{2}+3z^{2}a^{-2}-z^{2}a^{-4}-z^{2}-a^{4}+a^{2}+a^{-2}-a^{-4}+1$
Kauffman polynomial (db, data sources)	$az^{9}+z^{9}a^{-1}+4a^{2}z^{8}+4z^{8}a^{-2}+8z^{8}+5a^{3}z^{7}+13az^{7}+13z^{7}a^{-1}+5z^{7}a^{-3}+3a^{4}z^{6}+3z^{6}a^{-4}-6z^{6}+a^{5}z^{5}-8a^{3}z^{5}-31az^{5}-31z^{5}a^{-1}-8z^{5}a^{-3}+z^{5}a^{-5}-5a^{4}z^{4}-9a^{2}z^{4}-9z^{4}a^{-2}-5z^{4}a^{-4}-8z^{4}-2a^{5}z^{3}+5a^{3}z^{3}+25az^{3}+25z^{3}a^{-1}+5z^{3}a^{-3}-2z^{3}a^{-5}+3a^{4}z^{2}+6a^{2}z^{2}+6z^{2}a^{-2}+3z^{2}a^{-4}+6z^{2}+a^{5}z-2a^{3}z-8az-8za^{-1}-2za^{-3}+za^{-5}-a^{4}-a^{2}-a^{-2}-a^{-4}+1$
The A2 invariant	$-q^{16}+q^{12}-3q^{10}+2q^{8}-q^{4}+4q^{2}-1+4q^{-2}-q^{-4}+2q^{-8}-3q^{-10}+q^{-12}-q^{-16}$
The G2 invariant	$q^{80}-2q^{78}+5q^{76}-8q^{74}+9q^{72}-9q^{70}+q^{68}+14q^{66}-35q^{64}+56q^{62}-69q^{60}+55q^{58}-16q^{56}-50q^{54}+129q^{52}-183q^{50}+191q^{48}-130q^{46}+4q^{44}+139q^{42}-255q^{40}+293q^{38}-227q^{36}+79q^{34}+91q^{32}-219q^{30}+247q^{28}-166q^{26}+14q^{24}+137q^{22}-214q^{20}+173q^{18}-34q^{16}-147q^{14}+296q^{12}-335q^{10}+249q^{8}-55q^{6}-172q^{4}+360q^{2}-427+360q^{-2}-172q^{-4}-55q^{-6}+249q^{-8}-335q^{-10}+296q^{-12}-147q^{-14}-34q^{-16}+173q^{-18}-214q^{-20}+137q^{-22}+14q^{-24}-166q^{-26}+247q^{-28}-219q^{-30}+91q^{-32}+79q^{-34}-227q^{-36}+293q^{-38}-255q^{-40}+139q^{-42}+4q^{-44}-130q^{-46}+191q^{-48}-183q^{-50}+129q^{-52}-50q^{-54}-16q^{-56}+55q^{-58}-69q^{-60}+56q^{-62}-35q^{-64}+14q^{-66}+q^{-68}-9q^{-70}+9q^{-72}-8q^{-74}+5q^{-76}-2q^{-78}+q^{-80}$

Further Quantum Invariants

Further quantum knot invariants for 10_81.

A1 Invariants.

Weight	Invariant
1	$-q^{11}+2q^{9}-4q^{7}+4q^{5}-2q^{3}+2q+2q^{-1}-2q^{-3}+4q^{-5}-4q^{-7}+2q^{-9}-q^{-11}$
2	$q^{32}-2q^{30}+8q^{26}-10q^{24}-8q^{22}+26q^{20}-13q^{18}-27q^{16}+38q^{14}-38q^{10}+26q^{8}+15q^{6}-26q^{4}+q^{2}+21+q^{-2}-26q^{-4}+15q^{-6}+26q^{-8}-38q^{-10}+38q^{-14}-27q^{-16}-13q^{-18}+26q^{-20}-8q^{-22}-10q^{-24}+8q^{-26}-2q^{-30}+q^{-32}$
3	$-q^{63}+2q^{61}-4q^{57}-2q^{55}+12q^{53}+9q^{51}-26q^{49}-25q^{47}+41q^{45}+58q^{43}-47q^{41}-113q^{39}+41q^{37}+173q^{35}-3q^{33}-226q^{31}-68q^{29}+260q^{27}+140q^{25}-250q^{23}-215q^{21}+207q^{19}+260q^{17}-137q^{15}-277q^{13}+64q^{11}+255q^{9}+21q^{7}-215q^{5}-91q^{3}+159q+159q^{-1}-91q^{-3}-215q^{-5}+21q^{-7}+255q^{-9}+64q^{-11}-277q^{-13}-137q^{-15}+260q^{-17}+207q^{-19}-215q^{-21}-250q^{-23}+140q^{-25}+260q^{-27}-68q^{-29}-226q^{-31}-3q^{-33}+173q^{-35}+41q^{-37}-113q^{-39}-47q^{-41}+58q^{-43}+41q^{-45}-25q^{-47}-26q^{-49}+9q^{-51}+12q^{-53}-2q^{-55}-4q^{-57}+2q^{-61}-q^{-63}$
4	$q^{104}-2q^{102}+4q^{98}-2q^{96}-13q^{92}+q^{90}+31q^{88}+8q^{86}-9q^{84}-80q^{82}-34q^{80}+117q^{78}+121q^{76}+39q^{74}-273q^{72}-277q^{70}+153q^{68}+456q^{66}+436q^{64}-420q^{62}-907q^{60}-292q^{58}+744q^{56}+1385q^{54}+71q^{52}-1490q^{50}-1436q^{48}+256q^{46}+2270q^{44}+1356q^{42}-1174q^{40}-2491q^{38}-1058q^{36}+2127q^{34}+2496q^{32}+51q^{30}-2484q^{28}-2193q^{26}+1010q^{24}+2579q^{22}+1212q^{20}-1509q^{18}-2375q^{16}-229q^{14}+1776q^{12}+1719q^{10}-323q^{8}-1855q^{6}-1131q^{4}+741q^{2}+1799+741q^{-2}-1131q^{-4}-1855q^{-6}-323q^{-8}+1719q^{-10}+1776q^{-12}-229q^{-14}-2375q^{-16}-1509q^{-18}+1212q^{-20}+2579q^{-22}+1010q^{-24}-2193q^{-26}-2484q^{-28}+51q^{-30}+2496q^{-32}+2127q^{-34}-1058q^{-36}-2491q^{-38}-1174q^{-40}+1356q^{-42}+2270q^{-44}+256q^{-46}-1436q^{-48}-1490q^{-50}+71q^{-52}+1385q^{-54}+744q^{-56}-292q^{-58}-907q^{-60}-420q^{-62}+436q^{-64}+456q^{-66}+153q^{-68}-277q^{-70}-273q^{-72}+39q^{-74}+121q^{-76}+117q^{-78}-34q^{-80}-80q^{-82}-9q^{-84}+8q^{-86}+31q^{-88}+q^{-90}-13q^{-92}-2q^{-96}+4q^{-98}-2q^{-102}+q^{-104}$
5	$-q^{155}+2q^{153}-4q^{149}+2q^{147}+4q^{145}+q^{143}+3q^{141}-6q^{139}-24q^{137}-7q^{135}+34q^{133}+49q^{131}+30q^{129}-49q^{127}-139q^{125}-122q^{123}+73q^{121}+307q^{119}+323q^{117}-3q^{115}-536q^{113}-776q^{111}-304q^{109}+763q^{107}+1544q^{105}+1063q^{103}-721q^{101}-2541q^{99}-2552q^{97}-33q^{95}+3502q^{93}+4831q^{91}+1897q^{89}-3746q^{87}-7535q^{85}-5268q^{83}+2547q^{81}+9987q^{79}+9878q^{77}+624q^{75}-11060q^{73}-14906q^{71}-5906q^{69}+9898q^{67}+19206q^{65}+12485q^{63}-6200q^{61}-21361q^{59}-19110q^{57}+243q^{55}+20810q^{53}+24323q^{51}+6647q^{49}-17440q^{47}-26957q^{45}-13219q^{43}+12096q^{41}+26728q^{39}+18078q^{37}-5902q^{35}-23973q^{33}-20683q^{31}+113q^{29}+19538q^{27}+21025q^{25}+4588q^{23}-14499q^{21}-19760q^{19}-7842q^{17}+9640q^{15}+17570q^{13}+10084q^{11}-5387q^{9}-15327q^{7}-11708q^{5}+1734q^{3}+13340q+13340q^{-1}+1734q^{-3}-11708q^{-5}-15327q^{-7}-5387q^{-9}+10084q^{-11}+17570q^{-13}+9640q^{-15}-7842q^{-17}-19760q^{-19}-14499q^{-21}+4588q^{-23}+21025q^{-25}+19538q^{-27}+113q^{-29}-20683q^{-31}-23973q^{-33}-5902q^{-35}+18078q^{-37}+26728q^{-39}+12096q^{-41}-13219q^{-43}-26957q^{-45}-17440q^{-47}+6647q^{-49}+24323q^{-51}+20810q^{-53}+243q^{-55}-19110q^{-57}-21361q^{-59}-6200q^{-61}+12485q^{-63}+19206q^{-65}+9898q^{-67}-5906q^{-69}-14906q^{-71}-11060q^{-73}+624q^{-75}+9878q^{-77}+9987q^{-79}+2547q^{-81}-5268q^{-83}-7535q^{-85}-3746q^{-87}+1897q^{-89}+4831q^{-91}+3502q^{-93}-33q^{-95}-2552q^{-97}-2541q^{-99}-721q^{-101}+1063q^{-103}+1544q^{-105}+763q^{-107}-304q^{-109}-776q^{-111}-536q^{-113}-3q^{-115}+323q^{-117}+307q^{-119}+73q^{-121}-122q^{-123}-139q^{-125}-49q^{-127}+30q^{-129}+49q^{-131}+34q^{-133}-7q^{-135}-24q^{-137}-6q^{-139}+3q^{-141}+q^{-143}+4q^{-145}+2q^{-147}-4q^{-149}+2q^{-153}-q^{-155}$

A2 Invariants.

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

A3 Invariants.

Weight	Invariant
0,1,0	$q^{34}-2q^{32}+q^{30}+5q^{28}-10q^{26}+2q^{24}+15q^{22}-23q^{20}+3q^{18}+24q^{16}-31q^{14}+24q^{10}-22q^{8}-4q^{6}+19q^{4}+2q^{2}-2+2q^{-2}+19q^{-4}-4q^{-6}-22q^{-8}+24q^{-10}-31q^{-14}+24q^{-16}+3q^{-18}-23q^{-20}+15q^{-22}+2q^{-24}-10q^{-26}+5q^{-28}+q^{-30}-2q^{-32}+q^{-34}$
1,0,0	$-q^{21}-q^{17}+q^{15}-3q^{13}+3q^{11}-2q^{9}+2q^{7}-q^{5}+3q^{3}+q+q^{-1}+3q^{-3}-q^{-5}+2q^{-7}-2q^{-9}+3q^{-11}-3q^{-13}+q^{-15}-q^{-17}-q^{-21}$

B2 Invariants.

Weight	Invariant
0,1	$-q^{34}+2q^{32}-5q^{30}+9q^{28}-16q^{26}+22q^{24}-29q^{22}+33q^{20}-35q^{18}+32q^{16}-23q^{14}+12q^{12}+6q^{10}-22q^{8}+40q^{6}-53q^{4}+64q^{2}-68+64q^{-2}-53q^{-4}+40q^{-6}-22q^{-8}+6q^{-10}+12q^{-12}-23q^{-14}+32q^{-16}-35q^{-18}+33q^{-20}-29q^{-22}+22q^{-24}-16q^{-26}+9q^{-28}-5q^{-30}+2q^{-32}-q^{-34}$
1,0	$q^{56}-2q^{52}-2q^{50}+3q^{48}+7q^{46}-13q^{42}-9q^{40}+13q^{38}+22q^{36}-4q^{34}-31q^{32}-14q^{30}+28q^{28}+29q^{26}-16q^{24}-38q^{22}-4q^{20}+34q^{18}+15q^{16}-25q^{14}-22q^{12}+16q^{10}+24q^{8}-6q^{6}-22q^{4}+5q^{2}+27+5q^{-2}-22q^{-4}-6q^{-6}+24q^{-8}+16q^{-10}-22q^{-12}-25q^{-14}+15q^{-16}+34q^{-18}-4q^{-20}-38q^{-22}-16q^{-24}+29q^{-26}+28q^{-28}-14q^{-30}-31q^{-32}-4q^{-34}+22q^{-36}+13q^{-38}-9q^{-40}-13q^{-42}+7q^{-46}+3q^{-48}-2q^{-50}-2q^{-52}+q^{-56}$

G2 Invariants.

Weight

Invariant

1,0

$q^{80}-2q^{78}+5q^{76}-8q^{74}+9q^{72}-9q^{70}+q^{68}+14q^{66}-35q^{64}+56q^{62}-69q^{60}+55q^{58}-16q^{56}-50q^{54}+129q^{52}-183q^{50}+191q^{48}-130q^{46}+4q^{44}+139q^{42}-255q^{40}+293q^{38}-227q^{36}+79q^{34}+91q^{32}-219q^{30}+247q^{28}-166q^{26}+14q^{24}+137q^{22}-214q^{20}+173q^{18}-34q^{16}-147q^{14}+296q^{12}-335q^{10}+249q^{8}-55q^{6}-172q^{4}+360q^{2}-427+360q^{-2}-172q^{-4}-55q^{-6}+249q^{-8}-335q^{-10}+296q^{-12}-147q^{-14}-34q^{-16}+173q^{-18}-214q^{-20}+137q^{-22}+14q^{-24}-166q^{-26}+247q^{-28}-219q^{-30}+91q^{-32}+79q^{-34}-227q^{-36}+293q^{-38}-255q^{-40}+139q^{-42}+4q^{-44}-130q^{-46}+191q^{-48}-183q^{-50}+129q^{-52}-50q^{-54}-16q^{-56}+55q^{-58}-69q^{-60}+56q^{-62}-35q^{-64}+14q^{-66}+q^{-68}-9q^{-70}+9q^{-72}-8q^{-74}+5q^{-76}-2q^{-78}+q^{-80}$

.

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 81"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t^{3}+8t^{2}-20t+27-20t^{-1}+8t^{-2}-t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 85, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{5}+3q^{4}-7q^{3}+11q^{2}-13q+15-13q^{-1}+11q^{-2}-7q^{-3}+3q^{-4}-q^{-5}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$az^{9}+z^{9}a^{-1}+4a^{2}z^{8}+4z^{8}a^{-2}+8z^{8}+5a^{3}z^{7}+13az^{7}+13z^{7}a^{-1}+5z^{7}a^{-3}+3a^{4}z^{6}+3z^{6}a^{-4}-6z^{6}+a^{5}z^{5}-8a^{3}z^{5}-31az^{5}-31z^{5}a^{-1}-8z^{5}a^{-3}+z^{5}a^{-5}-5a^{4}z^{4}-9a^{2}z^{4}-9z^{4}a^{-2}-5z^{4}a^{-4}-8z^{4}-2a^{5}z^{3}+5a^{3}z^{3}+25az^{3}+25z^{3}a^{-1}+5z^{3}a^{-3}-2z^{3}a^{-5}+3a^{4}z^{2}+6a^{2}z^{2}+6z^{2}a^{-2}+3z^{2}a^{-4}+6z^{2}+a^{5}z-2a^{3}z-8az-8za^{-1}-2za^{-3}+za^{-5}-a^{4}-a^{2}-a^{-2}-a^{-4}+1$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {...}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {10_109, ...}

Vassiliev invariants

V2 and V3:

(3, 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 $12$ $0$ $72$ $110$ $18$ $0$ $0$ $0$ $0$ $288$ $0$ $1320$ $216$ ${\frac {15071}{10}}$ $-{\frac {1466}{15}}$ ${\frac {10942}{15}}$ ${\frac {289}{6}}$ ${\frac {991}{10}}$

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 81. 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                 5 1   -4 5               6 2     4 3             7 5       -2 1           8 6         2 -1         6 8           2 -3       5 7             -2 -5     2 6               4 -7   1 5                 -4 -9   2                   2 -11 1                     -1

Integral Khovanov Homology (db, data source)

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

The Coloured Jones Polynomials

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

$n$	$J_{n}$
2	$q^{15}-3q^{14}+2q^{13}+9q^{12}-21q^{11}+4q^{10}+43q^{9}-60q^{8}-10q^{7}+108q^{6}-98q^{5}-48q^{4}+172q^{3}-109q^{2}-89q+199-89q^{-1}-109q^{-2}+172q^{-3}-48q^{-4}-98q^{-5}+108q^{-6}-10q^{-7}-60q^{-8}+43q^{-9}+4q^{-10}-21q^{-11}+9q^{-12}+2q^{-13}-3q^{-14}+q^{-15}$
3	$-q^{30}+3q^{29}-2q^{28}-4q^{27}+q^{26}+17q^{25}-5q^{24}-39q^{23}+2q^{22}+83q^{21}+12q^{20}-144q^{19}-64q^{18}+237q^{17}+144q^{16}-320q^{15}-287q^{14}+395q^{13}+472q^{12}-440q^{11}-677q^{10}+430q^{9}+894q^{8}-387q^{7}-1074q^{6}+290q^{5}+1235q^{4}-196q^{3}-1308q^{2}+54q+1359+54q^{-1}-1308q^{-2}-196q^{-3}+1235q^{-4}+290q^{-5}-1074q^{-6}-387q^{-7}+894q^{-8}+430q^{-9}-677q^{-10}-440q^{-11}+472q^{-12}+395q^{-13}-287q^{-14}-320q^{-15}+144q^{-16}+237q^{-17}-64q^{-18}-144q^{-19}+12q^{-20}+83q^{-21}+2q^{-22}-39q^{-23}-5q^{-24}+17q^{-25}+q^{-26}-4q^{-27}-2q^{-28}+3q^{-29}-q^{-30}$
4	$q^{50}-3q^{49}+2q^{48}+4q^{47}-6q^{46}+3q^{45}-16q^{44}+16q^{43}+34q^{42}-29q^{41}-14q^{40}-87q^{39}+62q^{38}+185q^{37}-25q^{36}-96q^{35}-399q^{34}+58q^{33}+615q^{32}+278q^{31}-116q^{30}-1255q^{29}-429q^{28}+1230q^{27}+1314q^{26}+525q^{25}-2569q^{24}-1990q^{23}+1284q^{22}+3006q^{21}+2539q^{20}-3483q^{19}-4520q^{18}-33q^{17}+4439q^{16}+5724q^{15}-3114q^{14}-6965q^{13}-2568q^{12}+4730q^{11}+8927q^{10}-1545q^{9}-8332q^{8}-5289q^{7}+3864q^{6}+11073q^{5}+460q^{4}-8389q^{3}-7331q^{2}+2332q+11797+2332q^{-1}-7331q^{-2}-8389q^{-3}+460q^{-4}+11073q^{-5}+3864q^{-6}-5289q^{-7}-8332q^{-8}-1545q^{-9}+8927q^{-10}+4730q^{-11}-2568q^{-12}-6965q^{-13}-3114q^{-14}+5724q^{-15}+4439q^{-16}-33q^{-17}-4520q^{-18}-3483q^{-19}+2539q^{-20}+3006q^{-21}+1284q^{-22}-1990q^{-23}-2569q^{-24}+525q^{-25}+1314q^{-26}+1230q^{-27}-429q^{-28}-1255q^{-29}-116q^{-30}+278q^{-31}+615q^{-32}+58q^{-33}-399q^{-34}-96q^{-35}-25q^{-36}+185q^{-37}+62q^{-38}-87q^{-39}-14q^{-40}-29q^{-41}+34q^{-42}+16q^{-43}-16q^{-44}+3q^{-45}-6q^{-46}+4q^{-47}+2q^{-48}-3q^{-49}+q^{-50}$
5	$-q^{75}+3q^{74}-2q^{73}-4q^{72}+6q^{71}+2q^{70}-4q^{69}+5q^{68}-11q^{67}-22q^{66}+23q^{65}+43q^{64}+11q^{63}-14q^{62}-90q^{61}-112q^{60}+40q^{59}+238q^{58}+245q^{57}+2q^{56}-416q^{55}-645q^{54}-200q^{53}+710q^{52}+1312q^{51}+783q^{50}-897q^{49}-2429q^{48}-2020q^{47}+699q^{46}+3831q^{45}+4318q^{44}+432q^{43}-5363q^{42}-7663q^{41}-3090q^{40}+6098q^{39}+12133q^{38}+7872q^{37}-5472q^{36}-16917q^{35}-14774q^{34}+2252q^{33}+21133q^{32}+23676q^{31}+3836q^{30}-23638q^{29}-33459q^{28}-12909q^{27}+23384q^{26}+43029q^{25}+24403q^{24}-20125q^{23}-51135q^{22}-36996q^{21}+13867q^{20}+56767q^{19}+49718q^{18}-5493q^{17}-59785q^{16}-60976q^{15}-4204q^{14}+60057q^{13}+70514q^{12}+13932q^{11}-58298q^{10}-77413q^{9}-23291q^{8}+54796q^{7}+82432q^{6}+31414q^{5}-50368q^{4}-84899q^{3}-38762q^{2}+44856q+86051+44856q^{-1}-38762q^{-2}-84899q^{-3}-50368q^{-4}+31414q^{-5}+82432q^{-6}+54796q^{-7}-23291q^{-8}-77413q^{-9}-58298q^{-10}+13932q^{-11}+70514q^{-12}+60057q^{-13}-4204q^{-14}-60976q^{-15}-59785q^{-16}-5493q^{-17}+49718q^{-18}+56767q^{-19}+13867q^{-20}-36996q^{-21}-51135q^{-22}-20125q^{-23}+24403q^{-24}+43029q^{-25}+23384q^{-26}-12909q^{-27}-33459q^{-28}-23638q^{-29}+3836q^{-30}+23676q^{-31}+21133q^{-32}+2252q^{-33}-14774q^{-34}-16917q^{-35}-5472q^{-36}+7872q^{-37}+12133q^{-38}+6098q^{-39}-3090q^{-40}-7663q^{-41}-5363q^{-42}+432q^{-43}+4318q^{-44}+3831q^{-45}+699q^{-46}-2020q^{-47}-2429q^{-48}-897q^{-49}+783q^{-50}+1312q^{-51}+710q^{-52}-200q^{-53}-645q^{-54}-416q^{-55}+2q^{-56}+245q^{-57}+238q^{-58}+40q^{-59}-112q^{-60}-90q^{-61}-14q^{-62}+11q^{-63}+43q^{-64}+23q^{-65}-22q^{-66}-11q^{-67}+5q^{-68}-4q^{-69}+2q^{-70}+6q^{-71}-4q^{-72}-2q^{-73}+3q^{-74}-q^{-75}$
6	Not Available
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, 81]]
Out[2]=	PD[X[4, 2, 5, 1], X[8, 4, 9, 3], X[12, 6, 13, 5], X[16, 9, 17, 10], X[20, 17, 1, 18], X[18, 13, 19, 14], X[14, 19, 15, 20], X[10, 15, 11, 16], X[6, 12, 7, 11], X[2, 8, 3, 7]]
In[3]:=	GaussCode[Knot[10, 81]]
Out[3]=	GaussCode[1, -10, 2, -1, 3, -9, 10, -2, 4, -8, 9, -3, 6, -7, 8, -4, 5, -6, 7, -5]
In[4]:=	DTCode[Knot[10, 81]]
Out[4]=	DTCode[4, 8, 12, 2, 16, 6, 18, 10, 20, 14]
In[5]:=	br = BR[Knot[10, 81]]
Out[5]=	BR[5, {1, 1, -2, 1, 3, 2, 2, -4, -3, -3, -3, -4}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{5, 12}
In[7]:=	BraidIndex[Knot[10, 81]]
Out[7]=	5
In[8]:=	Show[DrawMorseLink[Knot[10, 81]]]
Out[8]=	-Graphics-
In[9]:=	(#[Knot[10, 81]]&) /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{NegativeAmphicheiral, 2, 3, 3, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 81]][t]
Out[10]=	-3 8 20 2 3 27 - t + -- - -- - 20 t + 8 t - t 2 t t
In[11]:=	Conway[Knot[10, 81]][z]
Out[11]=	2 4 6 1 + 3 z + 2 z - z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 81]}
In[13]:=	{KnotDet[Knot[10, 81]], KnotSignature[Knot[10, 81]]}
Out[13]=	{85, 0}
In[14]:=	Jones[Knot[10, 81]][q]
Out[14]=	-5 3 7 11 13 2 3 4 5 15 - q + -- - -- + -- - -- - 13 q + 11 q - 7 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, 81], Knot[10, 109]}
In[16]:=	A2Invariant[Knot[10, 81]][q]
Out[16]=	-16 -12 3 2 -4 4 2 4 8 10 -1 - q + q - --- + -- - q + -- + 4 q - q + 2 q - 3 q + 10 8 2 q q q 12 16 q - q
In[17]:=	HOMFLYPT[Knot[10, 81]][a, z]
Out[17]=	2 2 -4 -2 2 4 2 z 3 z 2 2 4 2 4 1 - a + a + a - a - z - -- + ---- + 3 a z - a z - 2 z + 4 2 a a 4 2 z 2 4 6 ---- + 2 a z - z 2 a
In[18]:=	Kauffman[Knot[10, 81]][a, z]
Out[18]=	-4 -2 2 4 z 2 z 8 z 3 5 1 - a - a - a - a + -- - --- - --- - 8 a z - 2 a z + a z + 5 3 a a a 2 2 3 3 3 2 3 z 6 z 2 2 4 2 2 z 5 z 25 z 6 z + ---- + ---- + 6 a z + 3 a z - ---- + ---- + ----- + 4 2 5 3 a a a a a 4 4 3 3 3 5 3 4 5 z 9 z 2 4 25 a z + 5 a z - 2 a z - 8 z - ---- - ---- - 9 a z - 4 2 a a 5 5 5 4 4 z 8 z 31 z 5 3 5 5 5 6 5 a z + -- - ---- - ----- - 31 a z - 8 a z + a z - 6 z + 5 3 a a a 6 7 7 8 3 z 4 6 5 z 13 z 7 3 7 8 4 z ---- + 3 a z + ---- + ----- + 13 a z + 5 a z + 8 z + ---- + 4 3 a 2 a a a 9 2 8 z 9 4 a z + -- + a z a
In[19]:=	{Vassiliev[2][Knot[10, 81]], Vassiliev[3][Knot[10, 81]]}
Out[19]=	{3, 0}
In[20]:=	Kh[Knot[10, 81]][q, t]
Out[20]=	8 1 2 1 5 2 6 5 - + 8 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 7 6 3 3 2 5 2 5 3 7 3 ---- + --- + 6 q t + 7 q t + 5 q t + 6 q t + 2 q t + 5 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, 81], 2][q]
Out[21]=	-15 3 2 9 21 4 43 60 10 108 98 199 + q - --- + --- + --- - --- + --- + -- - -- - -- + --- - -- - 14 13 12 11 10 9 8 7 6 5 q q q q q q q q q q 48 172 109 89 2 3 4 5 -- + --- - --- - -- - 89 q - 109 q + 172 q - 48 q - 98 q + 4 3 2 q q q q 6 7 8 9 10 11 12 13 108 q - 10 q - 60 q + 43 q + 4 q - 21 q + 9 q + 2 q - 14 15 3 q + q

See/edit the Rolfsen_Splice_Template.