9 2

9_1

9_3

Visit 9 2's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 2's page at Knotilus! Visit 9 2's page at the original Knot Atlas!

9 2 Quick Notes

9 2 Further Notes and Views

Knot presentations

Planar diagram presentation	X1425 X3,12,4,13 X5,18,6,1 X7,16,8,17 X9,14,10,15 X13,10,14,11 X15,8,16,9 X17,6,18,7 X11,2,12,3
Gauss code	-1, 9, -2, 1, -3, 8, -4, 7, -5, 6, -9, 2, -6, 5, -7, 4, -8, 3
Dowker-Thistlethwaite code	4 12 18 16 14 2 10 8 6
Conway Notation	[72]

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 1 Bridge index 2 Super bridge index [math]\displaystyle{ \{4,7\} }[/math] Nakanishi index 1 Maximal Thurston-Bennequin number [-12][1] Hyperbolic Volume 3.48666 A-Polynomial See Data:9 2/A-polynomial

[edit Notes for 9 2's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus [math]\displaystyle{ 1 }[/math] Topological 4 genus [math]\displaystyle{ 1 }[/math] Concordance genus [math]\displaystyle{ 1 }[/math] Rasmussen s-Invariant -2

[edit Notes for 9 2's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ 4 t-7+4 t^{-1} }[/math]
Conway polynomial	[math]\displaystyle{ 4 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 15, -2 }
Jones polynomial	[math]\displaystyle{ q^{-1} - q^{-2} +2 q^{-3} -2 q^{-4} +2 q^{-5} -2 q^{-6} +2 q^{-7} - q^{-8} + q^{-9} - q^{-10} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ -a^{10}+z^2 a^8+a^8+z^2 a^6+z^2 a^4+z^2 a^2+a^2 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ z^7 a^{11}-6 z^5 a^{11}+10 z^3 a^{11}-4 z a^{11}+z^8 a^{10}-6 z^6 a^{10}+11 z^4 a^{10}-7 z^2 a^{10}+a^{10}+2 z^7 a^9-10 z^5 a^9+13 z^3 a^9-4 z a^9+z^8 a^8-5 z^6 a^8+8 z^4 a^8-6 z^2 a^8+a^8+z^7 a^7-3 z^5 a^7+z^3 a^7+z^6 a^6-2 z^4 a^6+z^5 a^5-z^3 a^5+z^4 a^4+z^3 a^3+z^2 a^2-a^2 }[/math]
The A2 invariant	[math]\displaystyle{ -q^{32}-q^{30}+q^{24}+q^{22}+q^8+q^6+q^2 }[/math]
The G2 invariant	[math]\displaystyle{ q^{156}+q^{152}-q^{150}+q^{142}-2 q^{140}+q^{138}-q^{136}-q^{134}-2 q^{130}-q^{128}-q^{126}-q^{124}-q^{118}+q^{112}-q^{108}+q^{106}+q^{104}+2 q^{102}+q^{98}+q^{94}+q^{92}-2 q^{90}+q^{88}+q^{86}+q^{76}-q^{72}+q^{66}-q^{62}-q^{52}+q^{48}+q^{38}+q^{34}+q^{28}+q^{24}+q^{20}+q^{14}+q^{10} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 9_2.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ -q^{21}+q^{15}+q^5+q }[/math]
2	[math]\displaystyle{ q^{60}-q^{56}-q^{50}+q^{46}-q^{30}-q^{28}+q^{18}+q^{16}+q^{14}+q^8+q^2 }[/math]
3	[math]\displaystyle{ -q^{117}+q^{113}+q^{111}-q^{107}+q^{103}-q^{99}-q^{97}+q^{93}+q^{73}+q^{71}-q^{67}-q^{61}-q^{59}-q^{53}+q^{49}+q^{47}-q^{43}-q^{41}-q^{39}+q^{37}-q^{33}-q^{31}+q^{29}+2 q^{27}-q^{23}+q^{21}+2 q^{19}+q^{17}+q^{11}+q^3 }[/math]
4	[math]\displaystyle{ q^{192}-q^{188}-q^{186}-q^{184}+q^{182}+q^{180}+q^{178}-2 q^{174}+q^{170}+q^{168}+q^{166}-q^{164}-q^{162}-q^{160}+q^{156}-q^{134}-q^{132}+q^{128}+2 q^{126}-q^{122}+q^{118}+2 q^{116}-2 q^{112}-q^{110}+q^{106}-2 q^{102}-q^{100}+q^{96}+q^{94}+q^{90}+q^{88}+q^{86}-q^{82}-q^{80}+q^{76}-q^{72}-q^{70}-q^{68}-q^{66}+q^{62}-q^{60}-2 q^{58}-2 q^{56}+2 q^{52}+q^{50}-2 q^{46}+q^{44}+2 q^{42}+q^{40}-q^{38}-2 q^{36}+q^{34}+2 q^{32}+q^{30}-q^{26}+q^{24}+q^{22}+q^{20}+q^{18}+q^{14}+q^4 }[/math]
5	[math]\displaystyle{ -q^{285}+q^{281}+q^{279}+q^{277}-q^{273}-2 q^{271}-q^{269}+q^{265}+2 q^{263}+q^{261}-q^{259}-2 q^{257}-q^{255}+q^{251}+2 q^{249}+q^{247}-q^{243}-q^{241}-q^{239}+q^{235}+q^{213}+q^{211}-q^{207}-2 q^{205}-2 q^{203}+2 q^{199}+2 q^{197}-2 q^{193}-2 q^{191}-q^{189}+2 q^{187}+4 q^{185}+2 q^{183}-q^{181}-2 q^{179}-2 q^{177}+2 q^{173}+2 q^{171}-2 q^{167}-2 q^{165}-q^{163}+q^{159}+q^{157}-q^{155}-q^{153}-q^{151}+q^{147}+2 q^{145}+q^{143}-q^{139}-q^{137}+q^{135}+2 q^{133}+q^{131}-q^{127}+q^{123}-q^{119}-2 q^{117}-q^{115}+q^{113}+2 q^{111}+q^{109}-q^{105}-q^{103}-2 q^{101}+q^{97}+2 q^{95}+2 q^{93}+q^{91}-2 q^{89}-3 q^{87}-2 q^{85}+q^{81}+q^{79}-q^{77}-2 q^{75}-3 q^{73}-q^{71}+q^{69}+q^{67}-q^{63}+q^{57}-q^{53}+2 q^{49}+3 q^{47}+q^{45}-q^{43}-q^{41}-q^{39}+q^{37}+2 q^{35}+q^{33}+q^{23}+q^{21}+q^{19}+q^{17}+q^5 }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ -q^{32}-q^{30}+q^{24}+q^{22}+q^8+q^6+q^2 }[/math]
1,1	[math]\displaystyle{ q^{84}+2 q^{80}-2 q^{78}+2 q^{76}-4 q^{74}+2 q^{72}-4 q^{70}+4 q^{62}-q^{60}+4 q^{58}-4 q^{56}+2 q^{54}-4 q^{52}+2 q^{50}-2 q^{48}+2 q^{46}-2 q^{42}-2 q^{40}-2 q^{38}-2 q^{36}+2 q^{30}+2 q^{28}+2 q^{26}+2 q^{24}+q^{20}+2 q^{16}+2 q^{12}+2 q^8+q^4 }[/math]
2,0	[math]\displaystyle{ q^{82}+q^{80}+q^{78}-q^{76}-q^{74}-q^{72}-q^{70}-q^{68}-q^{66}+q^{64}+q^{62}+q^{60}-q^{44}-2 q^{42}-2 q^{40}-q^{38}+q^{32}+q^{30}+q^{28}+2 q^{26}+q^{24}+q^{20}+q^{18}+q^{16}+q^{12}+q^{10}+q^4 }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{66}+q^{62}-q^{58}-q^{56}-q^{54}-q^{52}-q^{50}-q^{46}-q^{42}+q^{38}+q^{36}+q^{34}+q^{32}+q^{30}+q^{16}+q^{12}+2 q^{10}+q^8+q^4 }[/math]
1,0,0	[math]\displaystyle{ -q^{43}-q^{41}-q^{39}+q^{33}+q^{31}+q^{29}+q^{11}+q^9+q^7+q^3 }[/math]

B2 Invariants.

Weight	Invariant
0,1	[math]\displaystyle{ -q^{66}-q^{62}-q^{58}+q^{56}-q^{54}+q^{52}+q^{50}+q^{46}+q^{42}-2 q^{40}+q^{38}-q^{36}+q^{34}-q^{32}+q^{30}+q^{16}+q^{12}+q^8+q^4 }[/math]
1,0	[math]\displaystyle{ q^{108}+q^{100}-q^{96}-q^{94}-q^{88}-q^{86}+q^{82}-q^{76}-q^{68}-q^{66}+q^{56}+q^{54}+q^{48}+q^{46}+q^{26}+q^{18}+q^{16}+q^{14}+q^6 }[/math]

G2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ q^{156}+q^{152}-q^{150}+q^{142}-2 q^{140}+q^{138}-q^{136}-q^{134}-2 q^{130}-q^{128}-q^{126}-q^{124}-q^{118}+q^{112}-q^{108}+q^{106}+q^{104}+2 q^{102}+q^{98}+q^{94}+q^{92}-2 q^{90}+q^{88}+q^{86}+q^{76}-q^{72}+q^{66}-q^{62}-q^{52}+q^{48}+q^{38}+q^{34}+q^{28}+q^{24}+q^{20}+q^{14}+q^{10} }[/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 2"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ 4 t-7+4 t^{-1} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ 4 z^2+1 }[/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{ \{1\} }[/math]

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 15, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ q^{-1} - q^{-2} +2 q^{-3} -2 q^{-4} +2 q^{-5} -2 q^{-6} +2 q^{-7} - q^{-8} + q^{-9} - q^{-10} }[/math]

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

[math]\displaystyle{ -a^{10}+z^2 a^8+a^8+z^2 a^6+z^2 a^4+z^2 a^2+a^2 }[/math]

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ z^7 a^{11}-6 z^5 a^{11}+10 z^3 a^{11}-4 z a^{11}+z^8 a^{10}-6 z^6 a^{10}+11 z^4 a^{10}-7 z^2 a^{10}+a^{10}+2 z^7 a^9-10 z^5 a^9+13 z^3 a^9-4 z a^9+z^8 a^8-5 z^6 a^8+8 z^4 a^8-6 z^2 a^8+a^8+z^7 a^7-3 z^5 a^7+z^3 a^7+z^6 a^6-2 z^4 a^6+z^5 a^5-z^3 a^5+z^4 a^4+z^3 a^3+z^2 a^2-a^2 }[/math]

Vassiliev invariants

V2 and V3:

(4, -10)

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{ 16 }[/math] [math]\displaystyle{ -80 }[/math] [math]\displaystyle{ 128 }[/math] [math]\displaystyle{ \frac{1304}{3} }[/math] [math]\displaystyle{ \frac{184}{3} }[/math] [math]\displaystyle{ -1280 }[/math] [math]\displaystyle{ -\frac{8000}{3} }[/math] [math]\displaystyle{ -\frac{1280}{3} }[/math] [math]\displaystyle{ -400 }[/math] [math]\displaystyle{ \frac{2048}{3} }[/math] [math]\displaystyle{ 3200 }[/math] [math]\displaystyle{ \frac{20864}{3} }[/math] [math]\displaystyle{ \frac{2944}{3} }[/math] [math]\displaystyle{ \frac{249422}{15} }[/math] [math]\displaystyle{ -\frac{856}{5} }[/math] [math]\displaystyle{ \frac{315368}{45} }[/math] [math]\displaystyle{ \frac{2482}{9} }[/math] [math]\displaystyle{ \frac{13742}{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]-2 is the signature of 9 2. 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
-1										1	1
-3									1	1	0
-5								1			1
-7							1	1			0
-9						1	1				0
-11					1	1					0
-13				1	1						0
-15				1							1
-17		1	1								0
-19											0
-21	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, 2]]
Out[2]=	9
In[3]:=	PD[Knot[9, 2]]
Out[3]=	PD[X[1, 4, 2, 5], X[3, 12, 4, 13], X[5, 18, 6, 1], X[7, 16, 8, 17], X[9, 14, 10, 15], X[13, 10, 14, 11], X[15, 8, 16, 9], X[17, 6, 18, 7], X[11, 2, 12, 3]]
In[4]:=	GaussCode[Knot[9, 2]]
Out[4]=	GaussCode[-1, 9, -2, 1, -3, 8, -4, 7, -5, 6, -9, 2, -6, 5, -7, 4, -8, 3]
In[5]:=	BR[Knot[9, 2]]
Out[5]=	BR[5, {-1, -1, -1, -2, 1, -2, -3, 2, -3, -4, 3, -4}]
In[6]:=	alex = Alexander[Knot[9, 2]][t]
Out[6]=	4 -7 + - + 4 t t
In[7]:=	Conway[Knot[9, 2]][z]
Out[7]=	2 1 + 4 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[7, 4], Knot[9, 2]}
In[9]:=	{KnotDet[Knot[9, 2]], KnotSignature[Knot[9, 2]]}
Out[9]=	{15, -2}
In[10]:=	J=Jones[Knot[9, 2]][q]
Out[10]=	-10 -9 -8 2 2 2 2 2 -2 1 -q + q - q + -- - -- + -- - -- + -- - q + - 7 6 5 4 3 q q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[9, 2], Knot[11, NonAlternating, 13]}
In[12]:=	A2Invariant[Knot[9, 2]][q]
Out[12]=	-32 -30 -24 -22 -8 -6 -2 -q - q + q + q + q + q + q
In[13]:=	Kauffman[Knot[9, 2]][a, z]
Out[13]=	2 8 10 9 11 2 2 8 2 10 2 -a + a + a - 4 a z - 4 a z + a z - 6 a z - 7 a z + 3 3 5 3 7 3 9 3 11 3 4 4 6 4 a z - a z + a z + 13 a z + 10 a z + a z - 2 a z + 8 4 10 4 5 5 7 5 9 5 11 5 6 6 8 a z + 11 a z + a z - 3 a z - 10 a z - 6 a z + a z - 8 6 10 6 7 7 9 7 11 7 8 8 10 8 5 a z - 6 a z + a z + 2 a z + a z + a z + a z
In[14]:=	{Vassiliev[2][Knot[9, 2]], Vassiliev[3][Knot[9, 2]]}
Out[14]=	{0, -10}
In[15]:=	Kh[Knot[9, 2]][q, t]
Out[15]=	-3 1 1 1 1 1 1 1 q + - + ------ + ------ + ------ + ------ + ------ + ------ + q 21 9 17 8 17 7 15 6 13 6 13 5 q t q t q t q t q t q t 1 1 1 1 1 1 1 1 ------ + ------ + ----- + ----- + ----- + ----- + ----- + ---- 11 5 11 4 9 4 9 3 7 3 7 2 5 2 3 q t q t q t q t q t q t q t q t