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]

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]=

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

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)

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

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

${\textrm {Include}}({\textrm {ColouredJonesM.mhtml}})$

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 17, 2005, 14:44:34)...
In[2]:=	Crossings[Knot[9, 8]]
Out[2]=	9
In[3]:=	PD[Knot[9, 8]]
Out[3]=	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[4]:=	GaussCode[Knot[9, 8]]
Out[4]=	GaussCode[-1, 9, -2, 1, -3, 8, -9, 2, -4, 7, -5, 6, -8, 3, -6, 5, -7, 4]
In[5]:=	BR[Knot[9, 8]]
Out[5]=	BR[5, {-1, -1, 2, -1, 2, 3, -2, -4, 3, -4}]
In[6]:=	alex = Alexander[Knot[9, 8]][t]
Out[6]=	2 8 2 -11 - -- + - + 8 t - 2 t 2 t t
In[7]:=	Conway[Knot[9, 8]][z]
Out[7]=	4 1 - 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[8, 14], Knot[9, 8], Knot[10, 131]}
In[9]:=	{KnotDet[Knot[9, 8]], KnotSignature[Knot[9, 8]]}
Out[9]=	{31, -2}
In[10]:=	J=Jones[Knot[9, 8]][q]
Out[10]=	-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[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[9, 8], Knot[11, NonAlternating, 60]}
In[12]:=	A2Invariant[Knot[9, 8]][q]
Out[12]=	-20 -18 -16 -12 2 -6 -4 2 4 10 -q - q + q + q + --- + q - q - q + q + q 10 q
In[13]:=	Kauffman[Knot[9, 8]][a, z]
Out[13]=	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[14]:=	{Vassiliev[2][Knot[9, 8]], Vassiliev[3][Knot[9, 8]]}
Out[14]=	{0, -2}
In[15]:=	Kh[Knot[9, 8]][q, t]
Out[15]=	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