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 [math]\displaystyle{ 0 }[/math] Topological 4 genus [math]\displaystyle{ 0 }[/math] Concordance genus [math]\displaystyle{ 0 }[/math] Rasmussen s-Invariant 0

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

Polynomial invariants

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

Further Quantum Invariants

Further quantum knot invariants for 9_46.

A1 Invariants.

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

A2 Invariants.

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

A3 Invariants.

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

A4 Invariants.

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

B2 Invariants.

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

D4 Invariants.

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

G2 Invariants.

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

.

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]=

[math]\displaystyle{ -2 t+5-2 t^{-1} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ 1-2 z^2 }[/math]

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]=

[math]\displaystyle{ \{3,t+1\} }[/math]

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 9, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ 2- q^{-1} + q^{-2} -2 q^{-3} + q^{-4} - q^{-5} + q^{-6} }[/math]

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

[math]\displaystyle{ a^6-z^2 a^4-a^4-z^2 a^2-a^2+2 }[/math]

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ a^5 z^7+a^3 z^7+a^6 z^6+2 a^4 z^6+a^2 z^6-5 a^5 z^5-5 a^3 z^5-5 a^6 z^4-9 a^4 z^4-4 a^2 z^4+7 a^5 z^3+8 a^3 z^3+a z^3+6 a^6 z^2+9 a^4 z^2+3 a^2 z^2-4 a^5 z-6 a^3 z-2 a z-a^6-a^4+a^2+2 }[/math]

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

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 [math]\displaystyle{ t^rq^j }[/math] are shown, along with their alternating sums [math]\displaystyle{ \chi }[/math] (fixed [math]\displaystyle{ j }[/math], alternation over [math]\displaystyle{ r }[/math]). The squares with yellow highlighting are those on the "critical diagonals", where [math]\displaystyle{ j-2r=s+1 }[/math] or [math]\displaystyle{ j-2r=s+1 }[/math], where [math]\displaystyle{ s= }[/math]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

Computer Talk

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

[math]\displaystyle{ \textrm{Include}(\textrm{ColouredJonesM.mhtml}) }[/math]

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