9 45

9_44

9_46

Visit 9 45's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 45's page at Knotilus! Visit 9 45's page at the original Knot Atlas!

9 45 Quick Notes

9 45 Further Notes and Views

Knot presentations

Planar diagram presentation	X4251 X10,6,11,5 X8394 X2,9,3,10 X7,14,8,15 X18,15,1,16 X16,11,17,12 X12,17,13,18 X13,6,14,7
Gauss code	1, -4, 3, -1, 2, 9, -5, -3, 4, -2, 7, -8, -9, 5, 6, -7, 8, -6
Dowker-Thistlethwaite code	4 8 10 -14 2 16 -6 18 12
Conway Notation	[211,21,2-]

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 2 Bridge index 3 Super bridge index [math]\displaystyle{ \{4,5\} }[/math] Nakanishi index 1 Maximal Thurston-Bennequin number [-10][1] Hyperbolic Volume 8.60203 A-Polynomial See Data:9 45/A-polynomial

[edit Notes for 9 45'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{ 2 }[/math] Rasmussen s-Invariant 2

[edit Notes for 9 45's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ -t^2+6 t-9+6 t^{-1} - t^{-2} }[/math]
Conway polynomial	[math]\displaystyle{ -z^4+2 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 23, -2 }
Jones polynomial	[math]\displaystyle{ 2 q^{-1} -3 q^{-2} +4 q^{-3} -4 q^{-4} +4 q^{-5} -3 q^{-6} +2 q^{-7} - q^{-8} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ -a^8+2 z^2 a^6+2 a^6-z^4 a^4-2 z^2 a^4-2 a^4+2 z^2 a^2+2 a^2 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ z^5 a^9-3 z^3 a^9+2 z a^9+2 z^6 a^8-6 z^4 a^8+4 z^2 a^8-a^8+z^7 a^7-5 z^3 a^7+2 z a^7+4 z^6 a^6-10 z^4 a^6+7 z^2 a^6-2 a^6+z^7 a^5-z^3 a^5+2 z^6 a^4-4 z^4 a^4+6 z^2 a^4-2 a^4+z^5 a^3+z^3 a^3+3 z^2 a^2-2 a^2 }[/math]
The A2 invariant	[math]\displaystyle{ -q^{26}-q^{24}+q^{22}+q^{18}+q^{16}-q^{14}-q^{10}+q^8+q^6+2 q^2 }[/math]
The G2 invariant	[math]\displaystyle{ q^{128}-q^{126}+3 q^{124}-4 q^{122}+2 q^{120}-5 q^{116}+9 q^{114}-10 q^{112}+8 q^{110}-3 q^{108}-7 q^{106}+10 q^{104}-12 q^{102}+8 q^{100}-3 q^{98}-6 q^{96}+9 q^{94}-7 q^{92}+2 q^{90}+6 q^{88}-9 q^{86}+10 q^{84}-4 q^{82}-2 q^{80}+9 q^{78}-12 q^{76}+15 q^{74}-9 q^{72}+3 q^{70}+7 q^{68}-12 q^{66}+14 q^{64}-12 q^{62}+5 q^{60}+2 q^{58}-9 q^{56}+9 q^{54}-8 q^{52}+q^{50}+6 q^{48}-10 q^{46}+6 q^{44}-q^{42}-7 q^{40}+11 q^{38}-11 q^{36}+8 q^{34}-5 q^{30}+9 q^{28}-8 q^{26}+8 q^{24}-q^{22}-q^{20}+2 q^{18}-2 q^{16}+3 q^{14}-q^{12}+2 q^{10}+q^8 }[/math]

Further Quantum Invariants

Further quantum knot invariants for 9_45.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ -q^{17}+q^{15}-q^{13}+q^{11}+q^5-q^3+2 q }[/math]
2	[math]\displaystyle{ q^{48}-q^{46}-2 q^{44}+3 q^{42}+q^{40}-4 q^{38}+2 q^{36}+3 q^{34}-4 q^{32}-q^{30}+3 q^{28}-2 q^{26}-2 q^{24}+2 q^{22}+q^{20}-2 q^{18}-q^{16}+5 q^{14}-q^{12}-3 q^{10}+5 q^8-3 q^4+2 q^2+1 }[/math]
3	[math]\displaystyle{ -q^{93}+q^{91}+2 q^{89}-4 q^{85}-3 q^{83}+5 q^{81}+6 q^{79}-3 q^{77}-9 q^{75}-q^{73}+11 q^{71}+6 q^{69}-9 q^{67}-10 q^{65}+5 q^{63}+13 q^{61}-3 q^{59}-13 q^{57}-q^{55}+13 q^{53}+4 q^{51}-11 q^{49}-4 q^{47}+9 q^{45}+4 q^{43}-7 q^{41}-7 q^{39}+2 q^{37}+7 q^{35}+q^{33}-9 q^{31}-6 q^{29}+9 q^{27}+12 q^{25}-6 q^{23}-14 q^{21}+4 q^{19}+15 q^{17}-10 q^{13}-3 q^{11}+8 q^9+4 q^7-4 q^5-q^3+2 q^{-1} }[/math]
4	[math]\displaystyle{ q^{152}-q^{150}-2 q^{148}+q^{144}+6 q^{142}+q^{140}-5 q^{138}-7 q^{136}-7 q^{134}+11 q^{132}+13 q^{130}+5 q^{128}-9 q^{126}-25 q^{124}-5 q^{122}+14 q^{120}+28 q^{118}+16 q^{116}-26 q^{114}-32 q^{112}-15 q^{110}+27 q^{108}+45 q^{106}+4 q^{104}-33 q^{102}-46 q^{100}+49 q^{96}+34 q^{94}-12 q^{92}-51 q^{90}-23 q^{88}+33 q^{86}+42 q^{84}+4 q^{82}-40 q^{80}-26 q^{78}+18 q^{76}+34 q^{74}+7 q^{72}-27 q^{70}-23 q^{68}+6 q^{66}+27 q^{64}+11 q^{62}-13 q^{60}-23 q^{58}-12 q^{56}+17 q^{54}+25 q^{52}+15 q^{50}-23 q^{48}-42 q^{46}-5 q^{44}+33 q^{42}+48 q^{40}-4 q^{38}-57 q^{36}-35 q^{34}+17 q^{32}+61 q^{30}+25 q^{28}-37 q^{26}-40 q^{24}-11 q^{22}+36 q^{20}+31 q^{18}-6 q^{16}-19 q^{14}-16 q^{12}+7 q^{10}+12 q^8+3 q^6-q^4-5 q^2-1+2 q^{-2} + q^{-4} }[/math]
5	[math]\displaystyle{ -q^{225}+q^{223}+2 q^{221}-q^{217}-3 q^{215}-4 q^{213}-q^{211}+7 q^{209}+9 q^{207}+5 q^{205}-3 q^{203}-14 q^{201}-18 q^{199}-7 q^{197}+16 q^{195}+28 q^{193}+24 q^{191}+q^{189}-32 q^{187}-48 q^{185}-30 q^{183}+17 q^{181}+59 q^{179}+67 q^{177}+25 q^{175}-47 q^{173}-97 q^{171}-77 q^{169}+8 q^{167}+96 q^{165}+125 q^{163}+56 q^{161}-67 q^{159}-151 q^{157}-122 q^{155}+12 q^{153}+144 q^{151}+170 q^{149}+58 q^{147}-109 q^{145}-196 q^{143}-116 q^{141}+61 q^{139}+187 q^{137}+159 q^{135}-9 q^{133}-165 q^{131}-175 q^{129}-31 q^{127}+133 q^{125}+168 q^{123}+56 q^{121}-102 q^{119}-153 q^{117}-61 q^{115}+76 q^{113}+131 q^{111}+57 q^{109}-59 q^{107}-108 q^{105}-49 q^{103}+50 q^{101}+92 q^{99}+44 q^{97}-39 q^{95}-78 q^{93}-47 q^{91}+22 q^{89}+72 q^{87}+63 q^{85}+4 q^{83}-63 q^{81}-87 q^{79}-47 q^{77}+41 q^{75}+112 q^{73}+98 q^{71}-9 q^{69}-128 q^{67}-155 q^{65}-44 q^{63}+121 q^{61}+202 q^{59}+108 q^{57}-94 q^{55}-222 q^{53}-164 q^{51}+41 q^{49}+209 q^{47}+200 q^{45}+22 q^{43}-170 q^{41}-199 q^{39}-67 q^{37}+101 q^{35}+170 q^{33}+94 q^{31}-42 q^{29}-119 q^{27}-85 q^{25}+q^{23}+64 q^{21}+64 q^{19}+21 q^{17}-25 q^{15}-38 q^{13}-13 q^{11}+4 q^9+13 q^7+13 q^5-6 q-3 q^{-1} +2 q^{-7} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ -q^{26}-q^{24}+q^{22}+q^{18}+q^{16}-q^{14}-q^{10}+q^8+q^6+2 q^2 }[/math]
1,1	[math]\displaystyle{ q^{68}-2 q^{66}+6 q^{64}-12 q^{62}+17 q^{60}-24 q^{58}+30 q^{56}-30 q^{54}+25 q^{52}-18 q^{50}+6 q^{48}+12 q^{46}-27 q^{44}+38 q^{42}-48 q^{40}+52 q^{38}-54 q^{36}+44 q^{34}-36 q^{32}+22 q^{30}-6 q^{28}-6 q^{26}+20 q^{24}-26 q^{22}+31 q^{20}-28 q^{18}+22 q^{16}-16 q^{14}+13 q^{12}-4 q^{10}+2 q^8+2 q^4+2 q^2 }[/math]
2,0	[math]\displaystyle{ q^{66}+q^{64}-3 q^{60}-2 q^{58}+q^{56}+2 q^{54}+3 q^{48}+2 q^{46}-2 q^{44}-4 q^{42}-q^{40}-2 q^{38}-2 q^{36}-q^{34}+2 q^{32}+2 q^{30}+q^{28}+2 q^{26}-q^{24}-q^{22}+2 q^{20}+2 q^{18}-2 q^{16}-q^{14}+3 q^{12}+2 q^{10}-q^8-q^6+3 q^4+q^2 }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{54}-q^{52}+q^{50}+q^{48}-3 q^{46}+q^{44}-q^{42}-4 q^{40}+2 q^{38}+q^{36}-2 q^{34}+2 q^{32}+2 q^{30}+q^{22}-2 q^{20}-2 q^{18}+3 q^{16}-2 q^{14}+5 q^{10}+3 q^4 }[/math]
1,0,0	[math]\displaystyle{ -q^{35}-q^{33}-q^{31}+q^{29}+2 q^{25}+q^{23}+q^{21}-q^{19}-q^{17}-q^{15}-q^{13}+q^{11}+q^9+2 q^7+2 q^3 }[/math]

A4 Invariants.

Weight	Invariant
0,1,0,0	[math]\displaystyle{ q^{72}+q^{70}+q^{64}-q^{62}-4 q^{60}-2 q^{58}+q^{56}-2 q^{54}-3 q^{52}+3 q^{50}+4 q^{48}-q^{46}-q^{44}+2 q^{42}-q^{40}-3 q^{38}+q^{36}+3 q^{34}-q^{32}+q^{30}+4 q^{28}-2 q^{26}-4 q^{24}+q^{20}-2 q^{18}+5 q^{14}+4 q^{12}+q^{10}+q^8+3 q^6 }[/math]
1,0,0,0	[math]\displaystyle{ -q^{44}-q^{42}-q^{40}-q^{38}+q^{36}+2 q^{32}+2 q^{30}+q^{28}+q^{26}-q^{24}-q^{22}-2 q^{20}-q^{18}-q^{16}+q^{14}+q^{12}+2 q^{10}+2 q^8+2 q^4 }[/math]

B2 Invariants.

Weight	Invariant
0,1	[math]\displaystyle{ -q^{54}+q^{52}-3 q^{50}+3 q^{48}-3 q^{46}+3 q^{44}-3 q^{42}+2 q^{40}-q^{36}+4 q^{34}-4 q^{32}+6 q^{30}-6 q^{28}+6 q^{26}-6 q^{24}+3 q^{22}-2 q^{20}+q^{16}-2 q^{14}+4 q^{12}-3 q^{10}+4 q^8-2 q^6+3 q^4 }[/math]
1,0	[math]\displaystyle{ q^{88}-q^{84}-q^{82}+2 q^{80}+2 q^{78}-2 q^{76}-3 q^{74}+3 q^{70}+q^{68}-4 q^{66}-3 q^{64}+2 q^{62}+3 q^{60}-3 q^{56}+2 q^{52}+q^{50}-2 q^{48}-q^{46}+2 q^{44}+2 q^{42}-q^{40}-3 q^{38}+q^{36}+3 q^{34}-3 q^{30}-q^{28}+3 q^{26}+2 q^{24}-2 q^{22}-3 q^{20}+2 q^{18}+4 q^{16}+q^{14}-2 q^{12}-q^{10}+q^8+3 q^6 }[/math]

D4 Invariants.

Weight	Invariant
1,0,0,0	[math]\displaystyle{ q^{74}-q^{72}+2 q^{70}-2 q^{68}+3 q^{66}-3 q^{64}+q^{62}-4 q^{60}+q^{58}-3 q^{56}-q^{54}-q^{50}+3 q^{48}-2 q^{46}+6 q^{44}-2 q^{42}+6 q^{40}-4 q^{38}+5 q^{36}-4 q^{34}+3 q^{32}-4 q^{30}-q^{28}-2 q^{26}-q^{24}+q^{22}-2 q^{20}+3 q^{18}-q^{16}+6 q^{14}+3 q^{10}-q^8+3 q^6 }[/math]

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{128}-q^{126}+3 q^{124}-4 q^{122}+2 q^{120}-5 q^{116}+9 q^{114}-10 q^{112}+8 q^{110}-3 q^{108}-7 q^{106}+10 q^{104}-12 q^{102}+8 q^{100}-3 q^{98}-6 q^{96}+9 q^{94}-7 q^{92}+2 q^{90}+6 q^{88}-9 q^{86}+10 q^{84}-4 q^{82}-2 q^{80}+9 q^{78}-12 q^{76}+15 q^{74}-9 q^{72}+3 q^{70}+7 q^{68}-12 q^{66}+14 q^{64}-12 q^{62}+5 q^{60}+2 q^{58}-9 q^{56}+9 q^{54}-8 q^{52}+q^{50}+6 q^{48}-10 q^{46}+6 q^{44}-q^{42}-7 q^{40}+11 q^{38}-11 q^{36}+8 q^{34}-5 q^{30}+9 q^{28}-8 q^{26}+8 q^{24}-q^{22}-q^{20}+2 q^{18}-2 q^{16}+3 q^{14}-q^{12}+2 q^{10}+q^8 }[/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 45"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ -t^2+6 t-9+6 t^{-1} - t^{-2} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ -z^4+2 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]=

{ 23, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

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

Vassiliev invariants

V2 and V3:

(2, -4)

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{ -32 }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ \frac{412}{3} }[/math] [math]\displaystyle{ \frac{68}{3} }[/math] [math]\displaystyle{ -256 }[/math] [math]\displaystyle{ -\frac{1856}{3} }[/math] [math]\displaystyle{ -\frac{320}{3} }[/math] [math]\displaystyle{ -96 }[/math] [math]\displaystyle{ \frac{256}{3} }[/math] [math]\displaystyle{ 512 }[/math] [math]\displaystyle{ \frac{3296}{3} }[/math] [math]\displaystyle{ \frac{544}{3} }[/math] [math]\displaystyle{ \frac{42751}{15} }[/math] [math]\displaystyle{ -\frac{284}{15} }[/math] [math]\displaystyle{ \frac{56284}{45} }[/math] [math]\displaystyle{ \frac{257}{9} }[/math] [math]\displaystyle{ \frac{2431}{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 45. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \	-7	-6	-5	-4	-3	-2	-1	0	χ
-1								2	2
-3							2	1	-1
-5						2	1		1
-7					2	2			0
-9				2	2				0
-11			1	2					1
-13		1	2						-1
-15		1							1
-17	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, 45]]
Out[2]=	9
In[3]:=	PD[Knot[9, 45]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 6, 11, 5], X[8, 3, 9, 4], X[2, 9, 3, 10], X[7, 14, 8, 15], X[18, 15, 1, 16], X[16, 11, 17, 12], X[12, 17, 13, 18], X[13, 6, 14, 7]]
In[4]:=	GaussCode[Knot[9, 45]]
Out[4]=	GaussCode[1, -4, 3, -1, 2, 9, -5, -3, 4, -2, 7, -8, -9, 5, 6, -7, 8, -6]
In[5]:=	BR[Knot[9, 45]]
Out[5]=	BR[4, {-1, -1, -2, 1, -2, -1, -3, 2, -3}]
In[6]:=	alex = Alexander[Knot[9, 45]][t]
Out[6]=	-2 6 2 -9 - t + - + 6 t - t t
In[7]:=	Conway[Knot[9, 45]][z]
Out[7]=	2 4 1 + 2 z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[9, 45]}
In[9]:=	{KnotDet[Knot[9, 45]], KnotSignature[Knot[9, 45]]}
Out[9]=	{23, -2}
In[10]:=	J=Jones[Knot[9, 45]][q]
Out[10]=	-8 2 3 4 4 4 3 2 -q + -- - -- + -- - -- + -- - -- + - 7 6 5 4 3 2 q q q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[9, 45]}
In[12]:=	A2Invariant[Knot[9, 45]][q]
Out[12]=	-26 -24 -22 -18 -16 -14 -10 -8 -6 2 -q - q + q + q + q - q - q + q + q + -- 2 q
In[13]:=	Kauffman[Knot[9, 45]][a, z]
Out[13]=	2 4 6 8 7 9 2 2 4 2 -2 a - 2 a - 2 a - a + 2 a z + 2 a z + 3 a z + 6 a z + 6 2 8 2 3 3 5 3 7 3 9 3 4 4 7 a z + 4 a z + a z - a z - 5 a z - 3 a z - 4 a z - 6 4 8 4 3 5 9 5 4 6 6 6 8 6 10 a z - 6 a z + a z + a z + 2 a z + 4 a z + 2 a z + 5 7 7 7 a z + a z
In[14]:=	{Vassiliev[2][Knot[9, 45]], Vassiliev[3][Knot[9, 45]]}
Out[14]=	{0, -4}
In[15]:=	Kh[Knot[9, 45]][q, t]
Out[15]=	-3 2 1 1 1 2 1 2 2 q + - + ------ + ------ + ------ + ------ + ------ + ------ + ----- + q 17 7 15 6 13 6 13 5 11 5 11 4 9 4 q t q t q t q t q t q t q t 2 2 2 2 1 2 ----- + ----- + ----- + ----- + ---- + ---- 9 3 7 3 7 2 5 2 5 3 q t q t q t q t q t q t