9 46

9_45

9_47

Visit 9 46's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 46's page at Knotilus! Visit 9 46's page at the original Knot Atlas! 9_46 is also known as the pretzel knot P(3,3,-3).

9 46 Further Notes and Views

Knot presentations

Planar diagram presentation	X4251 X7,12,8,13 X10,3,11,4 X2,11,3,12 X5,14,6,15 X13,6,14,7 X15,18,16,1 X9,17,10,16 X17,9,18,8
Gauss code	1, -4, 3, -1, -5, 6, -2, 9, -8, -3, 4, 2, -6, 5, -7, 8, -9, 7
Dowker-Thistlethwaite code	4 10 -14 -12 -16 2 -6 -18 -8
Conway Notation	[3,3,21-]

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 1 Bridge index 3 Super bridge index 4 Nakanishi index 2 Maximal Thurston-Bennequin number [-7][-1] Hyperbolic Volume 4.7517 A-Polynomial See Data:9 46/A-polynomial

[edit Notes for 9 46's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus $0$ Topological 4 genus $0$ Concordance genus $0$ Rasmussen s-Invariant 0

[edit Notes for 9 46's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-2t+5-2t^{-1}$
Conway polynomial	$1-2z^{2}$
2nd Alexander ideal (db, data sources)	$\{3,t+1\}$
Determinant and Signature	{ 9, 0 }
Jones polynomial	$2-q^{-1}+q^{-2}-2q^{-3}+q^{-4}-q^{-5}+q^{-6}$
HOMFLY-PT polynomial (db, data sources)	$a^{6}-z^{2}a^{4}-a^{4}-z^{2}a^{2}-a^{2}+2$
Kauffman polynomial (db, data sources)	$a^{5}z^{7}+a^{3}z^{7}+a^{6}z^{6}+2a^{4}z^{6}+a^{2}z^{6}-5a^{5}z^{5}-5a^{3}z^{5}-5a^{6}z^{4}-9a^{4}z^{4}-4a^{2}z^{4}+7a^{5}z^{3}+8a^{3}z^{3}+az^{3}+6a^{6}z^{2}+9a^{4}z^{2}+3a^{2}z^{2}-4a^{5}z-6a^{3}z-2az-a^{6}-a^{4}+a^{2}+2$
The A2 invariant	$q^{20}+q^{18}-q^{12}-q^{10}-q^{8}-q^{6}+q^{2}+2+2q^{-2}$
The G2 invariant	$q^{94}+q^{90}-q^{88}-q^{82}+2q^{80}+2q^{70}+q^{68}-3q^{66}+q^{62}+2q^{60}+2q^{58}-4q^{56}+3q^{52}+2q^{50}-2q^{48}-5q^{46}+3q^{42}+2q^{40}-4q^{38}-3q^{36}+3q^{32}-5q^{28}-4q^{26}+2q^{24}+2q^{22}-2q^{20}-q^{18}-3q^{16}+3q^{14}+2q^{12}-q^{10}+2q^{4}+3q^{2}+1+q^{-2}+q^{-4}+3q^{-8}+q^{-10}-q^{-12}+q^{-14}$

Further Quantum Invariants

Further quantum knot invariants for 9_46.

A1 Invariants.

Weight	Invariant
1	$q^{13}-q^{7}-q^{5}+q+2q^{-1}$
2	$q^{38}-q^{34}-q^{28}-q^{26}+q^{22}+q^{20}+q^{18}+2q^{16}-2q^{8}-2q^{6}-q^{4}+q^{2}+1+q^{-2}+q^{-4}+q^{-6}$
3	$q^{75}-q^{71}-q^{69}+q^{65}-q^{61}-q^{59}+q^{55}+2q^{53}+q^{51}+q^{45}+q^{43}-q^{41}-3q^{39}-q^{37}-q^{33}-2q^{31}-q^{29}+q^{27}+q^{25}+q^{23}+2q^{21}+3q^{19}+q^{17}+q^{15}-q^{9}-q^{7}-3q^{5}-2q^{3}+q+3q^{-1}+q^{-3}-3q^{-5}+2q^{-9}+2q^{-11}$
4	$q^{124}-q^{120}-q^{118}-q^{116}+q^{114}+q^{112}+q^{110}-2q^{106}-q^{104}+q^{100}+2q^{98}+2q^{96}+q^{94}-q^{92}-2q^{90}-q^{88}+q^{86}+q^{84}+q^{82}-2q^{80}-4q^{78}-3q^{76}+3q^{72}+q^{70}-2q^{66}-q^{64}+3q^{62}+4q^{60}+3q^{58}-2q^{54}+q^{52}+2q^{50}+2q^{48}-q^{46}-4q^{44}-2q^{42}-q^{40}-q^{38}-2q^{36}-3q^{34}-q^{32}+q^{30}+q^{28}+3q^{22}+3q^{20}+3q^{18}+2q^{16}-2q^{12}-q^{10}+q^{8}+2q^{6}+q^{4}-4q^{2}-3-q^{-2}+4q^{-4}+4q^{-6}-2q^{-8}-3q^{-10}-3q^{-12}+q^{-14}+3q^{-16}+q^{-18}+q^{-20}$
5	$q^{185}-q^{181}-q^{179}-q^{177}+q^{173}+2q^{171}+q^{169}-q^{165}-2q^{163}-2q^{161}+2q^{157}+2q^{155}+2q^{153}+2q^{151}-2q^{147}-3q^{145}-3q^{143}-q^{141}+2q^{139}+3q^{137}+2q^{135}-3q^{131}-5q^{129}-4q^{127}-q^{125}+4q^{123}+6q^{121}+4q^{119}+q^{117}-3q^{115}-4q^{113}-2q^{111}+3q^{109}+6q^{107}+6q^{105}+2q^{103}-3q^{101}-7q^{99}-6q^{97}+4q^{93}+4q^{91}+q^{89}-5q^{87}-8q^{85}-5q^{83}+q^{81}+4q^{79}+4q^{77}-4q^{73}-3q^{71}+q^{69}+6q^{67}+7q^{65}+3q^{63}-2q^{61}-2q^{59}+q^{57}+4q^{55}+3q^{53}-3q^{49}-2q^{47}-q^{45}-3q^{41}-4q^{39}-3q^{37}-q^{35}+q^{33}+q^{31}-q^{27}-q^{25}-q^{23}+q^{21}+3q^{19}+5q^{17}+5q^{15}+3q^{13}-4q^{11}-6q^{9}-5q^{7}+q^{5}+7q^{3}+10q+5q^{-1}-5q^{-3}-10q^{-5}-7q^{-7}+q^{-9}+7q^{-11}+7q^{-13}+q^{-15}-5q^{-17}-5q^{-19}-2q^{-21}+2q^{-25}+2q^{-27}+2q^{-29}$

A2 Invariants.

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

A3 Invariants.

Weight	Invariant
0,1,0	$q^{40}+q^{36}-q^{32}-2q^{30}-2q^{28}+q^{24}+4q^{22}+4q^{20}+4q^{18}-2q^{14}-5q^{12}-6q^{10}-4q^{8}-2q^{6}+2q^{4}+3q^{2}+5+3q^{-2}+2q^{-4}$
1,0,0	$q^{27}+q^{25}+q^{23}-q^{17}-q^{15}-q^{13}-q^{11}-q^{9}-q^{7}+q^{3}+2q+2q^{-1}+2q^{-3}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{54}+q^{52}+q^{50}+q^{48}+q^{46}-q^{44}-2q^{42}-3q^{40}-4q^{38}-4q^{36}-2q^{34}+2q^{32}+4q^{30}+7q^{28}+9q^{26}+8q^{24}+4q^{22}+q^{20}-4q^{18}-8q^{16}-10q^{14}-9q^{12}-7q^{10}-4q^{8}+q^{6}+4q^{4}+6q^{2}+6+6q^{-2}+3q^{-4}+2q^{-6}$
1,0,0,0	$q^{34}+q^{32}+q^{30}+q^{28}-q^{22}-q^{20}-q^{18}-q^{16}-q^{14}-q^{12}-q^{10}-q^{8}+q^{4}+2q^{2}+2+2q^{-2}+2q^{-4}$

B2 Invariants.

Weight	Invariant
0,1	$q^{40}+q^{36}+q^{32}-q^{24}-2q^{20}-2q^{16}-q^{12}+2q^{4}+q^{2}+1+q^{-2}+2q^{-4}$
1,0	$q^{66}+q^{58}-q^{54}-q^{52}-q^{48}-2q^{46}-q^{44}+q^{40}+q^{38}+2q^{36}+2q^{34}+3q^{32}+2q^{30}+2q^{28}-q^{22}-2q^{20}-4q^{18}-3q^{16}-2q^{14}-2q^{12}-2q^{10}-q^{8}+2q^{6}+q^{4}+2q^{2}+2+3q^{-2}+q^{-4}+2q^{-6}$

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{54}+q^{50}+q^{46}-q^{44}-q^{42}-q^{40}-q^{38}+2q^{32}+2q^{30}+4q^{28}+2q^{26}+2q^{24}-q^{22}-q^{20}-4q^{18}-4q^{16}-5q^{14}-4q^{12}-2q^{10}-q^{8}+2q^{6}+2q^{4}+4q^{2}+4+3q^{-2}+2q^{-4}+2q^{-6}$

G2 Invariants.

Weight	Invariant
1,0	$q^{94}+q^{90}-q^{88}-q^{82}+2q^{80}+2q^{70}+q^{68}-3q^{66}+q^{62}+2q^{60}+2q^{58}-4q^{56}+3q^{52}+2q^{50}-2q^{48}-5q^{46}+3q^{42}+2q^{40}-4q^{38}-3q^{36}+3q^{32}-5q^{28}-4q^{26}+2q^{24}+2q^{22}-2q^{20}-q^{18}-3q^{16}+3q^{14}+2q^{12}-q^{10}+2q^{4}+3q^{2}+1+q^{-2}+q^{-4}+3q^{-8}+q^{-10}-q^{-12}+q^{-14}$

.

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 46"];

In[4]:=

Alexander[K][t]

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

Out[4]=

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

In[5]:=

Conway[K][z]

Out[5]=

$1-2z^{2}$

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]=

$\{3,t+1\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 9, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$a^{5}z^{7}+a^{3}z^{7}+a^{6}z^{6}+2a^{4}z^{6}+a^{2}z^{6}-5a^{5}z^{5}-5a^{3}z^{5}-5a^{6}z^{4}-9a^{4}z^{4}-4a^{2}z^{4}+7a^{5}z^{3}+8a^{3}z^{3}+az^{3}+6a^{6}z^{2}+9a^{4}z^{2}+3a^{2}z^{2}-4a^{5}z-6a^{3}z-2az-a^{6}-a^{4}+a^{2}+2$

Vassiliev invariants

V2 and V3:

(-2, 3)

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 $-8$ $24$ $32$ ${\frac {20}{3}}$ ${\frac {52}{3}}$ $-192$ $-272$ $-64$ $-72$ $-{\frac {256}{3}}$ $288$ $-{\frac {160}{3}}$ $-{\frac {416}{3}}$ ${\frac {10409}{15}}$ ${\frac {244}{15}}$ ${\frac {8636}{45}}$ ${\frac {775}{9}}$ $-{\frac {151}{15}}$

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

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

Integral Khovanov Homology (db, data source)

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

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, 46]]
Out[2]=	9
In[3]:=	PD[Knot[9, 46]]
Out[3]=	PD[X[4, 2, 5, 1], X[7, 12, 8, 13], X[10, 3, 11, 4], X[2, 11, 3, 12], X[5, 14, 6, 15], X[13, 6, 14, 7], X[15, 18, 16, 1], X[9, 17, 10, 16], X[17, 9, 18, 8]]
In[4]:=	GaussCode[Knot[9, 46]]
Out[4]=	GaussCode[1, -4, 3, -1, -5, 6, -2, 9, -8, -3, 4, 2, -6, 5, -7, 8, -9, 7]
In[5]:=	BR[Knot[9, 46]]
Out[5]=	BR[4, {-1, 2, -1, 2, -3, -2, 1, -2, -3}]
In[6]:=	alex = Alexander[Knot[9, 46]][t]
Out[6]=	2 5 - - - 2 t t
In[7]:=	Conway[Knot[9, 46]][z]
Out[7]=	2 1 - 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[6, 1], Knot[9, 46], Knot[11, NonAlternating, 67], Knot[11, NonAlternating, 97], Knot[11, NonAlternating, 139]}
In[9]:=	{KnotDet[Knot[9, 46]], KnotSignature[Knot[9, 46]]}
Out[9]=	{9, 0}
In[10]:=	J=Jones[Knot[9, 46]][q]
Out[10]=	-6 -5 -4 2 -2 1 2 + q - q + q - -- + q - - 3 q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[9, 46]}
In[12]:=	A2Invariant[Knot[9, 46]][q]
Out[12]=	-20 -18 -12 -10 -8 -6 -2 2 2 + q + q - q - q - q - q + q + 2 q
In[13]:=	Kauffman[Knot[9, 46]][a, z]
Out[13]=	2 4 6 3 5 2 2 4 2 2 + a - a - a - 2 a z - 6 a z - 4 a z + 3 a z + 9 a z + 6 2 3 3 3 5 3 2 4 4 4 6 4 6 a z + a z + 8 a z + 7 a z - 4 a z - 9 a z - 5 a z - 3 5 5 5 2 6 4 6 6 6 3 7 5 7 5 a z - 5 a z + a z + 2 a z + a z + a z + a z
In[14]:=	{Vassiliev[2][Knot[9, 46]], Vassiliev[3][Knot[9, 46]]}
Out[14]=	{0, 3}
In[15]:=	Kh[Knot[9, 46]][q, t]
Out[15]=	1 1 1 1 1 1 1 1 - + 2 q + ------ + ----- + ----- + ----- + ----- + ----- + ---- q 13 6 9 5 9 4 7 3 5 3 3 2 3 q t q t q t q t q t q t q t