9 44

9_43

9_45

Visit 9 44's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 44's page at Knotilus! Visit 9 44's page at the original Knot Atlas!

9 44 Quick Notes

9 44 Further Notes and Views

Knot presentations

Planar diagram presentation	X1425 X5,10,6,11 X3948 X9,3,10,2 X14,8,15,7 X18,15,1,16 X16,11,17,12 X12,17,13,18 X6,14,7,13
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	[22,21,2-]

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 [-6][-3] Hyperbolic Volume 7.40677 A-Polynomial See Data:9 44/A-polynomial

[edit Notes for 9 44's three dimensional invariants] 9_44 has girth 4. See arXiv:math.GT/0508590 and a forthcoming paper by the same authors.

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 0

[edit Notes for 9 44's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ t^2-4 t+7-4 t^{-1} + t^{-2} }[/math]
Conway polynomial	[math]\displaystyle{ z^4+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 17, 0 }
Jones polynomial	[math]\displaystyle{ q^2-2 q+3-3 q^{-1} +3 q^{-2} -2 q^{-3} +2 q^{-4} - q^{-5} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ -z^2 a^4-a^4+z^4 a^2+3 z^2 a^2+3 a^2-2 z^2-2+ a^{-2} }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ a^3 z^7+a z^7+2 a^4 z^6+3 a^2 z^6+z^6+a^5 z^5-2 a^3 z^5-3 a z^5-7 a^4 z^4-10 a^2 z^4-3 z^4-3 a^5 z^3-a^3 z^3+4 a z^3+2 z^3 a^{-1} +5 a^4 z^2+10 a^2 z^2+z^2 a^{-2} +6 z^2+a^5 z+a^3 z-a z-z a^{-1} -a^4-3 a^2- a^{-2} -2 }[/math]
The A2 invariant	[math]\displaystyle{ -q^{16}+2 q^8+q^6+q^4-1- q^{-4} + q^{-6} + q^{-8} }[/math]
The G2 invariant	[math]\displaystyle{ q^{80}-q^{78}+2 q^{76}-3 q^{74}+q^{72}-4 q^{68}+6 q^{66}-5 q^{64}+3 q^{62}-q^{60}-4 q^{58}+5 q^{56}-4 q^{54}+2 q^{50}-4 q^{48}+4 q^{46}-4 q^{42}+7 q^{40}-5 q^{38}+3 q^{36}-2 q^{32}+6 q^{30}-4 q^{28}+6 q^{26}-2 q^{24}+2 q^{22}+4 q^{20}-4 q^{18}+4 q^{16}-2 q^{14}+q^{12}+3 q^{10}-4 q^8+2 q^6-4 q^2+5-6 q^{-2} +2 q^{-6} -6 q^{-8} +5 q^{-10} -3 q^{-12} + q^{-14} + q^{-16} -3 q^{-18} +2 q^{-20} + q^{-24} + q^{-26} + q^{-32} + q^{-38} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 9_44.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ -q^{11}+q^9+q^5+ q^{-1} - q^{-3} + q^{-5} }[/math]
2	[math]\displaystyle{ q^{32}-q^{30}-2 q^{28}+2 q^{26}+q^{24}-2 q^{22}+2 q^{18}-q^{16}-q^{14}+2 q^{12}-q^8+q^6+q^4-q^2-1+3 q^{-2} -2 q^{-6} +2 q^{-8} + q^{-10} - q^{-12} }[/math]
3	[math]\displaystyle{ -q^{63}+q^{61}+2 q^{59}-3 q^{55}-3 q^{53}+3 q^{51}+4 q^{49}-4 q^{45}-3 q^{43}+3 q^{41}+5 q^{39}-q^{37}-6 q^{35}-3 q^{33}+6 q^{31}+4 q^{29}-4 q^{27}-4 q^{25}+4 q^{23}+5 q^{21}-3 q^{19}-4 q^{17}+2 q^{15}+3 q^{13}-2 q^{11}-3 q^9+3 q^5+3 q^3-2 q-4 q^{-1} +2 q^{-3} +7 q^{-5} +3 q^{-7} -7 q^{-9} -4 q^{-11} +5 q^{-13} +4 q^{-15} -2 q^{-17} -5 q^{-19} +3 q^{-23} + q^{-25} - q^{-29} }[/math]
4	[math]\displaystyle{ q^{104}-q^{102}-2 q^{100}+q^{96}+5 q^{94}+q^{92}-3 q^{90}-5 q^{88}-6 q^{86}+5 q^{84}+7 q^{82}+5 q^{80}-10 q^{76}-7 q^{74}-q^{72}+9 q^{70}+14 q^{68}+q^{66}-11 q^{64}-16 q^{62}-5 q^{60}+15 q^{58}+17 q^{56}+2 q^{54}-18 q^{52}-18 q^{50}+5 q^{48}+21 q^{46}+14 q^{44}-9 q^{42}-20 q^{40}-5 q^{38}+14 q^{36}+13 q^{34}-4 q^{32}-15 q^{30}-5 q^{28}+9 q^{26}+8 q^{24}-2 q^{22}-9 q^{20}-3 q^{18}+7 q^{16}+6 q^{14}+2 q^{12}-4 q^{10}-7 q^8-q^6+6 q^4+13 q^2+3-15 q^{-2} -15 q^{-4} - q^{-6} +22 q^{-8} +20 q^{-10} -9 q^{-12} -25 q^{-14} -15 q^{-16} +15 q^{-18} +26 q^{-20} +5 q^{-22} -14 q^{-24} -18 q^{-26} -2 q^{-28} +13 q^{-30} +9 q^{-32} -7 q^{-36} -5 q^{-38} + q^{-40} +2 q^{-42} +2 q^{-44} - q^{-48} }[/math]
5	[math]\displaystyle{ -q^{155}+q^{153}+2 q^{151}-q^{147}-3 q^{145}-3 q^{143}-q^{141}+5 q^{139}+7 q^{137}+4 q^{135}-q^{133}-7 q^{131}-11 q^{129}-8 q^{127}+5 q^{125}+11 q^{123}+13 q^{121}+9 q^{119}-4 q^{117}-16 q^{115}-20 q^{113}-10 q^{111}+6 q^{109}+24 q^{107}+29 q^{105}+13 q^{103}-16 q^{101}-37 q^{99}-34 q^{97}-8 q^{95}+32 q^{93}+50 q^{91}+32 q^{89}-13 q^{87}-54 q^{85}-54 q^{83}-12 q^{81}+43 q^{79}+67 q^{77}+38 q^{75}-25 q^{73}-67 q^{71}-53 q^{69}+3 q^{67}+59 q^{65}+62 q^{63}+13 q^{61}-46 q^{59}-62 q^{57}-22 q^{55}+32 q^{53}+54 q^{51}+25 q^{49}-23 q^{47}-45 q^{45}-23 q^{43}+18 q^{41}+34 q^{39}+17 q^{37}-13 q^{35}-26 q^{33}-10 q^{31}+12 q^{29}+19 q^{27}+8 q^{25}-9 q^{23}-16 q^{21}-10 q^{19}+4 q^{17}+17 q^{15}+17 q^{13}+6 q^{11}-14 q^9-29 q^7-21 q^5+9 q^3+41 q+40 q^{-1} +5 q^{-3} -44 q^{-5} -63 q^{-7} -27 q^{-9} +42 q^{-11} +81 q^{-13} +49 q^{-15} -23 q^{-17} -82 q^{-19} -70 q^{-21} + q^{-23} +69 q^{-25} +78 q^{-27} +19 q^{-29} -44 q^{-31} -66 q^{-33} -36 q^{-35} +18 q^{-37} +47 q^{-39} +36 q^{-41} + q^{-43} -23 q^{-45} -26 q^{-47} -11 q^{-49} +6 q^{-51} +14 q^{-53} +9 q^{-55} -3 q^{-59} -4 q^{-61} -2 q^{-63} + q^{-65} + q^{-67} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ -q^{16}+2 q^8+q^6+q^4-1- q^{-4} + q^{-6} + q^{-8} }[/math]
1,1	[math]\displaystyle{ q^{44}-2 q^{42}+4 q^{40}-8 q^{38}+11 q^{36}-12 q^{34}+12 q^{32}-12 q^{30}+4 q^{28}+2 q^{26}-8 q^{24}+16 q^{22}-19 q^{20}+22 q^{18}-20 q^{16}+20 q^{14}-14 q^{12}+10 q^{10}-2 q^8-4 q^6+8 q^4-14 q^2+14-12 q^{-2} +11 q^{-4} -4 q^{-6} +4 q^{-8} +2 q^{-10} - q^{-12} -2 q^{-16} + q^{-20} }[/math]
2,0	[math]\displaystyle{ q^{42}-q^{38}-q^{36}+q^{32}-q^{30}-q^{28}+2 q^{18}+3 q^{16}+q^{14}+q^{12}-q^8-2 q^6-q^4-2 q^2+2 q^{-2} +3 q^{-4} +2 q^{-6} +2 q^{-10} -2 q^{-14} - q^{-16} + q^{-20} }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{34}-q^{32}-2 q^{26}+q^{24}-q^{22}-q^{20}+q^{18}+q^{16}+2 q^{12}+2 q^{10}+q^8+q^6+q^2-1- q^{-2} + q^{-4} -2 q^{-6} +2 q^{-10} + q^{-16} }[/math]
1,0,0	[math]\displaystyle{ -q^{21}-q^{17}+2 q^{11}+2 q^9+2 q^7+q^5-q-2 q^{-1} - q^{-5} + q^{-7} + q^{-9} + q^{-11} }[/math]

A4 Invariants.

Weight	Invariant
0,1,0,0	[math]\displaystyle{ q^{44}+q^{38}-q^{36}-3 q^{34}-2 q^{32}-q^{30}-3 q^{28}-q^{26}+3 q^{24}+5 q^{22}+2 q^{20}+3 q^{18}+4 q^{16}-q^{14}-2 q^{12}-q^8-q^6+2 q^4+3 q^2+1+ q^{-4} -3 q^{-8} - q^{-10} + q^{-12} + q^{-18} + q^{-20} + q^{-22} }[/math]
1,0,0,0	[math]\displaystyle{ -q^{26}-q^{22}-q^{20}+2 q^{14}+2 q^{12}+3 q^{10}+2 q^8+q^6-q^2-2-2 q^{-2} - q^{-6} + q^{-8} + q^{-10} + q^{-12} + q^{-14} }[/math]

B2 Invariants.

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

D4 Invariants.

Weight	Invariant
1,0,0,0	[math]\displaystyle{ q^{46}-q^{44}+q^{42}-2 q^{40}+2 q^{38}-2 q^{36}-2 q^{32}-q^{28}-q^{26}+q^{24}-q^{22}+3 q^{20}+5 q^{16}+5 q^{12}-q^{10}+3 q^8-q^6+q^4-2 q^2-1- q^{-2} -2 q^{-4} + q^{-6} -2 q^{-8} + q^{-10} - q^{-12} +3 q^{-14} + q^{-18} + q^{-22} }[/math]

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{80}-q^{78}+2 q^{76}-3 q^{74}+q^{72}-4 q^{68}+6 q^{66}-5 q^{64}+3 q^{62}-q^{60}-4 q^{58}+5 q^{56}-4 q^{54}+2 q^{50}-4 q^{48}+4 q^{46}-4 q^{42}+7 q^{40}-5 q^{38}+3 q^{36}-2 q^{32}+6 q^{30}-4 q^{28}+6 q^{26}-2 q^{24}+2 q^{22}+4 q^{20}-4 q^{18}+4 q^{16}-2 q^{14}+q^{12}+3 q^{10}-4 q^8+2 q^6-4 q^2+5-6 q^{-2} +2 q^{-6} -6 q^{-8} +5 q^{-10} -3 q^{-12} + q^{-14} + q^{-16} -3 q^{-18} +2 q^{-20} + q^{-24} + q^{-26} + q^{-32} + q^{-38} }[/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 44"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ t^2-4 t+7-4 t^{-1} + t^{-2} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

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

{ 17, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

[math]\displaystyle{ -z^2 a^4-a^4+z^4 a^2+3 z^2 a^2+3 a^2-2 z^2-2+ a^{-2} }[/math]

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ a^3 z^7+a z^7+2 a^4 z^6+3 a^2 z^6+z^6+a^5 z^5-2 a^3 z^5-3 a z^5-7 a^4 z^4-10 a^2 z^4-3 z^4-3 a^5 z^3-a^3 z^3+4 a z^3+2 z^3 a^{-1} +5 a^4 z^2+10 a^2 z^2+z^2 a^{-2} +6 z^2+a^5 z+a^3 z-a z-z a^{-1} -a^4-3 a^2- a^{-2} -2 }[/math]

Vassiliev invariants

V2 and V3:

(0, -1)

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{ 0 }[/math] [math]\displaystyle{ -8 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ -8 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ \frac{16}{3} }[/math] [math]\displaystyle{ \frac{64}{3} }[/math] [math]\displaystyle{ -8 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 48 }[/math] [math]\displaystyle{ -\frac{56}{3} }[/math] [math]\displaystyle{ \frac{104}{3} }[/math] [math]\displaystyle{ -\frac{16}{3} }[/math] [math]\displaystyle{ 0 }[/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 44. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \	-5	-4	-3	-2	-1	0	1	2	χ
5								1	1
3							1		-1
1						2	1		1
-1					2	2			0
-3				1	1				0
-5			1	2					1
-7		1	1						0
-9		1							1
-11	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, 44]]
Out[2]=	9
In[3]:=	PD[Knot[9, 44]]
Out[3]=	PD[X[1, 4, 2, 5], X[5, 10, 6, 11], X[3, 9, 4, 8], X[9, 3, 10, 2], X[14, 8, 15, 7], X[18, 15, 1, 16], X[16, 11, 17, 12], X[12, 17, 13, 18], X[6, 14, 7, 13]]
In[4]:=	GaussCode[Knot[9, 44]]
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, 44]]
Out[5]=	BR[4, {-1, -1, -1, -2, 1, 1, 3, -2, 3}]
In[6]:=	alex = Alexander[Knot[9, 44]][t]
Out[6]=	-2 4 2 7 + t - - - 4 t + t t
In[7]:=	Conway[Knot[9, 44]][z]
Out[7]=	4 1 + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[9, 44]}
In[9]:=	{KnotDet[Knot[9, 44]], KnotSignature[Knot[9, 44]]}
Out[9]=	{17, 0}
In[10]:=	J=Jones[Knot[9, 44]][q]
Out[10]=	-5 2 2 3 3 2 3 - q + -- - -- + -- - - - 2 q + q 4 3 2 q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[9, 44]}
In[12]:=	A2Invariant[Knot[9, 44]][q]
Out[12]=	-16 2 -6 -4 4 6 8 -1 - q + -- + q + q - q + q + q 8 q
In[13]:=	Kauffman[Knot[9, 44]][a, z]
Out[13]=	2 -2 2 4 z 3 5 2 z 2 2 -2 - a - 3 a - a - - - a z + a z + a z + 6 z + -- + 10 a z + a 2 a 3 4 2 2 z 3 3 3 5 3 4 2 4 5 a z + ---- + 4 a z - a z - 3 a z - 3 z - 10 a z - a 4 4 5 3 5 5 5 6 2 6 4 6 7 7 a z - 3 a z - 2 a z + a z + z + 3 a z + 2 a z + a z + 3 7 a z
In[14]:=	{Vassiliev[2][Knot[9, 44]], Vassiliev[3][Knot[9, 44]]}
Out[14]=	{0, -1}
In[15]:=	Kh[Knot[9, 44]][q, t]
Out[15]=	2 1 1 1 1 1 2 1 - + 2 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + q 11 5 9 4 7 4 7 3 5 3 5 2 3 2 q t q t q t q t q t q t q t 1 2 3 5 2 ---- + --- + q t + q t + q t 3 q t q t