9 8

9_7

9_9

Visit 9 8's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 8's page at Knotilus! Visit 9 8's page at the original Knot Atlas! 9 8 Quick Notes

9 8 Further Notes and Views

Knot presentations

Planar diagram presentation	X1425 X3849 X5,14,6,15 X9,1,10,18 X11,17,12,16 X15,13,16,12 X17,11,18,10 X13,6,14,7 X7283
Gauss code	-1, 9, -2, 1, -3, 8, -9, 2, -4, 7, -5, 6, -8, 3, -6, 5, -7, 4
Dowker-Thistlethwaite code	4 8 14 2 18 16 6 12 10
Conway Notation	[2412]

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 2 3-genus 2 Bridge index 2 Super bridge index $\{4,6\}$ Nakanishi index 1 Maximal Thurston-Bennequin number [-8][-3] Hyperbolic Volume 8.19235 A-Polynomial See Data:9 8/A-polynomial

[edit Notes for 9 8'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 9 8'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	$q^{3}-2q^{2}+3q-4+5q^{-1}-5q^{-2}+5q^{-3}-3q^{-4}+2q^{-5}-q^{-6}$
HOMFLY-PT polynomial (db, data sources)	$-a^{6}+2z^{2}a^{4}+2a^{4}-z^{4}a^{2}-z^{2}a^{2}-z^{4}-2z^{2}-1+z^{2}a^{-2}+a^{-2}$
Kauffman polynomial (db, data sources)	$a^{2}z^{8}+z^{8}+2a^{3}z^{7}+4az^{7}+2z^{7}a^{-1}+2a^{4}z^{6}+z^{6}a^{-2}-z^{6}+2a^{5}z^{5}-3a^{3}z^{5}-13az^{5}-8z^{5}a^{-1}+2a^{6}z^{4}-4a^{2}z^{4}-4z^{4}a^{-2}-6z^{4}+a^{7}z^{3}+2a^{3}z^{3}+11az^{3}+8z^{3}a^{-1}-2a^{6}z^{2}-3a^{4}z^{2}+2a^{2}z^{2}+4z^{2}a^{-2}+7z^{2}-a^{7}z-a^{5}z-a^{3}z-3az-2za^{-1}+a^{6}+2a^{4}-a^{-2}-1$
The A2 invariant	$-q^{20}-q^{18}+q^{16}+q^{12}+2q^{10}+q^{6}-q^{4}-q^{-2}+q^{-4}+q^{-10}$
The G2 invariant	$q^{100}-q^{98}+2q^{96}-2q^{94}-3q^{88}+4q^{86}-5q^{84}+4q^{82}-4q^{80}+3q^{76}-5q^{74}+6q^{72}-7q^{70}+5q^{68}-4q^{66}+3q^{62}-5q^{60}+9q^{58}-6q^{56}+5q^{54}-q^{52}-2q^{50}+7q^{48}-5q^{46}+4q^{44}+3q^{42}-4q^{40}+7q^{38}-3q^{36}-3q^{34}+11q^{32}-13q^{30}+10q^{28}-5q^{26}-5q^{24}+14q^{22}-17q^{20}+14q^{18}-10q^{16}+7q^{12}-12q^{10}+12q^{8}-9q^{6}+3q^{4}+3q^{2}-7+7q^{-2}-3q^{-4}-2q^{-6}+8q^{-8}-10q^{-10}+7q^{-12}-8q^{-16}+14q^{-18}-15q^{-20}+10q^{-22}-2q^{-24}-6q^{-26}+11q^{-28}-11q^{-30}+10q^{-32}-3q^{-34}-q^{-36}+3q^{-38}-4q^{-40}+3q^{-42}-q^{-44}+q^{-46}$

Further Quantum Invariants

Further quantum knot invariants for 9_8.

A1 Invariants.

Weight	Invariant
1	$-q^{13}+q^{11}-q^{9}+2q^{7}+q-q^{-1}+q^{-3}-q^{-5}+q^{-7}$
2	$q^{36}-q^{34}-q^{32}+2q^{30}-2q^{28}+3q^{24}-4q^{22}+4q^{18}-3q^{16}-q^{14}+3q^{12}+q^{10}-2q^{8}+3q^{4}-q^{2}-2+4q^{-2}-4q^{-6}+3q^{-8}+2q^{-10}-4q^{-12}+q^{-14}+3q^{-16}-2q^{-18}-q^{-20}+q^{-22}$
3	$-q^{69}+q^{67}+q^{65}-2q^{61}+2q^{57}-q^{51}-q^{49}-q^{47}+4q^{45}+2q^{43}-6q^{41}-5q^{39}+8q^{37}+6q^{35}-5q^{33}-10q^{31}+2q^{29}+9q^{27}+2q^{25}-5q^{23}-6q^{21}+3q^{19}+8q^{17}+q^{15}-9q^{13}-2q^{11}+8q^{9}+4q^{7}-8q^{5}-4q^{3}+8q+7q^{-1}-5q^{-3}-7q^{-5}+3q^{-7}+9q^{-9}-10q^{-13}-4q^{-15}+7q^{-17}+8q^{-19}-4q^{-21}-9q^{-23}+q^{-25}+8q^{-27}+3q^{-29}-6q^{-31}-5q^{-33}+3q^{-35}+4q^{-37}-2q^{-41}-q^{-43}+q^{-45}$
4	$q^{112}-q^{110}-q^{108}+4q^{102}-2q^{100}-2q^{98}-2q^{96}-2q^{94}+9q^{92}+q^{90}-q^{88}-7q^{86}-9q^{84}+12q^{82}+10q^{80}+5q^{78}-14q^{76}-24q^{74}+9q^{72}+23q^{70}+21q^{68}-15q^{66}-44q^{64}-6q^{62}+29q^{60}+42q^{58}-46q^{54}-25q^{52}+9q^{50}+40q^{48}+23q^{46}-20q^{44}-27q^{42}-17q^{40}+12q^{38}+24q^{36}+10q^{34}-6q^{32}-25q^{30}-14q^{28}+14q^{26}+25q^{24}+7q^{22}-21q^{20}-22q^{18}+9q^{16}+31q^{14}+9q^{12}-22q^{10}-27q^{8}+7q^{6}+36q^{4}+12q^{2}-19-33q^{-2}-4q^{-4}+34q^{-6}+21q^{-8}-5q^{-10}-32q^{-12}-20q^{-14}+17q^{-16}+22q^{-18}+18q^{-20}-13q^{-22}-26q^{-24}-7q^{-26}+5q^{-28}+26q^{-30}+11q^{-32}-10q^{-34}-16q^{-36}-19q^{-38}+11q^{-40}+18q^{-42}+10q^{-44}-2q^{-46}-21q^{-48}-7q^{-50}+4q^{-52}+12q^{-54}+11q^{-56}-7q^{-58}-7q^{-60}-5q^{-62}+q^{-64}+6q^{-66}+q^{-68}-2q^{-72}-q^{-74}+q^{-76}$
5	$-q^{165}+q^{163}+q^{161}-2q^{155}-2q^{153}+2q^{151}+4q^{149}+q^{147}-6q^{143}-7q^{141}+q^{139}+9q^{137}+10q^{135}+q^{133}-11q^{131}-17q^{129}-6q^{127}+18q^{125}+28q^{123}+8q^{121}-24q^{119}-37q^{117}-18q^{115}+31q^{113}+62q^{111}+26q^{109}-44q^{107}-84q^{105}-45q^{103}+49q^{101}+113q^{99}+72q^{97}-51q^{95}-138q^{93}-107q^{91}+36q^{89}+156q^{87}+136q^{85}-q^{83}-145q^{81}-163q^{79}-38q^{77}+119q^{75}+163q^{73}+73q^{71}-63q^{69}-142q^{67}-102q^{65}+12q^{63}+99q^{61}+100q^{59}+32q^{57}-45q^{55}-87q^{53}-63q^{51}+3q^{49}+61q^{47}+72q^{45}+31q^{43}-37q^{41}-79q^{39}-46q^{37}+29q^{35}+75q^{33}+53q^{31}-24q^{29}-82q^{27}-54q^{25}+31q^{23}+89q^{21}+60q^{19}-33q^{17}-100q^{15}-70q^{13}+34q^{11}+111q^{9}+87q^{7}-24q^{5}-118q^{3}-103q+4q^{-1}+113q^{-3}+123q^{-5}+23q^{-7}-97q^{-9}-131q^{-11}-53q^{-13}+66q^{-15}+128q^{-17}+83q^{-19}-27q^{-21}-110q^{-23}-98q^{-25}-12q^{-27}+71q^{-29}+97q^{-31}+48q^{-33}-29q^{-35}-77q^{-37}-64q^{-39}-11q^{-41}+39q^{-43}+61q^{-45}+41q^{-47}-q^{-49}-39q^{-51}-48q^{-53}-28q^{-55}+4q^{-57}+36q^{-59}+43q^{-61}+22q^{-63}-13q^{-65}-35q^{-67}-34q^{-69}-13q^{-71}+18q^{-73}+33q^{-75}+25q^{-77}-19q^{-81}-24q^{-83}-15q^{-85}+6q^{-87}+17q^{-89}+14q^{-91}+3q^{-93}-5q^{-95}-9q^{-97}-7q^{-99}+q^{-101}+4q^{-103}+3q^{-105}+q^{-107}-2q^{-111}-q^{-113}+q^{-115}$

A2 Invariants.

Weight	Invariant
1,0	$-q^{20}-q^{18}+q^{16}+q^{12}+2q^{10}+q^{6}-q^{4}-q^{-2}+q^{-4}+q^{-10}$
1,1	$q^{52}-2q^{50}+4q^{48}-6q^{46}+9q^{44}-12q^{42}+14q^{40}-16q^{38}+15q^{36}-18q^{34}+14q^{32}-16q^{30}+16q^{28}-14q^{26}+18q^{24}-10q^{22}+10q^{20}+2q^{18}-14q^{16}+26q^{14}-44q^{12}+52q^{10}-62q^{8}+64q^{6}-59q^{4}+56q^{2}-38+26q^{-2}-7q^{-4}-10q^{-6}+24q^{-8}-36q^{-10}+41q^{-12}-40q^{-14}+36q^{-16}-28q^{-18}+19q^{-20}-12q^{-22}+6q^{-24}-2q^{-26}+q^{-28}$
2,0	$q^{50}+q^{48}-2q^{44}-q^{42}-2q^{38}-2q^{36}+2q^{34}+3q^{32}-q^{30}-q^{28}+3q^{26}+3q^{24}-3q^{22}+q^{18}-q^{16}-q^{14}-q^{8}+2q^{6}+3q^{4}+3q^{-2}+q^{-4}-3q^{-6}-q^{-8}+q^{-10}+q^{-12}-q^{-14}-q^{-16}+2q^{-18}+q^{-20}-q^{-22}-q^{-24}+q^{-28}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{42}-q^{40}+2q^{36}-3q^{34}-3q^{32}+2q^{30}-2q^{28}-3q^{26}+5q^{24}+2q^{22}+4q^{18}+2q^{16}-q^{14}-q^{12}-4q^{6}+q^{4}+3q^{2}-2+q^{-2}+3q^{-4}-2q^{-6}+2q^{-10}-2q^{-12}+q^{-14}+q^{-16}-q^{-18}+q^{-20}$
1,0,0	$-q^{27}-q^{25}-q^{23}+q^{21}+2q^{17}+q^{15}+2q^{13}+q^{9}-q^{5}-q-q^{-3}+q^{-5}+q^{-9}+q^{-13}$

B2 Invariants.

Weight	Invariant
0,1	$-q^{42}+q^{40}-2q^{38}+2q^{36}-3q^{34}+3q^{32}-4q^{30}+4q^{28}-3q^{26}+3q^{24}+4q^{18}-4q^{16}+7q^{14}-7q^{12}+8q^{10}-8q^{8}+6q^{6}-5q^{4}+3q^{2}-2-q^{-2}+3q^{-4}-4q^{-6}+4q^{-8}-4q^{-10}+4q^{-12}-3q^{-14}+3q^{-16}-q^{-18}+q^{-20}$
1,0	$q^{68}-q^{64}-q^{62}+q^{60}+2q^{58}-3q^{54}-3q^{52}+3q^{48}+q^{46}-3q^{44}-3q^{42}+q^{40}+5q^{38}+q^{36}-2q^{34}-q^{32}+4q^{30}+3q^{28}-q^{26}-3q^{24}+q^{22}+3q^{20}-3q^{16}-q^{14}+2q^{12}-3q^{8}-q^{6}+3q^{4}+3q^{2}-1-4q^{-2}+5q^{-6}+3q^{-8}-3q^{-10}-4q^{-12}+q^{-14}+4q^{-16}+q^{-18}-3q^{-20}-2q^{-22}+2q^{-24}+2q^{-26}-q^{-28}-q^{-30}+q^{-34}$

G2 Invariants.

Weight

Invariant

1,0

$q^{100}-q^{98}+2q^{96}-2q^{94}-3q^{88}+4q^{86}-5q^{84}+4q^{82}-4q^{80}+3q^{76}-5q^{74}+6q^{72}-7q^{70}+5q^{68}-4q^{66}+3q^{62}-5q^{60}+9q^{58}-6q^{56}+5q^{54}-q^{52}-2q^{50}+7q^{48}-5q^{46}+4q^{44}+3q^{42}-4q^{40}+7q^{38}-3q^{36}-3q^{34}+11q^{32}-13q^{30}+10q^{28}-5q^{26}-5q^{24}+14q^{22}-17q^{20}+14q^{18}-10q^{16}+7q^{12}-12q^{10}+12q^{8}-9q^{6}+3q^{4}+3q^{2}-7+7q^{-2}-3q^{-4}-2q^{-6}+8q^{-8}-10q^{-10}+7q^{-12}-8q^{-16}+14q^{-18}-15q^{-20}+10q^{-22}-2q^{-24}-6q^{-26}+11q^{-28}-11q^{-30}+10q^{-32}-3q^{-34}-q^{-36}+3q^{-38}-4q^{-40}+3q^{-42}-q^{-44}+q^{-46}$

.

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["9 8"];

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]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$a^{2}z^{8}+z^{8}+2a^{3}z^{7}+4az^{7}+2z^{7}a^{-1}+2a^{4}z^{6}+z^{6}a^{-2}-z^{6}+2a^{5}z^{5}-3a^{3}z^{5}-13az^{5}-8z^{5}a^{-1}+2a^{6}z^{4}-4a^{2}z^{4}-4z^{4}a^{-2}-6z^{4}+a^{7}z^{3}+2a^{3}z^{3}+11az^{3}+8z^{3}a^{-1}-2a^{6}z^{2}-3a^{4}z^{2}+2a^{2}z^{2}+4z^{2}a^{-2}+7z^{2}-a^{7}z-a^{5}z-a^{3}z-3az-2za^{-1}+a^{6}+2a^{4}-a^{-2}-1$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {8_14, 10_131, ...}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {K11n60, ...}

Vassiliev invariants

V2 and V3:

(0, -2)

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 $0$ $-16$ $0$ $48$ $16$ $0$ $-{\frac {256}{3}}$ ${\frac {32}{3}}$ $-48$ $0$ $128$ $0$ $0$ $344$ $-{\frac {224}{3}}$ ${\frac {608}{3}}$ ${\frac {104}{3}}$ $24$

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 9 8. 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 χ 7                   1 1 5                 1   -1 3               2 1   1 1             2 1     -1 -1           3 2       1 -3         3 3         0 -5       2 2           0 -7     1 3             2 -9   1 2               -1 -11   1                 1 -13 1                   -1

Integral Khovanov Homology (db, data source)

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

The Coloured Jones Polynomials

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

$n$	$J_{n}$
2	$q^{10}-2q^{9}-q^{8}+6q^{7}-4q^{6}-6q^{5}+12q^{4}-3q^{3}-13q^{2}+16q+1-19q^{-1}+17q^{-2}+5q^{-3}-22q^{-4}+15q^{-5}+8q^{-6}-20q^{-7}+11q^{-8}+6q^{-9}-13q^{-10}+7q^{-11}+2q^{-12}-6q^{-13}+4q^{-14}-2q^{-16}+q^{-17}$
3	$q^{21}-2q^{20}-q^{19}+2q^{18}+5q^{17}-3q^{16}-9q^{15}+q^{14}+14q^{13}+2q^{12}-16q^{11}-9q^{10}+19q^{9}+14q^{8}-17q^{7}-20q^{6}+13q^{5}+24q^{4}-8q^{3}-26q^{2}+3q+26+4q^{-1}-25q^{-2}-9q^{-3}+22q^{-4}+16q^{-5}-21q^{-6}-19q^{-7}+15q^{-8}+26q^{-9}-14q^{-10}-24q^{-11}+6q^{-12}+27q^{-13}-7q^{-14}-17q^{-15}-q^{-16}+15q^{-17}-2q^{-18}-6q^{-19}+q^{-20}+2q^{-21}-3q^{-22}+2q^{-23}+3q^{-24}-3q^{-25}-3q^{-26}+2q^{-27}+4q^{-28}-3q^{-29}-q^{-30}+2q^{-32}-q^{-33}$
4	$q^{36}-2q^{35}-q^{34}+2q^{33}+q^{32}+6q^{31}-7q^{30}-7q^{29}+q^{27}+24q^{26}-6q^{25}-15q^{24}-11q^{23}-13q^{22}+43q^{21}+6q^{20}-7q^{19}-18q^{18}-43q^{17}+46q^{16}+12q^{15}+14q^{14}-3q^{13}-64q^{12}+34q^{11}-7q^{10}+27q^{9}+28q^{8}-60q^{7}+29q^{6}-44q^{5}+15q^{4}+55q^{3}-34q^{2}+42q-82-14q^{-1}+69q^{-2}-3q^{-3}+66q^{-4}-111q^{-5}-48q^{-6}+74q^{-7}+28q^{-8}+88q^{-9}-133q^{-10}-79q^{-11}+75q^{-12}+56q^{-13}+106q^{-14}-144q^{-15}-107q^{-16}+64q^{-17}+75q^{-18}+122q^{-19}-130q^{-20}-119q^{-21}+35q^{-22}+65q^{-23}+129q^{-24}-87q^{-25}-102q^{-26}+4q^{-27}+31q^{-28}+108q^{-29}-41q^{-30}-60q^{-31}-9q^{-32}-4q^{-33}+70q^{-34}-12q^{-35}-24q^{-36}-7q^{-37}-18q^{-38}+37q^{-39}-2q^{-40}-5q^{-41}-2q^{-42}-16q^{-43}+16q^{-44}+q^{-46}-8q^{-48}+5q^{-49}+q^{-51}-2q^{-53}+q^{-54}$
5	$q^{55}-2q^{54}-q^{53}+2q^{52}+q^{51}+2q^{50}+2q^{49}-5q^{48}-9q^{47}+5q^{45}+10q^{44}+13q^{43}-2q^{42}-20q^{41}-21q^{40}-4q^{39}+15q^{38}+32q^{37}+23q^{36}-12q^{35}-36q^{34}-35q^{33}-6q^{32}+31q^{31}+45q^{30}+23q^{29}-15q^{28}-42q^{27}-38q^{26}-q^{25}+25q^{24}+32q^{23}+23q^{22}-18q^{20}-23q^{19}-25q^{18}-21q^{17}+10q^{16}+48q^{15}+59q^{14}+26q^{13}-51q^{12}-104q^{11}-76q^{10}+36q^{9}+142q^{8}+136q^{7}-6q^{6}-166q^{5}-195q^{4}-42q^{3}+176q^{2}+256q+94-176q^{-1}-304q^{-2}-149q^{-3}+161q^{-4}+350q^{-5}+205q^{-6}-152q^{-7}-381q^{-8}-253q^{-9}+131q^{-10}+417q^{-11}+298q^{-12}-123q^{-13}-439q^{-14}-338q^{-15}+103q^{-16}+475q^{-17}+375q^{-18}-101q^{-19}-485q^{-20}-413q^{-21}+70q^{-22}+517q^{-23}+443q^{-24}-60q^{-25}-496q^{-26}-471q^{-27}+4q^{-28}+493q^{-29}+485q^{-30}+17q^{-31}-428q^{-32}-472q^{-33}-83q^{-34}+379q^{-35}+445q^{-36}+96q^{-37}-292q^{-38}-382q^{-39}-127q^{-40}+222q^{-41}+320q^{-42}+114q^{-43}-148q^{-44}-245q^{-45}-107q^{-46}+102q^{-47}+177q^{-48}+83q^{-49}-61q^{-50}-122q^{-51}-66q^{-52}+38q^{-53}+83q^{-54}+44q^{-55}-21q^{-56}-52q^{-57}-30q^{-58}+7q^{-59}+34q^{-60}+25q^{-61}-8q^{-62}-20q^{-63}-10q^{-64}-3q^{-65}+10q^{-66}+14q^{-67}-2q^{-68}-8q^{-69}-q^{-70}-4q^{-71}+2q^{-72}+6q^{-73}-q^{-74}-2q^{-75}-q^{-77}+2q^{-79}-q^{-80}$
6	$q^{78}-2q^{77}-q^{76}+2q^{75}+q^{74}+2q^{73}-2q^{72}+4q^{71}-7q^{70}-9q^{69}+3q^{68}+4q^{67}+11q^{66}+2q^{65}+18q^{64}-14q^{63}-27q^{62}-13q^{61}-7q^{60}+17q^{59}+9q^{58}+63q^{57}+4q^{56}-31q^{55}-35q^{54}-43q^{53}-12q^{52}-20q^{51}+102q^{50}+44q^{49}+13q^{48}-15q^{47}-46q^{46}-48q^{45}-98q^{44}+82q^{43}+28q^{42}+44q^{41}+31q^{40}+23q^{39}+q^{38}-125q^{37}+53q^{36}-56q^{35}-29q^{34}-23q^{33}+66q^{32}+108q^{31}-21q^{30}+156q^{29}-69q^{28}-138q^{27}-225q^{26}-62q^{25}+105q^{24}+102q^{23}+401q^{22}+125q^{21}-99q^{20}-423q^{19}-337q^{18}-121q^{17}+55q^{16}+614q^{15}+465q^{14}+170q^{13}-432q^{12}-571q^{11}-483q^{10}-223q^{9}+629q^{8}+765q^{7}+569q^{6}-215q^{5}-622q^{4}-805q^{3}-624q^{2}+431q+904+938q^{-1}+123q^{-2}-493q^{-3}-993q^{-4}-1007q^{-5}+126q^{-6}+895q^{-7}+1200q^{-8}+455q^{-9}-285q^{-10}-1064q^{-11}-1303q^{-12}-171q^{-13}+830q^{-14}+1370q^{-15}+721q^{-16}-101q^{-17}-1098q^{-18}-1519q^{-19}-405q^{-20}+789q^{-21}+1508q^{-22}+926q^{-23}+22q^{-24}-1150q^{-25}-1701q^{-26}-591q^{-27}+772q^{-28}+1639q^{-29}+1118q^{-30}+135q^{-31}-1179q^{-32}-1854q^{-33}-792q^{-34}+682q^{-35}+1690q^{-36}+1297q^{-37}+325q^{-38}-1063q^{-39}-1883q^{-40}-992q^{-41}+428q^{-42}+1519q^{-43}+1347q^{-44}+556q^{-45}-737q^{-46}-1654q^{-47}-1048q^{-48}+99q^{-49}+1105q^{-50}+1136q^{-51}+653q^{-52}-342q^{-53}-1185q^{-54}-847q^{-55}-108q^{-56}+635q^{-57}+727q^{-58}+526q^{-59}-82q^{-60}-695q^{-61}-496q^{-62}-119q^{-63}+307q^{-64}+343q^{-65}+296q^{-66}+q^{-67}-361q^{-68}-207q^{-69}-46q^{-70}+151q^{-71}+118q^{-72}+125q^{-73}+2q^{-74}-190q^{-75}-59q^{-76}-2q^{-77}+85q^{-78}+31q^{-79}+48q^{-80}-q^{-81}-106q^{-82}-9q^{-83}+4q^{-84}+46q^{-85}+7q^{-86}+23q^{-87}+q^{-88}-55q^{-89}+2q^{-90}-2q^{-91}+21q^{-92}+13q^{-94}+2q^{-95}-24q^{-96}+4q^{-97}-4q^{-98}+8q^{-99}-q^{-100}+5q^{-101}+q^{-102}-8q^{-103}+3q^{-104}-2q^{-105}+2q^{-106}+q^{-108}-2q^{-110}+q^{-111}$
7	$q^{105}-2q^{104}-q^{103}+2q^{102}+q^{101}+2q^{100}-2q^{99}+2q^{97}-7q^{96}-6q^{95}+2q^{94}+4q^{93}+13q^{92}+4q^{91}+9q^{89}-18q^{88}-23q^{87}-16q^{86}-9q^{85}+27q^{84}+25q^{83}+21q^{82}+41q^{81}-6q^{80}-34q^{79}-47q^{78}-74q^{77}-2q^{76}+20q^{75}+33q^{74}+98q^{73}+51q^{72}+22q^{71}-21q^{70}-120q^{69}-68q^{68}-47q^{67}-37q^{66}+88q^{65}+66q^{64}+87q^{63}+92q^{62}-60q^{61}-30q^{60}-55q^{59}-106q^{58}+8q^{57}-47q^{56}-9q^{55}+86q^{54}-21q^{53}+89q^{52}+111q^{51}+24q^{50}+113q^{49}-87q^{48}-194q^{47}-160q^{46}-282q^{45}-37q^{44}+189q^{43}+276q^{42}+518q^{41}+280q^{40}-60q^{39}-308q^{38}-739q^{37}-585q^{36}-217q^{35}+169q^{34}+857q^{33}+926q^{32}+616q^{31}+119q^{30}-826q^{29}-1166q^{28}-1039q^{27}-577q^{26}+584q^{25}+1262q^{24}+1432q^{23}+1110q^{22}-168q^{21}-1154q^{20}-1689q^{19}-1640q^{18}-385q^{17}+837q^{16}+1754q^{15}+2101q^{14}+1008q^{13}-356q^{12}-1635q^{11}-2413q^{10}-1596q^{9}-248q^{8}+1313q^{7}+2554q^{6}+2137q^{5}+897q^{4}-861q^{3}-2540q^{2}-2550q-1529+315q^{-1}+2369q^{-2}+2843q^{-3}+2130q^{-4}+271q^{-5}-2121q^{-6}-3031q^{-7}-2635q^{-8}-832q^{-9}+1789q^{-10}+3125q^{-11}+3072q^{-12}+1371q^{-13}-1466q^{-14}-3167q^{-15}-3425q^{-16}-1833q^{-17}+1156q^{-18}+3174q^{-19}+3717q^{-20}+2229q^{-21}-889q^{-22}-3187q^{-23}-3968q^{-24}-2553q^{-25}+684q^{-26}+3226q^{-27}+4189q^{-28}+2816q^{-29}-538q^{-30}-3284q^{-31}-4403q^{-32}-3058q^{-33}+419q^{-34}+3389q^{-35}+4638q^{-36}+3272q^{-37}-328q^{-38}-3462q^{-39}-4848q^{-40}-3538q^{-41}+154q^{-42}+3524q^{-43}+5082q^{-44}+3800q^{-45}+52q^{-46}-3449q^{-47}-5191q^{-48}-4102q^{-49}-422q^{-50}+3238q^{-51}+5228q^{-52}+4359q^{-53}+816q^{-54}-2844q^{-55}-5022q^{-56}-4488q^{-57}-1295q^{-58}+2267q^{-59}+4638q^{-60}+4470q^{-61}+1682q^{-62}-1637q^{-63}-4014q^{-64}-4185q^{-65}-1941q^{-66}+945q^{-67}+3259q^{-68}+3730q^{-69}+2007q^{-70}-391q^{-71}-2460q^{-72}-3070q^{-73}-1872q^{-74}-39q^{-75}+1705q^{-76}+2370q^{-77}+1579q^{-78}+275q^{-79}-1070q^{-80}-1693q^{-81}-1208q^{-82}-357q^{-83}+616q^{-84}+1104q^{-85}+816q^{-86}+339q^{-87}-303q^{-88}-659q^{-89}-498q^{-90}-258q^{-91}+136q^{-92}+358q^{-93}+242q^{-94}+156q^{-95}-44q^{-96}-161q^{-97}-78q^{-98}-92q^{-99}+13q^{-100}+72q^{-101}-12q^{-102}+26q^{-103}-14q^{-104}-15q^{-105}+62q^{-106}-2q^{-107}+6q^{-108}+3q^{-109}-59q^{-110}-11q^{-111}-22q^{-112}-6q^{-113}+65q^{-114}+20q^{-115}+12q^{-116}-q^{-117}-42q^{-118}-5q^{-119}-19q^{-120}-13q^{-121}+34q^{-122}+13q^{-123}+10q^{-124}+3q^{-125}-20q^{-126}-8q^{-128}-9q^{-129}+14q^{-130}+2q^{-131}+4q^{-132}+4q^{-133}-8q^{-134}+q^{-135}-3q^{-136}-2q^{-137}+5q^{-138}-q^{-139}+2q^{-141}-2q^{-142}-q^{-144}+2q^{-146}-q^{-147}$

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[9, 8]]
Out[2]=	PD[X[1, 4, 2, 5], X[3, 8, 4, 9], X[5, 14, 6, 15], X[9, 1, 10, 18], X[11, 17, 12, 16], X[15, 13, 16, 12], X[17, 11, 18, 10], X[13, 6, 14, 7], X[7, 2, 8, 3]]
In[3]:=	GaussCode[Knot[9, 8]]
Out[3]=	GaussCode[-1, 9, -2, 1, -3, 8, -9, 2, -4, 7, -5, 6, -8, 3, -6, 5, -7, 4]
In[4]:=	DTCode[Knot[9, 8]]
Out[4]=	DTCode[4, 8, 14, 2, 18, 16, 6, 12, 10]
In[5]:=	br = BR[Knot[9, 8]]
Out[5]=	BR[5, {-1, -1, 2, -1, 2, 3, -2, -4, 3, -4}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{5, 10}
In[7]:=	BraidIndex[Knot[9, 8]]
Out[7]=	5
In[8]:=	Show[DrawMorseLink[Knot[9, 8]]]
Out[8]=	-Graphics-
In[9]:=	(#[Knot[9, 8]]&) /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 2, 2, 2, {4, 6}, 1}
In[10]:=	alex = Alexander[Knot[9, 8]][t]
Out[10]=	2 8 2 -11 - -- + - + 8 t - 2 t 2 t t
In[11]:=	Conway[Knot[9, 8]][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[9, 8]], KnotSignature[Knot[9, 8]]}
Out[13]=	{31, -2}
In[14]:=	Jones[Knot[9, 8]][q]
Out[14]=	-6 2 3 5 5 5 2 3 -4 - q + -- - -- + -- - -- + - + 3 q - 2 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[9, 8], Knot[11, NonAlternating, 60]}
In[16]:=	A2Invariant[Knot[9, 8]][q]
Out[16]=	-20 -18 -16 -12 2 -6 -4 2 4 10 -q - q + q + q + --- + q - q - q + q + q 10 q
In[17]:=	HOMFLYPT[Knot[9, 8]][a, z]
Out[17]=	2 -2 4 6 2 z 2 2 4 2 4 2 4 -1 + a + 2 a - a - 2 z + -- - a z + 2 a z - z - a z 2 a
In[18]:=	Kauffman[Knot[9, 8]][a, z]
Out[18]=	2 -2 4 6 2 z 3 5 7 2 4 z -1 - a + 2 a + a - --- - 3 a z - a z - a z - a z + 7 z + ---- + a 2 a 3 2 2 4 2 6 2 8 z 3 3 3 7 3 2 a z - 3 a z - 2 a z + ---- + 11 a z + 2 a z + a z - a 4 5 4 4 z 2 4 6 4 8 z 5 3 5 6 z - ---- - 4 a z + 2 a z - ---- - 13 a z - 3 a z + 2 a a 6 7 5 5 6 z 4 6 2 z 7 3 7 8 2 8 2 a z - z + -- + 2 a z + ---- + 4 a z + 2 a z + z + a z 2 a a
In[19]:=	{Vassiliev[2][Knot[9, 8]], Vassiliev[3][Knot[9, 8]]}
Out[19]=	{0, -2}
In[20]:=	Kh[Knot[9, 8]][q, t]
Out[20]=	3 3 1 1 1 2 1 3 2 -- + - + ------ + ------ + ----- + ----- + ----- + ----- + ----- + 3 q 13 5 11 4 9 4 9 3 7 3 7 2 5 2 q q t q t q t q t q t q t q t 2 3 2 t 2 3 2 3 3 5 3 7 4 ---- + ---- + --- + 2 q t + q t + 2 q t + q t + q t + q t 5 3 q q t q t
In[21]:=	ColouredJones[Knot[9, 8], 2][q]
Out[21]=	-17 2 4 6 2 7 13 6 11 20 8 15 1 + q - --- + --- - --- + --- + --- - --- + -- + -- - -- + -- + -- - 16 14 13 12 11 10 9 8 7 6 5 q q q q q q q q q q q 22 5 17 19 2 3 4 5 6 -- + -- + -- - -- + 16 q - 13 q - 3 q + 12 q - 6 q - 4 q + 4 3 2 q q q q 7 8 9 10 6 q - q - 2 q + q

See/edit the Rolfsen_Splice_Template. Back to the top.

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