8 14

8_13

8_15

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8 14 Further Notes and Views

Knot presentations

Planar diagram presentation	X1425 X5,10,6,11 X3948 X9,3,10,2 X7,14,8,15 X11,16,12,1 X15,12,16,13 X13,6,14,7
Gauss code	-1, 4, -3, 1, -2, 8, -5, 3, -4, 2, -6, 7, -8, 5, -7, 6
Dowker-Thistlethwaite code	4 8 10 14 2 16 6 12
Conway Notation	[22112]

Minimum Braid Representative: Length is 9, width is 4. Braid index is 4.

A Morse Link Presentation:

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 2 Bridge index 2 Super bridge index $\{4,5\}$ Nakanishi index 1 Maximal Thurston-Bennequin number [-9][-1] Hyperbolic Volume 9.2178 A-Polynomial See Data:8 14/A-polynomial

[edit Notes for 8 14's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for 8 14's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-2t^{2}+8t-11+8t^{-1}-2t^{-2}$
Conway polynomial	$1-2z^{4}$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 31, -2 }
Jones polynomial	$q-2+4q^{-1}-5q^{-2}+6q^{-3}-5q^{-4}+4q^{-5}-3q^{-6}+q^{-7}$
HOMFLY-PT polynomial (db, data sources)	$z^{2}a^{6}-z^{4}a^{4}-z^{2}a^{4}-z^{4}a^{2}-z^{2}a^{2}+z^{2}+1$
Kauffman polynomial (db, data sources)	$z^{4}a^{8}-z^{2}a^{8}+3z^{5}a^{7}-5z^{3}a^{7}+za^{7}+3z^{6}a^{6}-4z^{4}a^{6}+z^{2}a^{6}+z^{7}a^{5}+4z^{5}a^{5}-8z^{3}a^{5}+3za^{5}+5z^{6}a^{4}-7z^{4}a^{4}+3z^{2}a^{4}+z^{7}a^{3}+3z^{5}a^{3}-6z^{3}a^{3}+3za^{3}+2z^{6}a^{2}-z^{4}a^{2}-z^{2}a^{2}+2z^{5}a-3z^{3}a+za+z^{4}-2z^{2}+1$
The A2 invariant	$q^{22}-q^{20}-q^{18}+q^{16}-q^{14}+q^{12}+q^{6}-q^{4}+2q^{2}+q^{-4}$
The G2 invariant	$q^{114}-2q^{112}+4q^{110}-6q^{108}+3q^{106}-6q^{102}+14q^{100}-16q^{98}+17q^{96}-11q^{94}-4q^{92}+17q^{90}-25q^{88}+25q^{86}-17q^{84}+4q^{82}+11q^{80}-17q^{78}+17q^{76}-9q^{74}-3q^{72}+13q^{70}-16q^{68}+5q^{66}+7q^{64}-19q^{62}+28q^{60}-26q^{58}+15q^{56}+q^{54}-21q^{52}+33q^{50}-37q^{48}+28q^{46}-11q^{44}-7q^{42}+21q^{40}-23q^{38}+19q^{36}-7q^{34}-6q^{32}+13q^{30}-12q^{28}+2q^{26}+12q^{24}-19q^{22}+22q^{20}-13q^{18}+12q^{14}-21q^{12}+24q^{10}-18q^{8}+8q^{6}+2q^{4}-9q^{2}+12-10q^{-2}+9q^{-4}-3q^{-6}+2q^{-10}-3q^{-12}+3q^{-14}-q^{-16}+q^{-18}$

Further Quantum Invariants

Further quantum knot invariants for 8_14.

A1 Invariants.

Weight	Invariant
1	$q^{15}-2q^{13}+q^{11}-q^{9}+q^{7}+q^{5}-q^{3}+2q-q^{-1}+q^{-3}$
2	$q^{42}-2q^{40}-2q^{38}+6q^{36}-q^{34}-6q^{32}+7q^{30}+q^{28}-8q^{26}+4q^{24}+2q^{22}-5q^{20}+3q^{16}+2q^{14}-5q^{12}+2q^{10}+7q^{8}-7q^{6}-q^{4}+8q^{2}-4-2q^{-2}+4q^{-4}-q^{-6}-q^{-8}+q^{-10}$
3	$q^{81}-2q^{79}-2q^{77}+3q^{75}+6q^{73}-q^{71}-13q^{69}-q^{67}+16q^{65}+7q^{63}-19q^{61}-15q^{59}+19q^{57}+22q^{55}-16q^{53}-25q^{51}+12q^{49}+26q^{47}-5q^{45}-25q^{43}+2q^{41}+19q^{39}+3q^{37}-15q^{35}-8q^{33}+6q^{31}+13q^{29}-18q^{25}-7q^{23}+20q^{21}+16q^{19}-20q^{17}-21q^{15}+19q^{13}+25q^{11}-12q^{9}-25q^{7}+6q^{5}+23q^{3}-q-16q^{-1}-2q^{-3}+11q^{-5}+3q^{-7}-6q^{-9}-2q^{-11}+3q^{-13}+q^{-15}-q^{-17}-q^{-19}+q^{-21}$
4	$q^{132}-2q^{130}-2q^{128}+3q^{126}+3q^{124}+6q^{122}-8q^{120}-13q^{118}-q^{116}+8q^{114}+31q^{112}-2q^{110}-33q^{108}-27q^{106}-2q^{104}+65q^{102}+34q^{100}-30q^{98}-68q^{96}-50q^{94}+74q^{92}+84q^{90}+9q^{88}-84q^{86}-101q^{84}+43q^{82}+102q^{80}+57q^{78}-60q^{76}-116q^{74}+3q^{72}+81q^{70}+73q^{68}-23q^{66}-91q^{64}-24q^{62}+41q^{60}+62q^{58}+8q^{56}-49q^{54}-42q^{52}-q^{50}+45q^{48}+42q^{46}-q^{44}-65q^{42}-46q^{40}+28q^{38}+75q^{36}+49q^{34}-74q^{32}-88q^{30}-6q^{28}+90q^{26}+99q^{24}-53q^{22}-104q^{20}-48q^{18}+64q^{16}+115q^{14}-7q^{12}-73q^{10}-68q^{8}+14q^{6}+85q^{4}+22q^{2}-24-49q^{-2}-15q^{-4}+37q^{-6}+17q^{-8}+2q^{-10}-19q^{-12}-12q^{-14}+11q^{-16}+4q^{-18}+4q^{-20}-4q^{-22}-4q^{-24}+3q^{-26}+q^{-30}-q^{-32}-q^{-34}+q^{-36}$
5	$q^{195}-2q^{193}-2q^{191}+3q^{189}+3q^{187}+3q^{185}-q^{183}-8q^{181}-13q^{179}-q^{177}+17q^{175}+23q^{173}+13q^{171}-16q^{169}-43q^{167}-44q^{165}+10q^{163}+70q^{161}+78q^{159}+23q^{157}-76q^{155}-138q^{153}-86q^{151}+67q^{149}+189q^{147}+165q^{145}-8q^{143}-220q^{141}-266q^{139}-76q^{137}+215q^{135}+352q^{133}+184q^{131}-165q^{129}-404q^{127}-295q^{125}+85q^{123}+414q^{121}+381q^{119}+7q^{117}-379q^{115}-432q^{113}-94q^{111}+316q^{109}+431q^{107}+164q^{105}-241q^{103}-402q^{101}-194q^{99}+161q^{97}+341q^{95}+211q^{93}-89q^{91}-279q^{89}-199q^{87}+30q^{85}+203q^{83}+196q^{81}+28q^{79}-146q^{77}-184q^{75}-83q^{73}+84q^{71}+187q^{69}+147q^{67}-27q^{65}-200q^{63}-214q^{61}-37q^{59}+202q^{57}+287q^{55}+114q^{53}-199q^{51}-361q^{49}-193q^{47}+170q^{45}+411q^{43}+284q^{41}-114q^{39}-432q^{37}-367q^{35}+39q^{33}+407q^{31}+416q^{29}+57q^{27}-339q^{25}-427q^{23}-145q^{21}+245q^{19}+392q^{17}+203q^{15}-130q^{13}-316q^{11}-226q^{9}+33q^{7}+228q^{5}+205q^{3}+31q-132q^{-1}-161q^{-3}-62q^{-5}+63q^{-7}+109q^{-9}+60q^{-11}-19q^{-13}-61q^{-15}-44q^{-17}-q^{-19}+29q^{-21}+28q^{-23}+5q^{-25}-13q^{-27}-13q^{-29}-3q^{-31}+2q^{-33}+6q^{-35}+4q^{-37}-3q^{-39}-2q^{-41}+q^{-43}+q^{-49}-q^{-51}-q^{-53}+q^{-55}$

A2 Invariants.

Weight	Invariant
1,0	$q^{22}-q^{20}-q^{18}+q^{16}-q^{14}+q^{12}+q^{6}-q^{4}+2q^{2}+q^{-4}$
1,1	$q^{60}-4q^{58}+10q^{56}-20q^{54}+32q^{52}-48q^{50}+66q^{48}-78q^{46}+83q^{44}-78q^{42}+64q^{40}-36q^{38}-q^{36}+42q^{34}-84q^{32}+116q^{30}-145q^{28}+156q^{26}-154q^{24}+138q^{22}-105q^{20}+68q^{18}-26q^{16}-12q^{14}+47q^{12}-68q^{10}+78q^{8}-76q^{6}+70q^{4}-56q^{2}+42-28q^{-2}+19q^{-4}-10q^{-6}+6q^{-8}-2q^{-10}+q^{-12}$
2,0	$q^{56}-q^{54}-2q^{52}+3q^{48}+2q^{46}-4q^{44}+4q^{40}+q^{38}-4q^{36}-q^{34}+3q^{32}-2q^{30}-4q^{28}+q^{24}-2q^{22}+3q^{20}+3q^{18}-q^{16}+q^{14}+4q^{12}-5q^{8}+5q^{4}-q^{2}-3+2q^{-2}+3q^{-4}-q^{-8}+q^{-12}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{48}-2q^{46}+3q^{42}-5q^{40}+2q^{38}+5q^{36}-6q^{34}+q^{32}+5q^{30}-4q^{28}-q^{26}+3q^{24}-q^{22}-2q^{20}-2q^{18}+3q^{16}-q^{14}-4q^{12}+7q^{10}+q^{8}-5q^{6}+6q^{4}+q^{2}-3+3q^{-2}+q^{-4}-q^{-6}+q^{-8}$
1,0,0	$q^{29}-q^{27}-q^{23}+q^{21}-q^{19}+q^{17}+q^{7}-q^{5}+2q^{3}+q^{-1}+q^{-5}$

A4 Invariants.

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

B2 Invariants.

Weight	Invariant
0,1	$q^{48}-2q^{46}+4q^{44}-5q^{42}+5q^{40}-6q^{38}+5q^{36}-4q^{34}+q^{32}+q^{30}-4q^{28}+7q^{26}-9q^{24}+11q^{22}-10q^{20}+10q^{18}-7q^{16}+5q^{14}-2q^{12}-q^{10}+3q^{8}-5q^{6}+6q^{4}-5q^{2}+5-3q^{-2}+3q^{-4}-q^{-6}+q^{-8}$
1,0	$q^{78}-2q^{74}-2q^{72}+2q^{70}+4q^{68}-q^{66}-5q^{64}-2q^{62}+6q^{60}+5q^{58}-3q^{56}-6q^{54}+5q^{50}+2q^{48}-4q^{46}-3q^{44}+3q^{42}+3q^{40}-2q^{38}-4q^{36}+q^{34}+4q^{32}-5q^{28}-q^{26}+4q^{24}+2q^{22}-4q^{20}-3q^{18}+4q^{16}+6q^{14}-q^{12}-6q^{10}-q^{8}+6q^{6}+4q^{4}-2q^{2}-4+3q^{-4}+2q^{-6}-q^{-8}-q^{-10}+q^{-14}$

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{66}-2q^{64}+2q^{62}-3q^{60}+4q^{58}-5q^{56}+4q^{54}-4q^{52}+5q^{50}-3q^{48}+q^{46}+q^{42}+3q^{40}-5q^{38}+5q^{36}-6q^{34}+8q^{32}-9q^{30}+6q^{28}-9q^{26}+7q^{24}-5q^{22}+3q^{20}-3q^{18}+2q^{16}+3q^{14}-q^{12}+3q^{10}-4q^{8}+6q^{6}-3q^{4}+4q^{2}-3+4q^{-2}-q^{-4}+2q^{-6}-q^{-8}+q^{-10}$

G2 Invariants.

Weight

Invariant

1,0

$q^{114}-2q^{112}+4q^{110}-6q^{108}+3q^{106}-6q^{102}+14q^{100}-16q^{98}+17q^{96}-11q^{94}-4q^{92}+17q^{90}-25q^{88}+25q^{86}-17q^{84}+4q^{82}+11q^{80}-17q^{78}+17q^{76}-9q^{74}-3q^{72}+13q^{70}-16q^{68}+5q^{66}+7q^{64}-19q^{62}+28q^{60}-26q^{58}+15q^{56}+q^{54}-21q^{52}+33q^{50}-37q^{48}+28q^{46}-11q^{44}-7q^{42}+21q^{40}-23q^{38}+19q^{36}-7q^{34}-6q^{32}+13q^{30}-12q^{28}+2q^{26}+12q^{24}-19q^{22}+22q^{20}-13q^{18}+12q^{14}-21q^{12}+24q^{10}-18q^{8}+8q^{6}+2q^{4}-9q^{2}+12-10q^{-2}+9q^{-4}-3q^{-6}+2q^{-10}-3q^{-12}+3q^{-14}-q^{-16}+q^{-18}$

.

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["8 14"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-2t^{2}+8t-11+8t^{-1}-2t^{-2}$

In[5]:=

Conway[K][z]

Out[5]=

$1-2z^{4}$

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]=

{ 31, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{4}a^{8}-z^{2}a^{8}+3z^{5}a^{7}-5z^{3}a^{7}+za^{7}+3z^{6}a^{6}-4z^{4}a^{6}+z^{2}a^{6}+z^{7}a^{5}+4z^{5}a^{5}-8z^{3}a^{5}+3za^{5}+5z^{6}a^{4}-7z^{4}a^{4}+3z^{2}a^{4}+z^{7}a^{3}+3z^{5}a^{3}-6z^{3}a^{3}+3za^{3}+2z^{6}a^{2}-z^{4}a^{2}-z^{2}a^{2}+2z^{5}a-3z^{3}a+za+z^{4}-2z^{2}+1$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {9_8, 10_131, ...}

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

Vassiliev invariants

V2 and V3:

(0, 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 $0$ $0$ $0$ $16$ $16$ $0$ $-64$ $-32$ $-32$ $0$ $0$ $0$ $0$ $136$ ${\frac {32}{3}}$ ${\frac {320}{3}}$ ${\frac {56}{3}}$ $8$

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

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

Integral Khovanov Homology (db, data source)

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

The Coloured Jones Polynomials

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

$n$	$J_{n}$
2	$q^{4}-2q^{3}+6q-8-2q^{-1}+18q^{-2}-17q^{-3}-8q^{-4}+32q^{-5}-22q^{-6}-15q^{-7}+39q^{-8}-21q^{-9}-18q^{-10}+34q^{-11}-14q^{-12}-16q^{-13}+22q^{-14}-5q^{-15}-10q^{-16}+9q^{-17}-3q^{-19}+q^{-20}$
3	$q^{9}-2q^{8}+2q^{6}+3q^{5}-7q^{4}-4q^{3}+11q^{2}+11q-20-18q^{-1}+26q^{-2}+35q^{-3}-37q^{-4}-49q^{-5}+39q^{-6}+72q^{-7}-43q^{-8}-89q^{-9}+40q^{-10}+108q^{-11}-39q^{-12}-116q^{-13}+29q^{-14}+126q^{-15}-26q^{-16}-123q^{-17}+15q^{-18}+119q^{-19}-8q^{-20}-107q^{-21}-2q^{-22}+92q^{-23}+12q^{-24}-76q^{-25}-16q^{-26}+55q^{-27}+21q^{-28}-38q^{-29}-19q^{-30}+21q^{-31}+17q^{-32}-12q^{-33}-10q^{-34}+4q^{-35}+5q^{-36}-3q^{-38}+q^{-39}$
4	$q^{16}-2q^{15}+2q^{13}-q^{12}+4q^{11}-9q^{10}+10q^{8}-q^{7}+11q^{6}-32q^{5}-7q^{4}+31q^{3}+14q^{2}+31q-84-41q^{-1}+56q^{-2}+60q^{-3}+94q^{-4}-155q^{-5}-123q^{-6}+51q^{-7}+126q^{-8}+216q^{-9}-206q^{-10}-235q^{-11}-5q^{-12}+177q^{-13}+368q^{-14}-215q^{-15}-331q^{-16}-87q^{-17}+191q^{-18}+491q^{-19}-189q^{-20}-378q^{-21}-161q^{-22}+172q^{-23}+555q^{-24}-146q^{-25}-375q^{-26}-207q^{-27}+131q^{-28}+548q^{-29}-89q^{-30}-321q^{-31}-228q^{-32}+66q^{-33}+481q^{-34}-21q^{-35}-225q^{-36}-220q^{-37}-12q^{-38}+362q^{-39}+35q^{-40}-108q^{-41}-175q^{-42}-71q^{-43}+218q^{-44}+52q^{-45}-15q^{-46}-100q^{-47}-81q^{-48}+94q^{-49}+34q^{-50}+23q^{-51}-36q^{-52}-50q^{-53}+27q^{-54}+9q^{-55}+17q^{-56}-5q^{-57}-17q^{-58}+4q^{-59}+5q^{-61}-3q^{-63}+q^{-64}$
5	$q^{25}-2q^{24}+2q^{22}-q^{21}+2q^{19}-5q^{18}-q^{17}+9q^{16}+q^{15}-4q^{14}-3q^{13}-15q^{12}-q^{11}+27q^{10}+24q^{9}-3q^{8}-33q^{7}-58q^{6}-18q^{5}+69q^{4}+103q^{3}+46q^{2}-79q-183-117q^{-1}+98q^{-2}+266q^{-3}+220q^{-4}-56q^{-5}-378q^{-6}-376q^{-7}+8q^{-8}+452q^{-9}+553q^{-10}+133q^{-11}-525q^{-12}-766q^{-13}-274q^{-14}+540q^{-15}+949q^{-16}+492q^{-17}-534q^{-18}-1134q^{-19}-680q^{-20}+475q^{-21}+1267q^{-22}+890q^{-23}-407q^{-24}-1375q^{-25}-1043q^{-26}+307q^{-27}+1429q^{-28}+1203q^{-29}-234q^{-30}-1460q^{-31}-1282q^{-32}+130q^{-33}+1443q^{-34}+1376q^{-35}-60q^{-36}-1420q^{-37}-1385q^{-38}-37q^{-39}+1342q^{-40}+1414q^{-41}+114q^{-42}-1252q^{-43}-1378q^{-44}-210q^{-45}+1113q^{-46}+1334q^{-47}+304q^{-48}-952q^{-49}-1248q^{-50}-390q^{-51}+758q^{-52}+1126q^{-53}+465q^{-54}-547q^{-55}-981q^{-56}-505q^{-57}+348q^{-58}+788q^{-59}+518q^{-60}-161q^{-61}-607q^{-62}-472q^{-63}+19q^{-64}+408q^{-65}+409q^{-66}+78q^{-67}-258q^{-68}-304q^{-69}-118q^{-70}+117q^{-71}+219q^{-72}+124q^{-73}-46q^{-74}-131q^{-75}-94q^{-76}-5q^{-77}+66q^{-78}+72q^{-79}+16q^{-80}-32q^{-81}-39q^{-82}-13q^{-83}+6q^{-84}+18q^{-85}+17q^{-86}-5q^{-87}-10q^{-88}-3q^{-89}+5q^{-92}-3q^{-94}+q^{-95}$
6	$q^{36}-2q^{35}+2q^{33}-q^{32}-2q^{30}+6q^{29}-6q^{28}-2q^{27}+11q^{26}-3q^{25}-4q^{24}-12q^{23}+14q^{22}-13q^{21}-q^{20}+40q^{19}+5q^{18}-14q^{17}-51q^{16}+7q^{15}-45q^{14}+8q^{13}+129q^{12}+70q^{11}+2q^{10}-142q^{9}-77q^{8}-188q^{7}-27q^{6}+310q^{5}+305q^{4}+181q^{3}-206q^{2}-279q-616-306q^{-1}+459q^{-2}+767q^{-3}+751q^{-4}+33q^{-5}-420q^{-6}-1391q^{-7}-1106q^{-8}+223q^{-9}+1230q^{-10}+1757q^{-11}+882q^{-12}-107q^{-13}-2225q^{-14}-2424q^{-15}-703q^{-16}+1269q^{-17}+2849q^{-18}+2286q^{-19}+909q^{-20}-2673q^{-21}-3841q^{-22}-2180q^{-23}+676q^{-24}+3567q^{-25}+3777q^{-26}+2400q^{-27}-2542q^{-28}-4884q^{-29}-3703q^{-30}-313q^{-31}+3737q^{-32}+4898q^{-33}+3853q^{-34}-2036q^{-35}-5373q^{-36}-4845q^{-37}-1279q^{-38}+3519q^{-39}+5492q^{-40}+4912q^{-41}-1457q^{-42}-5422q^{-43}-5490q^{-44}-1996q^{-45}+3131q^{-46}+5656q^{-47}+5517q^{-48}-923q^{-49}-5177q^{-50}-5734q^{-51}-2486q^{-52}+2625q^{-53}+5496q^{-54}+5772q^{-55}-364q^{-56}-4641q^{-57}-5654q^{-58}-2868q^{-59}+1907q^{-60}+4985q^{-61}+5739q^{-62}+325q^{-63}-3718q^{-64}-5196q^{-65}-3161q^{-66}+902q^{-67}+4022q^{-68}+5343q^{-69}+1087q^{-70}-2391q^{-71}-4253q^{-72}-3204q^{-73}-237q^{-74}+2629q^{-75}+4445q^{-76}+1641q^{-77}-917q^{-78}-2861q^{-79}-2767q^{-80}-1116q^{-81}+1119q^{-82}+3080q^{-83}+1669q^{-84}+228q^{-85}-1373q^{-86}-1851q^{-87}-1365q^{-88}-12q^{-89}+1623q^{-90}+1162q^{-91}+686q^{-92}-288q^{-93}-834q^{-94}-1018q^{-95}-458q^{-96}+564q^{-97}+505q^{-98}+552q^{-99}+154q^{-100}-156q^{-101}-495q^{-102}-377q^{-103}+91q^{-104}+85q^{-105}+247q^{-106}+155q^{-107}+75q^{-108}-151q^{-109}-169q^{-110}-5q^{-111}-35q^{-112}+59q^{-113}+58q^{-114}+67q^{-115}-28q^{-116}-46q^{-117}-2q^{-118}-25q^{-119}+6q^{-120}+9q^{-121}+26q^{-122}-5q^{-123}-10q^{-124}+4q^{-125}-7q^{-126}+5q^{-129}-3q^{-131}+q^{-132}$
7	$q^{49}-2q^{48}+2q^{46}-q^{45}-2q^{43}+2q^{42}+5q^{41}-7q^{40}+7q^{38}-3q^{37}-2q^{36}-12q^{35}+q^{34}+20q^{33}-11q^{32}+6q^{31}+21q^{30}-4q^{29}-8q^{28}-51q^{27}-25q^{26}+35q^{25}+52q^{23}+83q^{22}+19q^{21}-14q^{20}-157q^{19}-164q^{18}-29q^{17}-7q^{16}+199q^{15}+318q^{14}+214q^{13}+87q^{12}-336q^{11}-566q^{10}-439q^{9}-289q^{8}+340q^{7}+874q^{6}+925q^{5}+725q^{4}-269q^{3}-1203q^{2}-1538q-1475-152q^{-1}+1434q^{-2}+2356q^{-3}+2577q^{-4}+949q^{-5}-1355q^{-6}-3130q^{-7}-4081q^{-8}-2342q^{-9}+847q^{-10}+3813q^{-11}+5765q^{-12}+4216q^{-13}+336q^{-14}-4003q^{-15}-7550q^{-16}-6681q^{-17}-2147q^{-18}+3744q^{-19}+9111q^{-20}+9280q^{-21}+4616q^{-22}-2657q^{-23}-10290q^{-24}-12076q^{-25}-7532q^{-26}+1100q^{-27}+10901q^{-28}+14515q^{-29}+10638q^{-30}+1136q^{-31}-10925q^{-32}-16688q^{-33}-13698q^{-34}-3553q^{-35}+10439q^{-36}+18233q^{-37}+16479q^{-38}+6129q^{-39}-9558q^{-40}-19338q^{-41}-18830q^{-42}-8479q^{-43}+8428q^{-44}+19907q^{-45}+20697q^{-46}+10636q^{-47}-7272q^{-48}-20178q^{-49}-22038q^{-50}-12343q^{-51}+6107q^{-52}+20069q^{-53}+23014q^{-54}+13796q^{-55}-5089q^{-56}-19909q^{-57}-23577q^{-58}-14790q^{-59}+4117q^{-60}+19454q^{-61}+23913q^{-62}+15709q^{-63}-3258q^{-64}-19044q^{-65}-23985q^{-66}-16270q^{-67}+2364q^{-68}+18291q^{-69}+23889q^{-70}+16880q^{-71}-1413q^{-72}-17493q^{-73}-23547q^{-74}-17258q^{-75}+318q^{-76}+16274q^{-77}+22936q^{-78}+17653q^{-79}+962q^{-80}-14797q^{-81}-21998q^{-82}-17841q^{-83}-2380q^{-84}+12872q^{-85}+20621q^{-86}+17836q^{-87}+3934q^{-88}-10582q^{-89}-18810q^{-90}-17494q^{-91}-5449q^{-92}+8006q^{-93}+16494q^{-94}+16696q^{-95}+6798q^{-96}-5250q^{-97}-13766q^{-98}-15435q^{-99}-7771q^{-100}+2596q^{-101}+10761q^{-102}+13609q^{-103}+8222q^{-104}-179q^{-105}-7700q^{-106}-11405q^{-107}-8068q^{-108}-1650q^{-109}+4794q^{-110}+8892q^{-111}+7352q^{-112}+2891q^{-113}-2354q^{-114}-6441q^{-115}-6116q^{-116}-3371q^{-117}+463q^{-118}+4126q^{-119}+4701q^{-120}+3337q^{-121}+685q^{-122}-2330q^{-123}-3209q^{-124}-2765q^{-125}-1283q^{-126}+960q^{-127}+1949q^{-128}+2102q^{-129}+1361q^{-130}-212q^{-131}-977q^{-132}-1342q^{-133}-1140q^{-134}-211q^{-135}+339q^{-136}+772q^{-137}+850q^{-138}+295q^{-139}-45q^{-140}-353q^{-141}-509q^{-142}-244q^{-143}-123q^{-144}+110q^{-145}+308q^{-146}+175q^{-147}+108q^{-148}-28q^{-149}-134q^{-150}-71q^{-151}-92q^{-152}-42q^{-153}+67q^{-154}+53q^{-155}+53q^{-156}+12q^{-157}-31q^{-158}+2q^{-159}-25q^{-160}-25q^{-161}+6q^{-162}+9q^{-163}+17q^{-164}+4q^{-165}-10q^{-166}+4q^{-167}-7q^{-169}+5q^{-172}-3q^{-174}+q^{-175}$

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[8, 14]]
Out[2]=	PD[X[1, 4, 2, 5], X[5, 10, 6, 11], X[3, 9, 4, 8], X[9, 3, 10, 2], X[7, 14, 8, 15], X[11, 16, 12, 1], X[15, 12, 16, 13], X[13, 6, 14, 7]]
In[3]:=	GaussCode[Knot[8, 14]]
Out[3]=	GaussCode[-1, 4, -3, 1, -2, 8, -5, 3, -4, 2, -6, 7, -8, 5, -7, 6]
In[4]:=	DTCode[Knot[8, 14]]
Out[4]=	DTCode[4, 8, 10, 14, 2, 16, 6, 12]
In[5]:=	br = BR[Knot[8, 14]]
Out[5]=	BR[4, {-1, -1, -1, -2, 1, -2, 3, -2, 3}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{4, 9}
In[7]:=	BraidIndex[Knot[8, 14]]
Out[7]=	4
In[8]:=	Show[DrawMorseLink[Knot[8, 14]]]
Out[8]=	-Graphics-
In[9]:=	(#[Knot[8, 14]]&) /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 1, 2, 2, {4, 5}, 1}
In[10]:=	alex = Alexander[Knot[8, 14]][t]
Out[10]=	2 8 2 -11 - -- + - + 8 t - 2 t 2 t t
In[11]:=	Conway[Knot[8, 14]][z]
Out[11]=	4 1 - 2 z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[8, 14], Knot[9, 8], Knot[10, 131]}
In[13]:=	{KnotDet[Knot[8, 14]], KnotSignature[Knot[8, 14]]}
Out[13]=	{31, -2}
In[14]:=	Jones[Knot[8, 14]][q]
Out[14]=	-7 3 4 5 6 5 4 -2 + q - -- + -- - -- + -- - -- + - + q 6 5 4 3 2 q q q q q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[8, 14]}
In[16]:=	A2Invariant[Knot[8, 14]][q]
Out[16]=	-22 -20 -18 -16 -14 -12 -6 -4 2 4 q - q - q + q - q + q + q - q + -- + q 2 q
In[17]:=	HOMFLYPT[Knot[8, 14]][a, z]
Out[17]=	2 2 2 4 2 6 2 2 4 4 4 1 + z - a z - a z + a z - a z - a z
In[18]:=	Kauffman[Knot[8, 14]][a, z]
Out[18]=	3 5 7 2 2 2 4 2 6 2 1 + a z + 3 a z + 3 a z + a z - 2 z - a z + 3 a z + a z - 8 2 3 3 3 5 3 7 3 4 2 4 4 4 a z - 3 a z - 6 a z - 8 a z - 5 a z + z - a z - 7 a z - 6 4 8 4 5 3 5 5 5 7 5 2 6 4 a z + a z + 2 a z + 3 a z + 4 a z + 3 a z + 2 a z + 4 6 6 6 3 7 5 7 5 a z + 3 a z + a z + a z
In[19]:=	{Vassiliev[2][Knot[8, 14]], Vassiliev[3][Knot[8, 14]]}
Out[19]=	{0, 0}
In[20]:=	Kh[Knot[8, 14]][q, t]
Out[20]=	2 3 1 2 1 2 2 3 2 -- + - + ------ + ------ + ------ + ------ + ----- + ----- + ----- + 3 q 15 6 13 5 11 5 11 4 9 4 9 3 7 3 q q t q t q t q t q t q t q t 3 3 2 3 t 3 2 ----- + ----- + ---- + ---- + - + q t + q t 7 2 5 2 5 3 q q t q t q t q t
In[21]:=	ColouredJones[Knot[8, 14], 2][q]
Out[21]=	-20 3 9 10 5 22 16 14 34 18 21 -8 + q - --- + --- - --- - --- + --- - --- - --- + --- - --- - -- + 19 17 16 15 14 13 12 11 10 9 q q q q q q q q q q 39 15 22 32 8 17 18 2 3 4 -- - -- - -- + -- - -- - -- + -- - - + 6 q - 2 q + q 8 7 6 5 4 3 2 q q q q q q q q

See/edit the Rolfsen_Splice_Template.