9 4

9_3

9_5

Visit 9 4's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 4's page at Knotilus! Visit 9 4's page at the original Knot Atlas! 9 4 Quick Notes

9 4 Further Notes and Views

Knot presentations

Planar diagram presentation	X1627 X3,12,4,13 X7,18,8,1 X9,16,10,17 X15,10,16,11 X17,8,18,9 X5,14,6,15 X11,2,12,3 X13,4,14,5
Gauss code	-1, 8, -2, 9, -7, 1, -3, 6, -4, 5, -8, 2, -9, 7, -5, 4, -6, 3
Dowker-Thistlethwaite code	6 12 14 18 16 2 4 10 8
Conway Notation	[54]

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 2 Bridge index 2 Super bridge index $\{4,7\}$ Nakanishi index 1 Maximal Thurston-Bennequin number [-14][3] Hyperbolic Volume 5.55652 A-Polynomial See Data:9 4/A-polynomial

[edit Notes for 9 4's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for 9 4's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$3t^{2}-5t+5-5t^{-1}+3t^{-2}$
Conway polynomial	$3z^{4}+7z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 21, -4 }
Jones polynomial	$q^{-2}-q^{-3}+2q^{-4}-3q^{-5}+4q^{-6}-3q^{-7}+3q^{-8}-2q^{-9}+q^{-10}-q^{-11}$
HOMFLY-PT polynomial (db, data sources)	$-z^{2}a^{10}-2a^{10}+z^{4}a^{8}+3z^{2}a^{8}+2a^{8}+z^{4}a^{6}+2z^{2}a^{6}+z^{4}a^{4}+3z^{2}a^{4}+a^{4}$
Kauffman polynomial (db, data sources)	$z^{5}a^{13}-4z^{3}a^{13}+3za^{13}+z^{6}a^{12}-3z^{4}a^{12}+z^{2}a^{12}+z^{7}a^{11}-3z^{5}a^{11}+2z^{3}a^{11}-za^{11}+z^{8}a^{10}-5z^{6}a^{10}+11z^{4}a^{10}-10z^{2}a^{10}+2a^{10}+2z^{7}a^{9}-8z^{5}a^{9}+12z^{3}a^{9}-4za^{9}+z^{8}a^{8}-5z^{6}a^{8}+11z^{4}a^{8}-7z^{2}a^{8}+2a^{8}+z^{7}a^{7}-3z^{5}a^{7}+4z^{3}a^{7}+z^{6}a^{6}-2z^{4}a^{6}+z^{2}a^{6}+z^{5}a^{5}-2z^{3}a^{5}+z^{4}a^{4}-3z^{2}a^{4}+a^{4}$
The A2 invariant	$-q^{34}-q^{32}-q^{30}-q^{28}+q^{26}+q^{24}+q^{22}+q^{20}+q^{16}+q^{10}+q^{6}$
The G2 invariant	$q^{176}+q^{172}-q^{170}+q^{168}-q^{164}+2q^{162}-2q^{160}+2q^{158}-2q^{156}+q^{152}-2q^{150}+3q^{148}-5q^{146}+2q^{144}-q^{142}-3q^{140}+2q^{138}-5q^{136}+q^{134}+q^{132}-3q^{130}-3q^{126}-q^{124}+3q^{122}-5q^{120}+2q^{118}-q^{116}-q^{114}+5q^{112}-3q^{110}+4q^{108}-q^{106}+3q^{104}+2q^{102}-3q^{100}+6q^{98}-3q^{96}+4q^{94}+q^{92}-2q^{90}+3q^{88}-q^{86}+q^{82}-3q^{80}+q^{78}-2q^{74}+4q^{72}-3q^{70}+2q^{68}-q^{64}+q^{62}-2q^{60}+3q^{58}-q^{56}+q^{54}+2q^{48}-q^{46}+2q^{44}-q^{42}+q^{40}+q^{38}-q^{36}+q^{34}+q^{30}$

Further Quantum Invariants

Further quantum knot invariants for 9_4.

A1 Invariants.

Weight	Invariant
1	$-q^{23}-q^{19}+q^{17}+q^{13}+q^{11}-q^{9}+q^{7}+q^{3}$
2	$q^{64}+q^{58}-q^{56}-2q^{54}+q^{52}-2q^{48}+q^{46}+q^{44}-2q^{42}+q^{38}-q^{36}+q^{30}-2q^{28}+3q^{24}-q^{22}+2q^{18}+q^{12}+q^{6}$
3	$-q^{123}+q^{113}+q^{111}-q^{107}+q^{105}+2q^{103}-3q^{99}-2q^{97}+2q^{95}+2q^{93}-3q^{89}+3q^{85}+2q^{83}-3q^{81}-2q^{79}+2q^{77}+3q^{75}-2q^{73}-2q^{71}-2q^{65}-q^{63}-q^{61}+2q^{57}-2q^{55}-2q^{53}+q^{51}+4q^{49}-4q^{45}+3q^{41}+2q^{39}-q^{37}-q^{35}+q^{31}+q^{29}+q^{21}+q^{19}+q^{17}+q^{9}$
4	$q^{200}-q^{192}-q^{188}+q^{184}-q^{182}-2q^{178}-q^{176}+3q^{174}+2q^{172}+3q^{170}-2q^{168}-4q^{166}-q^{164}+q^{162}+6q^{160}+q^{158}-3q^{156}-4q^{154}-5q^{152}+3q^{150}+4q^{148}+4q^{146}-8q^{142}-3q^{140}+2q^{138}+7q^{136}+6q^{134}-5q^{132}-6q^{130}-2q^{128}+7q^{126}+8q^{124}-2q^{122}-5q^{120}-3q^{118}+3q^{116}+4q^{114}-q^{112}-2q^{110}-2q^{108}-2q^{102}-q^{100}-q^{98}-q^{96}+q^{94}+4q^{92}-q^{90}-3q^{88}-4q^{86}-q^{84}+8q^{82}+2q^{80}-3q^{78}-9q^{76}-5q^{74}+9q^{72}+6q^{70}+2q^{68}-6q^{66}-8q^{64}+q^{62}+4q^{60}+6q^{58}+q^{56}-5q^{54}-2q^{52}-q^{50}+4q^{48}+4q^{46}-q^{42}-3q^{40}+q^{38}+2q^{36}+q^{34}+q^{32}-2q^{30}+q^{26}+q^{24}+2q^{22}+q^{12}$
5	$-q^{295}+q^{287}+q^{285}-q^{277}+2q^{273}+q^{271}-q^{267}-3q^{265}-3q^{263}+3q^{259}+3q^{257}+2q^{255}-2q^{253}-5q^{251}-5q^{249}+5q^{245}+8q^{243}+5q^{241}-q^{239}-6q^{237}-8q^{235}-4q^{233}+3q^{231}+7q^{229}+8q^{227}+3q^{225}-3q^{223}-10q^{221}-11q^{219}-3q^{217}+6q^{215}+13q^{213}+11q^{211}-q^{209}-15q^{207}-16q^{205}-6q^{203}+8q^{201}+19q^{199}+14q^{197}-4q^{195}-18q^{193}-16q^{191}-q^{189}+16q^{187}+19q^{185}+4q^{183}-11q^{181}-16q^{179}-6q^{177}+10q^{175}+12q^{173}+5q^{171}-3q^{169}-8q^{167}-4q^{165}+3q^{163}+5q^{161}+q^{159}-q^{157}-q^{155}-q^{153}+q^{151}+q^{149}-q^{147}-q^{143}-q^{141}-q^{139}+2q^{137}+6q^{135}+2q^{133}-2q^{131}-7q^{129}-11q^{127}-2q^{125}+11q^{123}+14q^{121}+3q^{119}-12q^{117}-22q^{115}-11q^{113}+10q^{111}+25q^{109}+16q^{107}-7q^{105}-21q^{103}-20q^{101}-3q^{99}+16q^{97}+20q^{95}+8q^{93}-8q^{91}-16q^{89}-12q^{87}+10q^{83}+11q^{81}+4q^{79}-4q^{77}-7q^{75}-5q^{73}+4q^{69}+5q^{67}+2q^{65}-q^{63}-3q^{61}-2q^{59}+q^{57}+2q^{55}+3q^{53}+q^{51}-2q^{49}-q^{47}+q^{43}+2q^{41}+2q^{39}-q^{37}-q^{35}+q^{29}+2q^{27}+q^{25}+q^{15}$

A2 Invariants.

Weight	Invariant
1,0	$-q^{34}-q^{32}-q^{30}-q^{28}+q^{26}+q^{24}+q^{22}+q^{20}+q^{16}+q^{10}+q^{6}$
1,1	$q^{92}+2q^{88}-2q^{86}+4q^{84}-4q^{82}+4q^{80}-8q^{78}+5q^{76}-8q^{74}+8q^{72}-6q^{70}+3q^{68}+2q^{66}-2q^{64}+8q^{62}-12q^{60}+12q^{58}-14q^{56}+10q^{54}-15q^{52}+10q^{50}-10q^{48}+6q^{46}-3q^{44}-2q^{42}+2q^{40}-4q^{38}+7q^{36}-4q^{34}+6q^{32}+7q^{28}-2q^{26}+4q^{24}-2q^{22}+4q^{20}-2q^{18}+2q^{16}+q^{12}$
2,0	$q^{86}+q^{84}+2q^{82}+q^{80}+q^{78}-q^{76}-2q^{74}-2q^{72}-2q^{70}-2q^{68}-2q^{66}+q^{62}+q^{60}-q^{58}-q^{52}-q^{50}-q^{44}-q^{42}+q^{40}+q^{36}+3q^{34}+2q^{32}+q^{30}+2q^{28}+2q^{26}-q^{22}+q^{20}+q^{18}+q^{12}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{74}+q^{70}+q^{68}-3q^{60}-q^{58}-q^{56}-4q^{54}-q^{52}-q^{48}-q^{46}+q^{44}+q^{40}+q^{38}+3q^{36}+q^{34}+3q^{30}-q^{26}+2q^{24}+q^{22}+q^{18}+q^{16}+q^{12}$
1,0,0	$-q^{45}-q^{43}-2q^{41}-q^{39}-q^{37}+q^{35}+q^{33}+2q^{31}+q^{29}+q^{27}+q^{21}+q^{17}+q^{13}+q^{9}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{96}+q^{94}+q^{92}+2q^{90}+3q^{88}+q^{86}+q^{84}+q^{82}-q^{80}-5q^{78}-6q^{76}-5q^{74}-6q^{72}-6q^{70}-2q^{68}+2q^{62}+2q^{60}+q^{58}+2q^{54}+2q^{52}+2q^{50}+2q^{48}+4q^{46}+2q^{44}+q^{40}+q^{38}+q^{36}+q^{32}+2q^{30}+q^{28}+q^{26}+q^{24}+q^{22}+q^{18}$
1,0,0,0	$-q^{56}-q^{54}-2q^{52}-2q^{50}-q^{48}-q^{46}+q^{44}+q^{42}+2q^{40}+2q^{38}+q^{36}+q^{34}+q^{26}+q^{22}+q^{20}+q^{16}+q^{12}$

B2 Invariants.

Weight	Invariant
0,1	$-q^{74}-q^{70}+q^{68}-2q^{66}+2q^{64}-2q^{62}+q^{60}-q^{58}+q^{56}-q^{52}+2q^{50}-3q^{48}+3q^{46}-3q^{44}+4q^{42}-3q^{40}+3q^{38}-q^{36}+q^{34}-q^{30}+2q^{28}-q^{26}+2q^{24}-q^{22}+2q^{20}-q^{18}+q^{16}+q^{12}$
1,0	$q^{120}+q^{112}+q^{110}-q^{106}+q^{102}+q^{100}-2q^{98}-3q^{96}-q^{94}+q^{92}-3q^{88}-2q^{86}+q^{82}-q^{78}+q^{74}-q^{70}-q^{68}+q^{66}+q^{64}-q^{60}+q^{58}+2q^{56}+q^{54}-q^{52}+3q^{48}+2q^{46}-q^{44}-q^{42}+2q^{38}+q^{36}-q^{32}+q^{28}+q^{26}+q^{18}$

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{102}+q^{98}+2q^{94}-q^{92}+q^{90}-2q^{88}+q^{86}-2q^{84}-2q^{80}-q^{78}-2q^{76}-3q^{74}-q^{72}-3q^{70}-4q^{66}+2q^{64}-2q^{62}+4q^{60}-q^{58}+4q^{56}+4q^{52}+q^{50}+2q^{48}+2q^{42}-q^{40}+q^{38}-q^{36}+2q^{34}+2q^{30}+2q^{26}+q^{22}+q^{18}$

G2 Invariants.

Weight

Invariant

1,0

$q^{176}+q^{172}-q^{170}+q^{168}-q^{164}+2q^{162}-2q^{160}+2q^{158}-2q^{156}+q^{152}-2q^{150}+3q^{148}-5q^{146}+2q^{144}-q^{142}-3q^{140}+2q^{138}-5q^{136}+q^{134}+q^{132}-3q^{130}-3q^{126}-q^{124}+3q^{122}-5q^{120}+2q^{118}-q^{116}-q^{114}+5q^{112}-3q^{110}+4q^{108}-q^{106}+3q^{104}+2q^{102}-3q^{100}+6q^{98}-3q^{96}+4q^{94}+q^{92}-2q^{90}+3q^{88}-q^{86}+q^{82}-3q^{80}+q^{78}-2q^{74}+4q^{72}-3q^{70}+2q^{68}-q^{64}+q^{62}-2q^{60}+3q^{58}-q^{56}+q^{54}+2q^{48}-q^{46}+2q^{44}-q^{42}+q^{40}+q^{38}-q^{36}+q^{34}+q^{30}$

.

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 4"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$3t^{2}-5t+5-5t^{-1}+3t^{-2}$

In[5]:=

Conway[K][z]

Out[5]=

$3z^{4}+7z^{2}+1$

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]=

{ 21, -4 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^{-2}-q^{-3}+2q^{-4}-3q^{-5}+4q^{-6}-3q^{-7}+3q^{-8}-2q^{-9}+q^{-10}-q^{-11}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-z^{2}a^{10}-2a^{10}+z^{4}a^{8}+3z^{2}a^{8}+2a^{8}+z^{4}a^{6}+2z^{2}a^{6}+z^{4}a^{4}+3z^{2}a^{4}+a^{4}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{5}a^{13}-4z^{3}a^{13}+3za^{13}+z^{6}a^{12}-3z^{4}a^{12}+z^{2}a^{12}+z^{7}a^{11}-3z^{5}a^{11}+2z^{3}a^{11}-za^{11}+z^{8}a^{10}-5z^{6}a^{10}+11z^{4}a^{10}-10z^{2}a^{10}+2a^{10}+2z^{7}a^{9}-8z^{5}a^{9}+12z^{3}a^{9}-4za^{9}+z^{8}a^{8}-5z^{6}a^{8}+11z^{4}a^{8}-7z^{2}a^{8}+2a^{8}+z^{7}a^{7}-3z^{5}a^{7}+4z^{3}a^{7}+z^{6}a^{6}-2z^{4}a^{6}+z^{2}a^{6}+z^{5}a^{5}-2z^{3}a^{5}+z^{4}a^{4}-3z^{2}a^{4}+a^{4}$

Vassiliev invariants

V2 and V3:

(7, -19)

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 $28$ $-152$ $392$ ${\frac {3122}{3}}$ ${\frac {502}{3}}$ $-4256$ $-{\frac {23696}{3}}$ $-{\frac {4160}{3}}$ $-1144$ ${\frac {10976}{3}}$ $11552$ ${\frac {87416}{3}}$ ${\frac {14056}{3}}$ ${\frac {1820137}{30}}$ ${\frac {7526}{15}}$ ${\frac {1136714}{45}}$ ${\frac {9335}{18}}$ ${\frac {101737}{30}}$

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 $s=$-4 is the signature of 9 4. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -3                   1 1 -5                 1 1 0 -7               1     1 -9             2 1     -1 -11           2 1       1 -13         1 2         1 -15       2 2           0 -17       1             1 -19   1 2               -1 -21                     0 -23 1                   -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-5$ $i=-3$ $r=-9$ ${\mathbb {Z} }$ $r=-8$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-7$ ${\mathbb {Z} }^{2}$ $r=-6$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-5$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-3$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-2$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-1$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=0$ ${\mathbb {Z} }$ ${\mathbb {Z} }$

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, 4]]
Out[2]=	9
In[3]:=	PD[Knot[9, 4]]
Out[3]=	PD[X[1, 6, 2, 7], X[3, 12, 4, 13], X[7, 18, 8, 1], X[9, 16, 10, 17], X[15, 10, 16, 11], X[17, 8, 18, 9], X[5, 14, 6, 15], X[11, 2, 12, 3], X[13, 4, 14, 5]]
In[4]:=	GaussCode[Knot[9, 4]]
Out[4]=	GaussCode[-1, 8, -2, 9, -7, 1, -3, 6, -4, 5, -8, 2, -9, 7, -5, 4, -6, 3]
In[5]:=	BR[Knot[9, 4]]
Out[5]=	BR[4, {-1, -1, -1, -1, -1, -2, 1, -2, -3, 2, -3}]
In[6]:=	alex = Alexander[Knot[9, 4]][t]
Out[6]=	3 5 2 5 + -- - - - 5 t + 3 t 2 t t
In[7]:=	Conway[Knot[9, 4]][z]
Out[7]=	2 4 1 + 7 z + 3 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[9, 4]}
In[9]:=	{KnotDet[Knot[9, 4]], KnotSignature[Knot[9, 4]]}
Out[9]=	{21, -4}
In[10]:=	J=Jones[Knot[9, 4]][q]
Out[10]=	-11 -10 2 3 3 4 3 2 -3 -2 -q + q - -- + -- - -- + -- - -- + -- - q + q 9 8 7 6 5 4 q q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[9, 4]}
In[12]:=	A2Invariant[Knot[9, 4]][q]
Out[12]=	-34 -32 -30 -28 -26 -24 -22 -20 -16 -10 -q - q - q - q + q + q + q + q + q + q + -6 q
In[13]:=	Kauffman[Knot[9, 4]][a, z]
Out[13]=	4 8 10 9 11 13 4 2 6 2 a + 2 a + 2 a - 4 a z - a z + 3 a z - 3 a z + a z - 8 2 10 2 12 2 5 3 7 3 9 3 7 a z - 10 a z + a z - 2 a z + 4 a z + 12 a z + 11 3 13 3 4 4 6 4 8 4 10 4 2 a z - 4 a z + a z - 2 a z + 11 a z + 11 a z - 12 4 5 5 7 5 9 5 11 5 13 5 6 6 3 a z + a z - 3 a z - 8 a z - 3 a z + a z + a z - 8 6 10 6 12 6 7 7 9 7 11 7 8 8 5 a z - 5 a z + a z + a z + 2 a z + a z + a z + 10 8 a z
In[14]:=	{Vassiliev[2][Knot[9, 4]], Vassiliev[3][Knot[9, 4]]}
Out[14]=	{0, -19}
In[15]:=	Kh[Knot[9, 4]][q, t]
Out[15]=	-5 -3 1 1 2 1 2 2 q + q + ------ + ------ + ------ + ------ + ------ + ------ + 23 9 19 8 19 7 17 6 15 6 15 5 q t q t q t q t q t q t 1 2 2 1 2 1 1 1 ------ + ------ + ------ + ------ + ----- + ----- + ----- + ---- 13 5 13 4 11 4 11 3 9 3 9 2 7 2 5 q t q t q t q t q t q t q t q t