10 114

10_113

10_115

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10 114 Further Notes and Views

Knot presentations

Planar diagram presentation	X6271 X8394 X18,13,19,14 X20,11,1,12 X12,19,13,20 X2,16,3,15 X4,17,5,18 X10,6,11,5 X14,7,15,8 X16,10,17,9
Gauss code	1, -6, 2, -7, 8, -1, 9, -2, 10, -8, 4, -5, 3, -9, 6, -10, 7, -3, 5, -4
Dowker-Thistlethwaite code	6 8 10 14 16 20 18 2 4 12
Conway Notation	[8*30]

Minimum Braid Representative: Length is 11, width is 4. Braid index is 4.

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 [-7][-5] Hyperbolic Volume 15.3049 A-Polynomial See Data:10 114/A-polynomial

[edit Notes for 10 114'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 10 114's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-2t^{3}+10t^{2}-21t+27-21t^{-1}+10t^{-2}-2t^{-3}$
Conway polynomial	$-2z^{6}-2z^{4}+z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 93, 0 }
Jones polynomial	$q^{4}-4q^{3}+8q^{2}-12q+15-15q^{-1}+15q^{-2}-11q^{-3}+7q^{-4}-4q^{-5}+q^{-6}$
HOMFLY-PT polynomial (db, data sources)	$-a^{2}z^{6}-z^{6}+a^{4}z^{4}-2a^{2}z^{4}+z^{4}a^{-2}-2z^{4}+a^{4}z^{2}+z^{2}a^{-2}-z^{2}-a^{4}+2a^{2}$
Kauffman polynomial (db, data sources)	$3a^{3}z^{9}+3az^{9}+6a^{4}z^{8}+14a^{2}z^{8}+8z^{8}+4a^{5}z^{7}+2a^{3}z^{7}+8az^{7}+10z^{7}a^{-1}+a^{6}z^{6}-17a^{4}z^{6}-35a^{2}z^{6}+8z^{6}a^{-2}-9z^{6}-11a^{5}z^{5}-21a^{3}z^{5}-27az^{5}-13z^{5}a^{-1}+4z^{5}a^{-3}-2a^{6}z^{4}+14a^{4}z^{4}+26a^{2}z^{4}-8z^{4}a^{-2}+z^{4}a^{-4}+z^{4}+7a^{5}z^{3}+18a^{3}z^{3}+18az^{3}+5z^{3}a^{-1}-2z^{3}a^{-3}-3a^{4}z^{2}-5a^{2}z^{2}+2z^{2}a^{-2}-2a^{3}z-3az-za^{-1}-a^{4}-2a^{2}$
The A2 invariant	$q^{18}-2q^{16}-3q^{10}+4q^{8}+2q^{4}+2q^{2}-2+3q^{-2}-3q^{-4}+q^{-6}+q^{-8}-2q^{-10}+q^{-12}$
The G2 invariant	$q^{94}-3q^{92}+7q^{90}-14q^{88}+17q^{86}-18q^{84}+7q^{82}+23q^{80}-59q^{78}+102q^{76}-118q^{74}+87q^{72}-8q^{70}-116q^{68}+233q^{66}-288q^{64}+240q^{62}-90q^{60}-114q^{58}+292q^{56}-369q^{54}+304q^{52}-124q^{50}-102q^{48}+262q^{46}-301q^{44}+192q^{42}+8q^{40}-188q^{38}+283q^{36}-240q^{34}+79q^{32}+135q^{30}-323q^{28}+407q^{26}-345q^{24}+160q^{22}+105q^{20}-340q^{18}+471q^{16}-440q^{14}+265q^{12}-9q^{10}-238q^{8}+371q^{6}-346q^{4}+186q^{2}+42-216q^{-2}+264q^{-4}-170q^{-6}-16q^{-8}+194q^{-10}-288q^{-12}+259q^{-14}-123q^{-16}-57q^{-18}+212q^{-20}-287q^{-22}+265q^{-24}-164q^{-26}+35q^{-28}+77q^{-30}-152q^{-32}+168q^{-34}-142q^{-36}+92q^{-38}-29q^{-40}-22q^{-42}+51q^{-44}-63q^{-46}+54q^{-48}-34q^{-50}+16q^{-52}+q^{-54}-8q^{-56}+10q^{-58}-10q^{-60}+6q^{-62}-3q^{-64}+q^{-66}$

Further Quantum Invariants

Further quantum knot invariants for 10_114.

A1 Invariants.

Weight	Invariant
1	$q^{13}-3q^{11}+3q^{9}-4q^{7}+4q^{5}+3q^{-1}-4q^{-3}+4q^{-5}-3q^{-7}+q^{-9}$
2	$q^{38}-3q^{36}-q^{34}+12q^{32}-8q^{30}-17q^{28}+27q^{26}+4q^{24}-39q^{22}+24q^{20}+24q^{18}-43q^{16}+6q^{14}+33q^{12}-23q^{10}-14q^{8}+24q^{6}+9q^{4}-26q^{2}+38q^{-2}-24q^{-4}-26q^{-6}+43q^{-8}-8q^{-10}-30q^{-12}+25q^{-14}+2q^{-16}-14q^{-18}+8q^{-20}+q^{-22}-3q^{-24}+q^{-26}$
3	$q^{75}-3q^{73}-q^{71}+8q^{69}+7q^{67}-16q^{65}-30q^{63}+23q^{61}+66q^{59}-2q^{57}-112q^{55}-54q^{53}+140q^{51}+142q^{49}-124q^{47}-235q^{45}+52q^{43}+303q^{41}+54q^{39}-318q^{37}-163q^{35}+279q^{33}+257q^{31}-217q^{29}-302q^{27}+125q^{25}+321q^{23}-50q^{21}-307q^{19}-25q^{17}+276q^{15}+96q^{13}-226q^{11}-167q^{9}+162q^{7}+241q^{5}-70q^{3}-291q-46q^{-1}+317q^{-3}+160q^{-5}-286q^{-7}-261q^{-9}+216q^{-11}+304q^{-13}-116q^{-15}-291q^{-17}+27q^{-19}+230q^{-21}+25q^{-23}-151q^{-25}-37q^{-27}+81q^{-29}+30q^{-31}-41q^{-33}-15q^{-35}+23q^{-37}+2q^{-39}-9q^{-41}-2q^{-43}+5q^{-45}+q^{-47}-3q^{-49}+q^{-51}$

A2 Invariants.

Weight	Invariant
1,0	$q^{18}-2q^{16}-3q^{10}+4q^{8}+2q^{4}+2q^{2}-2+3q^{-2}-3q^{-4}+q^{-6}+q^{-8}-2q^{-10}+q^{-12}$
1,1	$q^{52}-6q^{50}+20q^{48}-52q^{46}+119q^{44}-240q^{42}+424q^{40}-670q^{38}+971q^{36}-1274q^{34}+1510q^{32}-1624q^{30}+1560q^{28}-1284q^{26}+786q^{24}-128q^{22}-620q^{20}+1380q^{18}-2070q^{16}+2604q^{14}-2922q^{12}+2992q^{10}-2792q^{8}+2358q^{6}-1732q^{4}+1018q^{2}-276-380q^{-2}+881q^{-4}-1196q^{-6}+1332q^{-8}-1326q^{-10}+1202q^{-12}-1022q^{-14}+830q^{-16}-640q^{-18}+468q^{-20}-328q^{-22}+220q^{-24}-134q^{-26}+73q^{-28}-38q^{-30}+18q^{-32}-6q^{-34}+q^{-36}$
2,0	$q^{48}-2q^{46}-2q^{44}+5q^{42}+3q^{40}-4q^{38}-7q^{36}+7q^{34}+12q^{32}-13q^{30}-12q^{28}+10q^{26}+9q^{24}-12q^{22}-12q^{20}+16q^{18}+8q^{16}-14q^{14}+11q^{10}-6q^{8}-q^{6}+14q^{4}-2q^{2}-7+8q^{-2}+13q^{-4}-17q^{-6}-15q^{-8}+19q^{-10}+6q^{-12}-18q^{-14}-2q^{-16}+14q^{-18}+3q^{-20}-9q^{-22}-2q^{-24}+6q^{-26}-q^{-28}-2q^{-30}+q^{-32}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{40}-3q^{38}+q^{36}+5q^{34}-11q^{32}+10q^{30}+9q^{28}-24q^{26}+17q^{24}+7q^{22}-34q^{20}+17q^{18}+12q^{16}-26q^{14}+13q^{12}+17q^{10}-8q^{8}-3q^{6}+5q^{4}+12q^{2}-15-9q^{-2}+31q^{-4}-19q^{-6}-16q^{-8}+33q^{-10}-14q^{-12}-16q^{-14}+22q^{-16}-5q^{-18}-10q^{-20}+9q^{-22}-3q^{-26}+q^{-28}$
1,0,0	$q^{23}-2q^{21}+q^{19}-3q^{17}+q^{15}-3q^{13}+4q^{11}+3q^{7}+q^{5}+q^{3}+q-2q^{-1}+3q^{-3}-3q^{-5}+2q^{-7}-2q^{-9}+2q^{-11}-2q^{-13}+q^{-15}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{50}-2q^{48}-2q^{46}+6q^{44}-3q^{42}-8q^{40}+12q^{38}+9q^{36}-15q^{34}-2q^{32}+19q^{30}-7q^{28}-31q^{26}+3q^{24}+23q^{22}-21q^{20}-16q^{18}+36q^{16}+12q^{14}-21q^{12}+15q^{10}+20q^{8}-19q^{6}-12q^{4}+20q^{2}+1-25q^{-2}+10q^{-4}+26q^{-6}-18q^{-8}-15q^{-10}+20q^{-12}+6q^{-14}-18q^{-16}-3q^{-18}+12q^{-20}+2q^{-22}-8q^{-24}+6q^{-28}-2q^{-30}-2q^{-32}+q^{-34}$
1,0,0,0	$q^{28}-2q^{26}+q^{24}-2q^{22}-2q^{20}+q^{18}-3q^{16}+4q^{14}+3q^{10}+2q^{8}+q^{6}+q^{4}+1-2q^{-2}+3q^{-4}-3q^{-6}+2q^{-8}-q^{-10}-q^{-12}+2q^{-14}-2q^{-16}+q^{-18}$

B2 Invariants.

Weight	Invariant
0,1	$q^{40}-3q^{38}+7q^{36}-13q^{34}+21q^{32}-30q^{30}+35q^{28}-40q^{26}+39q^{24}-35q^{22}+24q^{20}-9q^{18}-10q^{16}+32q^{14}-49q^{12}+67q^{10}-74q^{8}+79q^{6}-71q^{4}+60q^{2}-43+23q^{-2}-3q^{-4}-15q^{-6}+28q^{-8}-37q^{-10}+40q^{-12}-38q^{-14}+32q^{-16}-25q^{-18}+18q^{-20}-11q^{-22}+6q^{-24}-3q^{-26}+q^{-28}$
1,0	$q^{66}-3q^{62}-3q^{60}+4q^{58}+9q^{56}-2q^{54}-16q^{52}-6q^{50}+23q^{48}+21q^{46}-17q^{44}-35q^{42}+q^{40}+40q^{38}+18q^{36}-36q^{34}-37q^{32}+15q^{30}+40q^{28}+2q^{26}-36q^{24}-13q^{22}+27q^{20}+22q^{18}-17q^{16}-19q^{14}+15q^{12}+26q^{10}-9q^{8}-28q^{6}+3q^{4}+32q^{2}+6-33q^{-2}-18q^{-4}+30q^{-6}+30q^{-8}-20q^{-10}-40q^{-12}+2q^{-14}+40q^{-16}+17q^{-18}-27q^{-20}-29q^{-22}+8q^{-24}+27q^{-26}+8q^{-28}-15q^{-30}-14q^{-32}+3q^{-34}+10q^{-36}+3q^{-38}-3q^{-40}-3q^{-42}+q^{-46}$

D4 Invariants.

Weight

Invariant

1,0,0,0

$q^{54}-3q^{52}+4q^{50}-7q^{48}+12q^{46}-17q^{44}+22q^{42}-25q^{40}+31q^{38}-32q^{36}+30q^{34}-31q^{32}+26q^{30}-23q^{28}+7q^{26}-2q^{24}-8q^{22}+22q^{20}-33q^{18}+46q^{16}-45q^{14}+60q^{12}-60q^{10}+59q^{8}-54q^{6}+53q^{4}-43q^{2}+29-21q^{-2}+13q^{-4}+4q^{-6}-14q^{-8}+17q^{-10}-25q^{-12}+33q^{-14}-31q^{-16}+26q^{-18}-29q^{-20}+27q^{-22}-18q^{-24}+14q^{-26}-14q^{-28}+10q^{-30}-4q^{-32}+3q^{-34}-3q^{-36}+q^{-38}$

G2 Invariants.

Weight

Invariant

1,0

$q^{94}-3q^{92}+7q^{90}-14q^{88}+17q^{86}-18q^{84}+7q^{82}+23q^{80}-59q^{78}+102q^{76}-118q^{74}+87q^{72}-8q^{70}-116q^{68}+233q^{66}-288q^{64}+240q^{62}-90q^{60}-114q^{58}+292q^{56}-369q^{54}+304q^{52}-124q^{50}-102q^{48}+262q^{46}-301q^{44}+192q^{42}+8q^{40}-188q^{38}+283q^{36}-240q^{34}+79q^{32}+135q^{30}-323q^{28}+407q^{26}-345q^{24}+160q^{22}+105q^{20}-340q^{18}+471q^{16}-440q^{14}+265q^{12}-9q^{10}-238q^{8}+371q^{6}-346q^{4}+186q^{2}+42-216q^{-2}+264q^{-4}-170q^{-6}-16q^{-8}+194q^{-10}-288q^{-12}+259q^{-14}-123q^{-16}-57q^{-18}+212q^{-20}-287q^{-22}+265q^{-24}-164q^{-26}+35q^{-28}+77q^{-30}-152q^{-32}+168q^{-34}-142q^{-36}+92q^{-38}-29q^{-40}-22q^{-42}+51q^{-44}-63q^{-46}+54q^{-48}-34q^{-50}+16q^{-52}+q^{-54}-8q^{-56}+10q^{-58}-10q^{-60}+6q^{-62}-3q^{-64}+q^{-66}$

.

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["10 114"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-2t^{3}+10t^{2}-21t+27-21t^{-1}+10t^{-2}-2t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 93, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^{4}-4q^{3}+8q^{2}-12q+15-15q^{-1}+15q^{-2}-11q^{-3}+7q^{-4}-4q^{-5}+q^{-6}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$3a^{3}z^{9}+3az^{9}+6a^{4}z^{8}+14a^{2}z^{8}+8z^{8}+4a^{5}z^{7}+2a^{3}z^{7}+8az^{7}+10z^{7}a^{-1}+a^{6}z^{6}-17a^{4}z^{6}-35a^{2}z^{6}+8z^{6}a^{-2}-9z^{6}-11a^{5}z^{5}-21a^{3}z^{5}-27az^{5}-13z^{5}a^{-1}+4z^{5}a^{-3}-2a^{6}z^{4}+14a^{4}z^{4}+26a^{2}z^{4}-8z^{4}a^{-2}+z^{4}a^{-4}+z^{4}+7a^{5}z^{3}+18a^{3}z^{3}+18az^{3}+5z^{3}a^{-1}-2z^{3}a^{-3}-3a^{4}z^{2}-5a^{2}z^{2}+2z^{2}a^{-2}-2a^{3}z-3az-za^{-1}-a^{4}-2a^{2}$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a93, ...}

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

Vassiliev invariants

V2 and V3:

(1, -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 $4$ $-8$ $8$ ${\frac {110}{3}}$ ${\frac {58}{3}}$ $-32$ $-{\frac {272}{3}}$ $-{\frac {32}{3}}$ $-40$ ${\frac {32}{3}}$ $32$ ${\frac {440}{3}}$ ${\frac {232}{3}}$ ${\frac {8191}{30}}$ $-{\frac {1462}{15}}$ ${\frac {9302}{45}}$ ${\frac {1217}{18}}$ ${\frac {1951}{30}}$

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

\   r    \    j   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 9                     1 1 7                   3   -3 5                 5 1   4 3               7 3     -4 1             8 5       3 -1           8 8         0 -3         7 7           0 -5       4 8             4 -7     3 7               -4 -9   1 4                 3 -11   3                   -3 -13 1                     1

Integral Khovanov Homology (db, data source)

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

The Coloured Jones Polynomials

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

$n$	$J_{n}$
2	$q^{12}-4q^{11}+4q^{10}+8q^{9}-26q^{8}+20q^{7}+31q^{6}-81q^{5}+42q^{4}+82q^{3}-150q^{2}+44q+144-188q^{-1}+18q^{-2}+179q^{-3}-173q^{-4}-20q^{-5}+170q^{-6}-117q^{-7}-47q^{-8}+121q^{-9}-50q^{-10}-47q^{-11}+58q^{-12}-7q^{-13}-24q^{-14}+14q^{-15}+2q^{-16}-4q^{-17}+q^{-18}$
3	$q^{24}-4q^{23}+4q^{22}+4q^{21}-6q^{20}-11q^{19}+15q^{18}+25q^{17}-44q^{16}-37q^{15}+86q^{14}+76q^{13}-162q^{12}-151q^{11}+262q^{10}+281q^{9}-365q^{8}-469q^{7}+437q^{6}+701q^{5}-453q^{4}-946q^{3}+412q^{2}+1147q-296-1309q^{-1}+167q^{-2}+1368q^{-3}+15q^{-4}-1388q^{-5}-162q^{-6}+1309q^{-7}+337q^{-8}-1208q^{-9}-463q^{-10}+1027q^{-11}+594q^{-12}-837q^{-13}-659q^{-14}+600q^{-15}+679q^{-16}-363q^{-17}-637q^{-18}+158q^{-19}+524q^{-20}+9q^{-21}-388q^{-22}-93q^{-23}+237q^{-24}+120q^{-25}-122q^{-26}-95q^{-27}+43q^{-28}+62q^{-29}-12q^{-30}-27q^{-31}+9q^{-33}+2q^{-34}-4q^{-35}+q^{-36}$
4	$q^{40}-4q^{39}+4q^{38}+4q^{37}-10q^{36}+9q^{35}-16q^{34}+19q^{33}+11q^{32}-52q^{31}+54q^{30}-25q^{29}+58q^{28}-30q^{27}-239q^{26}+184q^{25}+155q^{24}+303q^{23}-242q^{22}-1004q^{21}+139q^{20}+803q^{19}+1465q^{18}-229q^{17}-2840q^{16}-1111q^{15}+1405q^{14}+4180q^{13}+1296q^{12}-4984q^{11}-4227q^{10}+395q^{9}+7366q^{8}+4897q^{7}-5521q^{6}-7825q^{5}-2743q^{4}+8870q^{3}+8925q^{2}-3804q-9709-6410q^{-1}+8019q^{-2}+11304q^{-3}-1105q^{-4}-9327q^{-5}-8912q^{-6}+5828q^{-7}+11623q^{-8}+1385q^{-9}-7529q^{-10}-10036q^{-11}+3156q^{-12}+10543q^{-13}+3532q^{-14}-4917q^{-15}-10112q^{-16}+133q^{-17}+8295q^{-18}+5263q^{-19}-1562q^{-20}-8871q^{-21}-2794q^{-22}+4794q^{-23}+5699q^{-24}+1853q^{-25}-5907q^{-26}-4273q^{-27}+864q^{-28}+4074q^{-29}+3720q^{-30}-2150q^{-31}-3397q^{-32}-1602q^{-33}+1349q^{-34}+3137q^{-35}+346q^{-36}-1278q^{-37}-1696q^{-38}-416q^{-39}+1365q^{-40}+757q^{-41}+83q^{-42}-686q^{-43}-586q^{-44}+230q^{-45}+275q^{-46}+248q^{-47}-84q^{-48}-209q^{-49}-12q^{-50}+16q^{-51}+74q^{-52}+12q^{-53}-33q^{-54}-3q^{-55}-5q^{-56}+9q^{-57}+2q^{-58}-4q^{-59}+q^{-60}$
5	$q^{60}-4q^{59}+4q^{58}+4q^{57}-10q^{56}+5q^{55}+4q^{54}-12q^{53}+5q^{52}+13q^{51}-12q^{50}+14q^{49}+25q^{48}-47q^{47}-76q^{46}-35q^{45}+88q^{44}+235q^{43}+221q^{42}-139q^{41}-681q^{40}-717q^{39}+142q^{38}+1397q^{37}+1861q^{36}+430q^{35}-2480q^{34}-4230q^{33}-2019q^{32}+3533q^{31}+7938q^{30}+5819q^{29}-3588q^{28}-13168q^{27}-12644q^{26}+1323q^{25}+18846q^{24}+22958q^{23}+5034q^{22}-23319q^{21}-36246q^{20}-16501q^{19}+24395q^{18}+50614q^{17}+32984q^{16}-20091q^{15}-63424q^{14}-52999q^{13}+9792q^{12}+72019q^{11}+73678q^{10}+5716q^{9}-74673q^{8}-91983q^{7}-24148q^{6}+71094q^{5}+105530q^{4}+42813q^{3}-62759q^{2}-113105q-58967+51245q^{-1}+115139q^{-2}+71610q^{-3}-39254q^{-4}-112765q^{-5}-79863q^{-6}+27453q^{-7}+107579q^{-8}+85150q^{-9}-17118q^{-10}-100836q^{-11}-87747q^{-12}+7185q^{-13}+93094q^{-14}+89436q^{-15}+2241q^{-16}-84237q^{-17}-89921q^{-18}-12620q^{-19}+73812q^{-20}+89682q^{-21}+23379q^{-22}-61069q^{-23}-87225q^{-24}-34862q^{-25}+45796q^{-26}+82137q^{-27}+45115q^{-28}-28292q^{-29}-72840q^{-30}-53030q^{-31}+9722q^{-32}+59629q^{-33}+56447q^{-34}+7696q^{-35}-42787q^{-36}-54326q^{-37}-21831q^{-38}+24422q^{-39}+46683q^{-40}+30397q^{-41}-7141q^{-42}-34561q^{-43}-32603q^{-44}-6610q^{-45}+20774q^{-46}+28908q^{-47}+14745q^{-48}-7856q^{-49}-21262q^{-50}-17244q^{-51}-1599q^{-52}+12367q^{-53}+15061q^{-54}+6743q^{-55}-4692q^{-56}-10497q^{-57}-7781q^{-58}-384q^{-59}+5600q^{-60}+6372q^{-61}+2581q^{-62}-1991q^{-63}-3928q^{-64}-2730q^{-65}-76q^{-66}+1889q^{-67}+1959q^{-68}+674q^{-69}-600q^{-70}-1010q^{-71}-639q^{-72}+25q^{-73}+438q^{-74}+366q^{-75}+78q^{-76}-124q^{-77}-144q^{-78}-80q^{-79}+22q^{-80}+67q^{-81}+23q^{-82}-9q^{-83}-9q^{-84}-8q^{-85}-5q^{-86}+9q^{-87}+2q^{-88}-4q^{-89}+q^{-90}$
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, 114]]
Out[2]=	PD[X[6, 2, 7, 1], X[8, 3, 9, 4], X[18, 13, 19, 14], X[20, 11, 1, 12], X[12, 19, 13, 20], X[2, 16, 3, 15], X[4, 17, 5, 18], X[10, 6, 11, 5], X[14, 7, 15, 8], X[16, 10, 17, 9]]
In[3]:=	GaussCode[Knot[10, 114]]
Out[3]=	GaussCode[1, -6, 2, -7, 8, -1, 9, -2, 10, -8, 4, -5, 3, -9, 6, -10, 7, -3, 5, -4]
In[4]:=	DTCode[Knot[10, 114]]
Out[4]=	DTCode[6, 8, 10, 14, 16, 20, 18, 2, 4, 12]
In[5]:=	br = BR[Knot[10, 114]]
Out[5]=	BR[4, {-1, -1, -2, 1, 3, -2, 3, -2, 3, -2, 3}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{4, 11}
In[7]:=	BraidIndex[Knot[10, 114]]
Out[7]=	4
In[8]:=	Show[DrawMorseLink[Knot[10, 114]]]
Out[8]=	-Graphics-
In[9]:=	(#[Knot[10, 114]]&) /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 1, 3, 3, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 114]][t]
Out[10]=	2 10 21 2 3 27 - -- + -- - -- - 21 t + 10 t - 2 t 3 2 t t t
In[11]:=	Conway[Knot[10, 114]][z]
Out[11]=	2 4 6 1 + z - 2 z - 2 z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 114], Knot[11, Alternating, 93]}
In[13]:=	{KnotDet[Knot[10, 114]], KnotSignature[Knot[10, 114]]}
Out[13]=	{93, 0}
In[14]:=	Jones[Knot[10, 114]][q]
Out[14]=	-6 4 7 11 15 15 2 3 4 15 + q - -- + -- - -- + -- - -- - 12 q + 8 q - 4 q + q 5 4 3 2 q q q q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[10, 114]}
In[16]:=	A2Invariant[Knot[10, 114]][q]
Out[16]=	-18 2 3 4 2 2 2 4 6 8 10 -2 + q - --- - --- + -- + -- + -- + 3 q - 3 q + q + q - 2 q + 16 10 8 4 2 q q q q q 12 q
In[17]:=	HOMFLYPT[Knot[10, 114]][a, z]
Out[17]=	2 4 2 4 2 z 4 2 4 z 2 4 4 4 6 2 6 2 a - a - z + -- + a z - 2 z + -- - 2 a z + a z - z - a z 2 2 a a
In[18]:=	Kauffman[Knot[10, 114]][a, z]
Out[18]=	2 3 2 4 z 3 2 z 2 2 4 2 2 z -2 a - a - - - 3 a z - 2 a z + ---- - 5 a z - 3 a z - ---- + a 2 3 a a 3 4 4 5 z 3 3 3 5 3 4 z 8 z 2 4 ---- + 18 a z + 18 a z + 7 a z + z + -- - ---- + 26 a z + a 4 2 a a 5 5 4 4 6 4 4 z 13 z 5 3 5 5 5 14 a z - 2 a z + ---- - ----- - 27 a z - 21 a z - 11 a z - 3 a a 6 7 6 8 z 2 6 4 6 6 6 10 z 7 9 z + ---- - 35 a z - 17 a z + a z + ----- + 8 a z + 2 a a 3 7 5 7 8 2 8 4 8 9 3 9 2 a z + 4 a z + 8 z + 14 a z + 6 a z + 3 a z + 3 a z
In[19]:=	{Vassiliev[2][Knot[10, 114]], Vassiliev[3][Knot[10, 114]]}
Out[19]=	{1, -1}
In[20]:=	Kh[Knot[10, 114]][q, t]
Out[20]=	8 1 3 1 4 3 7 4 - + 8 q + ------ + ------ + ----- + ----- + ----- + ----- + ----- + q 13 6 11 5 9 5 9 4 7 4 7 3 5 3 q t q t q t q t q t q t q t 8 7 7 8 3 3 2 5 2 ----- + ----- + ---- + --- + 5 q t + 7 q t + 3 q t + 5 q t + 5 2 3 2 3 q t q t q t q t 5 3 7 3 9 4 q t + 3 q t + q t
In[21]:=	ColouredJones[Knot[10, 114], 2][q]
Out[21]=	-18 4 2 14 24 7 58 47 50 121 47 144 + q - --- + --- + --- - --- - --- + --- - --- - --- + --- - -- - 17 16 15 14 13 12 11 10 9 8 q q q q q q q q q q 117 170 20 173 179 18 188 2 3 --- + --- - -- - --- + --- + -- - --- + 44 q - 150 q + 82 q + 7 6 5 4 3 2 q q q q q q q 4 5 6 7 8 9 10 11 12 42 q - 81 q + 31 q + 20 q - 26 q + 8 q + 4 q - 4 q + q

See/edit the Rolfsen_Splice_Template.