10 133

10_132

10_134

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10 133 Quick Notes

10 133 Further Notes and Views

Knot presentations

Planar diagram presentation	X1425 X3849 X14,9,15,10 X5,13,6,12 X13,7,14,6 X18,11,19,12 X20,15,1,16 X16,19,17,20 X10,17,11,18 X7283
Gauss code	-1, 10, -2, 1, -4, 5, -10, 2, 3, -9, 6, 4, -5, -3, 7, -8, 9, -6, 8, -7
Dowker-Thistlethwaite code	4 8 12 2 -14 -18 6 -20 -10 -16
Conway Notation	[23,21,2-]

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 2 Bridge index 3 Super bridge index Missing Nakanishi index 1 Maximal Thurston-Bennequin number [-11][1] Hyperbolic Volume 7.7983 A-Polynomial See Data:10 133/A-polynomial

[edit Notes for 10 133'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 10 133's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ -t^2+5 t-7+5 t^{-1} - t^{-2} }[/math]
Conway polynomial	[math]\displaystyle{ -z^4+z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 19, -2 }
Jones polynomial	[math]\displaystyle{ q^{-1} - q^{-2} +3 q^{-3} -3 q^{-4} +3 q^{-5} -3 q^{-6} +2 q^{-7} -2 q^{-8} + q^{-9} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ z^2 a^8+a^8-z^4 a^6-3 z^2 a^6-3 a^6+2 z^2 a^4+2 a^4+z^2 a^2+a^2 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ z^6 a^{10}-4 z^4 a^{10}+3 z^2 a^{10}+2 z^7 a^9-9 z^5 a^9+10 z^3 a^9-3 z a^9+z^8 a^8-3 z^6 a^8+z^2 a^8+a^8+3 z^7 a^7-13 z^5 a^7+16 z^3 a^7-7 z a^7+z^8 a^6-4 z^6 a^6+6 z^4 a^6-6 z^2 a^6+3 a^6+z^7 a^5-4 z^5 a^5+7 z^3 a^5-4 z a^5+2 z^4 a^4-3 z^2 a^4+2 a^4+z^3 a^3+z^2 a^2-a^2 }[/math]
The A2 invariant	[math]\displaystyle{ q^{28}-2 q^{20}-q^{18}-q^{16}+q^{12}+q^{10}+2 q^8+q^6+q^2 }[/math]
The G2 invariant	[math]\displaystyle{ q^{142}-q^{140}+2 q^{138}-3 q^{136}+q^{134}-q^{132}-3 q^{130}+6 q^{128}-6 q^{126}+4 q^{124}-q^{122}-2 q^{120}+5 q^{118}-4 q^{116}+2 q^{114}+3 q^{112}-3 q^{110}+4 q^{108}+q^{106}-3 q^{104}+8 q^{102}-6 q^{100}+3 q^{98}+q^{96}-4 q^{94}+5 q^{92}-6 q^{90}+3 q^{88}-4 q^{86}-q^{82}-5 q^{80}+q^{78}-4 q^{76}-3 q^{70}+q^{66}-4 q^{64}+6 q^{62}-5 q^{60}+2 q^{58}+4 q^{56}-5 q^{54}+7 q^{52}-2 q^{50}+q^{48}+3 q^{46}-2 q^{44}+q^{42}+2 q^{40}+2 q^{36}+q^{34}-q^{32}+2 q^{30}-q^{28}+q^{26}+q^{24}-q^{22}+2 q^{20}+q^{14}+q^{10} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 10_133.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ q^{19}-q^{17}-q^{13}+2 q^5+q }[/math]
2	[math]\displaystyle{ q^{54}-q^{52}-2 q^{50}+2 q^{48}+q^{46}-2 q^{44}+2 q^{40}-q^{36}+q^{34}+q^{32}-2 q^{30}+q^{26}-2 q^{24}-q^{22}+q^{20}-2 q^{16}+q^{14}+2 q^{12}+q^6+q^4+q^2 }[/math]
3	[math]\displaystyle{ q^{105}-q^{103}-2 q^{101}+3 q^{97}+3 q^{95}-3 q^{93}-4 q^{91}+4 q^{87}+3 q^{85}-2 q^{83}-4 q^{81}-q^{79}+3 q^{77}+4 q^{75}-2 q^{73}-6 q^{71}-2 q^{69}+6 q^{67}+4 q^{65}-5 q^{63}-4 q^{61}+5 q^{59}+5 q^{57}-3 q^{55}-3 q^{53}+3 q^{51}+3 q^{49}-4 q^{47}-3 q^{45}+q^{43}+3 q^{41}-2 q^{37}-5 q^{35}+2 q^{33}+6 q^{31}+q^{29}-8 q^{27}-6 q^{25}+7 q^{23}+7 q^{21}-3 q^{19}-6 q^{17}+q^{15}+5 q^{13}+2 q^{11}-2 q^9+q^5+2 q^3 }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ q^{28}-2 q^{20}-q^{18}-q^{16}+q^{12}+q^{10}+2 q^8+q^6+q^2 }[/math]
1,1	[math]\displaystyle{ q^{76}-2 q^{74}+4 q^{72}-8 q^{70}+11 q^{68}-14 q^{66}+14 q^{64}-12 q^{62}+8 q^{60}-8 q^{56}+14 q^{54}-19 q^{52}+22 q^{50}-20 q^{48}+20 q^{46}-16 q^{44}+12 q^{42}-6 q^{40}+6 q^{36}-10 q^{34}+12 q^{32}-14 q^{30}+6 q^{28}-8 q^{26}-2 q^{24}-2 q^{20}+2 q^{18}+4 q^{16}+2 q^{14}+6 q^{12}+4 q^8+q^4 }[/math]
2,0	[math]\displaystyle{ q^{72}-q^{68}-q^{66}+q^{62}-q^{60}-q^{58}+q^{52}+q^{50}+2 q^{48}+3 q^{46}+q^{44}-2 q^{40}-q^{38}-2 q^{36}-2 q^{34}-2 q^{32}-q^{30}-q^{28}-q^{26}-2 q^{22}+2 q^{20}+2 q^{18}+2 q^{16}+q^{14}+2 q^{12}+3 q^{10}+q^8+q^4 }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{60}-q^{58}-2 q^{52}-q^{48}+q^{46}+2 q^{44}+3 q^{42}+3 q^{40}+3 q^{38}-3 q^{34}-4 q^{32}-6 q^{30}-4 q^{28}-3 q^{26}+2 q^{22}+2 q^{20}+3 q^{18}+3 q^{16}+q^{14}+2 q^{12}+2 q^{10}+q^8+q^4 }[/math]
1,0,0	[math]\displaystyle{ q^{37}+q^{33}-2 q^{27}-2 q^{25}-2 q^{23}-q^{21}+q^{17}+2 q^{15}+q^{13}+2 q^{11}+q^9+q^7+q^3 }[/math]

A4 Invariants.

Weight	Invariant
0,1,0,0	[math]\displaystyle{ q^{78}+q^{72}-q^{70}-3 q^{68}-2 q^{66}-2 q^{64}-4 q^{62}-q^{60}+4 q^{58}+7 q^{56}+6 q^{54}+9 q^{52}+9 q^{50}+2 q^{48}-3 q^{46}-5 q^{44}-11 q^{42}-13 q^{40}-9 q^{38}-7 q^{36}-4 q^{34}+5 q^{30}+5 q^{28}+4 q^{26}+5 q^{24}+5 q^{22}+2 q^{20}+2 q^{18}+2 q^{16}+2 q^{14}+2 q^{12}+q^{10}+q^6 }[/math]
1,0,0,0	[math]\displaystyle{ q^{46}+q^{42}+q^{40}-2 q^{34}-2 q^{32}-3 q^{30}-2 q^{28}-q^{26}+q^{22}+2 q^{20}+2 q^{18}+q^{16}+2 q^{14}+q^{12}+q^{10}+q^8+q^4 }[/math]

B2 Invariants.

Weight	Invariant
0,1	[math]\displaystyle{ q^{60}-q^{58}+2 q^{56}-2 q^{54}+2 q^{52}-2 q^{50}+q^{48}-q^{46}+q^{42}-3 q^{40}+3 q^{38}-4 q^{36}+3 q^{34}-4 q^{32}+2 q^{30}-2 q^{28}+q^{26}+2 q^{20}-q^{18}+3 q^{16}-q^{14}+2 q^{12}+q^8+q^4 }[/math]
1,0	[math]\displaystyle{ q^{98}-q^{94}-q^{92}+q^{90}+q^{88}-2 q^{86}-2 q^{84}+q^{82}+2 q^{80}-q^{78}-2 q^{76}+3 q^{72}+2 q^{70}+2 q^{64}+2 q^{62}+q^{60}-q^{58}-q^{56}-q^{52}-4 q^{50}-3 q^{48}-q^{46}-2 q^{42}-3 q^{40}+2 q^{36}+q^{34}-q^{32}+q^{30}+2 q^{28}+3 q^{26}+q^{20}+2 q^{18}+q^{16}+q^{14}+q^6 }[/math]

D4 Invariants.

Weight	Invariant
1,0,0,0	[math]\displaystyle{ q^{82}-q^{80}+q^{78}-2 q^{76}+2 q^{74}-3 q^{72}-2 q^{68}+q^{66}+q^{62}+3 q^{60}+2 q^{58}+6 q^{56}+q^{54}+4 q^{52}-3 q^{50}+q^{48}-7 q^{46}-3 q^{44}-8 q^{42}-4 q^{40}-5 q^{38}-q^{36}+q^{32}+4 q^{30}+2 q^{28}+4 q^{26}+q^{24}+4 q^{22}+2 q^{18}+q^{16}+2 q^{14}+q^{12}+q^{10}+q^6 }[/math]

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{142}-q^{140}+2 q^{138}-3 q^{136}+q^{134}-q^{132}-3 q^{130}+6 q^{128}-6 q^{126}+4 q^{124}-q^{122}-2 q^{120}+5 q^{118}-4 q^{116}+2 q^{114}+3 q^{112}-3 q^{110}+4 q^{108}+q^{106}-3 q^{104}+8 q^{102}-6 q^{100}+3 q^{98}+q^{96}-4 q^{94}+5 q^{92}-6 q^{90}+3 q^{88}-4 q^{86}-q^{82}-5 q^{80}+q^{78}-4 q^{76}-3 q^{70}+q^{66}-4 q^{64}+6 q^{62}-5 q^{60}+2 q^{58}+4 q^{56}-5 q^{54}+7 q^{52}-2 q^{50}+q^{48}+3 q^{46}-2 q^{44}+q^{42}+2 q^{40}+2 q^{36}+q^{34}-q^{32}+2 q^{30}-q^{28}+q^{26}+q^{24}-q^{22}+2 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["10 133"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ -t^2+5 t-7+5 t^{-1} - t^{-2} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ -z^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]=

{ 19, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

[math]\displaystyle{ z^2 a^8+a^8-z^4 a^6-3 z^2 a^6-3 a^6+2 z^2 a^4+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^6 a^{10}-4 z^4 a^{10}+3 z^2 a^{10}+2 z^7 a^9-9 z^5 a^9+10 z^3 a^9-3 z a^9+z^8 a^8-3 z^6 a^8+z^2 a^8+a^8+3 z^7 a^7-13 z^5 a^7+16 z^3 a^7-7 z a^7+z^8 a^6-4 z^6 a^6+6 z^4 a^6-6 z^2 a^6+3 a^6+z^7 a^5-4 z^5 a^5+7 z^3 a^5-4 z a^5+2 z^4 a^4-3 z^2 a^4+2 a^4+z^3 a^3+z^2 a^2-a^2 }[/math]

Vassiliev invariants

V2 and V3:

(1, 0)

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{ 4 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 8 }[/math] [math]\displaystyle{ -\frac{34}{3} }[/math] [math]\displaystyle{ \frac{34}{3} }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ -32 }[/math] [math]\displaystyle{ -32 }[/math] [math]\displaystyle{ \frac{32}{3} }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ -\frac{136}{3} }[/math] [math]\displaystyle{ \frac{136}{3} }[/math] [math]\displaystyle{ -\frac{1649}{30} }[/math] [math]\displaystyle{ -\frac{1022}{15} }[/math] [math]\displaystyle{ \frac{13022}{45} }[/math] [math]\displaystyle{ \frac{113}{18} }[/math] [math]\displaystyle{ \frac{1231}{30} }[/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 10 133. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \	-8	-7	-6	-5	-4	-3	-2	-1	0	χ
-1									1	1
-3								1	1	0
-5							2			2
-7						1	1			0
-9					2	2				0
-11				1	1					0
-13			1	2						-1
-15		1	1							0
-17		1								-1
-19	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[10, 133]]
Out[2]=	10
In[3]:=	PD[Knot[10, 133]]
Out[3]=	PD[X[1, 4, 2, 5], X[3, 8, 4, 9], X[14, 9, 15, 10], X[5, 13, 6, 12], X[13, 7, 14, 6], X[18, 11, 19, 12], X[20, 15, 1, 16], X[16, 19, 17, 20], X[10, 17, 11, 18], X[7, 2, 8, 3]]
In[4]:=	GaussCode[Knot[10, 133]]
Out[4]=	GaussCode[-1, 10, -2, 1, -4, 5, -10, 2, 3, -9, 6, 4, -5, -3, 7, -8, 9, -6, 8, -7]
In[5]:=	BR[Knot[10, 133]]
Out[5]=	BR[4, {-1, -1, -1, -2, 1, 1, -2, -3, 2, -3, -3}]
In[6]:=	alex = Alexander[Knot[10, 133]][t]
Out[6]=	-2 5 2 -7 - t + - + 5 t - t t
In[7]:=	Conway[Knot[10, 133]][z]
Out[7]=	2 4 1 + z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[7, 6], Knot[10, 133]}
In[9]:=	{KnotDet[Knot[10, 133]], KnotSignature[Knot[10, 133]]}
Out[9]=	{19, -2}
In[10]:=	J=Jones[Knot[10, 133]][q]
Out[10]=	-9 2 2 3 3 3 3 -2 1 q - -- + -- - -- + -- - -- + -- - q + - 8 7 6 5 4 3 q q q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[10, 133], Knot[11, NonAlternating, 27]}
In[12]:=	A2Invariant[Knot[10, 133]][q]
Out[12]=	-28 2 -18 -16 -12 -10 2 -6 -2 q - --- - q - q + q + q + -- + q + q 20 8 q q
In[13]:=	Kauffman[Knot[10, 133]][a, z]
Out[13]=	2 4 6 8 5 7 9 2 2 4 2 -a + 2 a + 3 a + a - 4 a z - 7 a z - 3 a z + a z - 3 a z - 6 2 8 2 10 2 3 3 5 3 7 3 9 3 6 a z + a z + 3 a z + a z + 7 a z + 16 a z + 10 a z + 4 4 6 4 10 4 5 5 7 5 9 5 2 a z + 6 a z - 4 a z - 4 a z - 13 a z - 9 a z - 6 6 8 6 10 6 5 7 7 7 9 7 6 8 8 8 4 a z - 3 a z + a z + a z + 3 a z + 2 a z + a z + a z
In[14]:=	{Vassiliev[2][Knot[10, 133]], Vassiliev[3][Knot[10, 133]]}
Out[14]=	{0, 0}
In[15]:=	Kh[Knot[10, 133]][q, t]
Out[15]=	-3 1 1 1 1 1 1 2 q + - + ------ + ------ + ------ + ------ + ------ + ------ + q 19 8 17 7 15 7 15 6 13 6 13 5 q t q t q t q t q t q t 1 1 2 2 1 1 2 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