9 42

9_41

9_43

Visit 9 42's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 42's page at Knotilus! Visit 9 42's page at the original Knot Atlas!

9 42 Quick Notes

9_42 is Alexander Stoimenow's favourite knot!

Alsacian chair, alsacian museum, Strasbourg, France

Knot presentations

Planar diagram presentation	X1425 X5,10,6,11 X3948 X9,3,10,2 X16,12,17,11 X14,7,15,8 X6,15,7,16 X18,14,1,13 X12,18,13,17
Gauss code	-1, 4, -3, 1, -2, -7, 6, 3, -4, 2, 5, -9, 8, -6, 7, -5, 9, -8
Dowker-Thistlethwaite code	4 8 10 -14 2 -16 -18 -6 -12
Conway Notation	[22,3,2-]

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 2 Bridge index 3 Super bridge index 4 Nakanishi index 1 Maximal Thurston-Bennequin number [-3][-5] Hyperbolic Volume 4.05686 A-Polynomial See Data:9 42/A-polynomial

[edit Notes for 9 42'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 0

[edit Notes for 9 42's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ -t^2+2 t-1+2 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	{ 7, 2 }
Jones polynomial	[math]\displaystyle{ q^3-q^2+q-1+ q^{-1} - q^{-2} + q^{-3} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ -z^4+a^2 z^2+z^2 a^{-2} -4 z^2+2 a^2+2 a^{-2} -3 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ a z^7+z^7 a^{-1} +a^2 z^6+z^6 a^{-2} +2 z^6-5 a z^5-5 z^5 a^{-1} -5 a^2 z^4-5 z^4 a^{-2} -10 z^4+6 a z^3+6 z^3 a^{-1} +6 a^2 z^2+6 z^2 a^{-2} +12 z^2-2 a z-2 z a^{-1} -2 a^2-2 a^{-2} -3 }[/math]
The A2 invariant	[math]\displaystyle{ q^{10}+q^8+q^6-q^2-1- q^{-2} + q^{-6} + q^{-8} + q^{-10} }[/math]
The G2 invariant	[math]\displaystyle{ q^{46}+q^{42}+2 q^{32}+q^{26}+q^{24}+q^{22}+q^{20}-q^{18}+q^{16}+q^{14}-q^{12}+q^{10}-q^8-q^4-2 q^2-1- q^{-2} - q^{-4} -2 q^{-6} - q^{-8} - q^{-10} + q^{-12} - q^{-14} - q^{-16} + q^{-20} + q^{-22} + q^{-24} + q^{-26} +3 q^{-30} + q^{-34} + q^{-36} + q^{-40} + q^{-46} - q^{-50} - q^{-54} + q^{-56} - q^{-60} + q^{-62} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 9_42.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ q^7+ q^{-7} }[/math]
2	[math]\displaystyle{ q^{22}-q^{18}+q^6+1+ q^{-6} - q^{-18} + q^{-22} }[/math]
3	[math]\displaystyle{ q^{45}-q^{41}-q^{39}+q^{35}+q^{23}+q^{21}-q^{19}-q^{17}+q^{13}-q^9+q^5+q^3+ q^{-3} + q^{-5} - q^{-13} - q^{-15} + q^{-21} +2 q^{-23} - q^{-27} +2 q^{-31} -3 q^{-35} - q^{-37} + q^{-39} + q^{-41} }[/math]
4	[math]\displaystyle{ q^{76}-q^{72}-q^{70}-q^{68}+q^{66}+q^{64}+q^{62}-q^{58}+q^{48}+q^{46}-2 q^{42}-2 q^{40}+q^{36}+2 q^{34}-q^{30}-q^{28}+2 q^{24}+q^{22}+q^{20}-q^{18}-2 q^{16}+q^{12}+2 q^{10}-2 q^6-q^4+2+ q^{-2} - q^{-4} + q^{-8} +2 q^{-10} + q^{-12} - q^{-14} + q^{-20} - q^{-24} - q^{-34} -2 q^{-36} + q^{-40} +2 q^{-42} +2 q^{-44} - q^{-46} - q^{-48} -2 q^{-50} + q^{-52} +3 q^{-54} -2 q^{-60} - q^{-62} + q^{-68} }[/math]
5	[math]\displaystyle{ q^{115}-q^{111}-q^{109}-q^{107}+q^{103}+2 q^{101}+q^{99}-q^{95}-q^{93}-q^{91}+q^{87}+q^{81}+q^{79}-q^{75}-3 q^{73}-2 q^{71}+2 q^{67}+3 q^{65}+2 q^{63}-2 q^{59}-2 q^{57}-q^{55}+q^{53}+2 q^{51}+2 q^{49}+q^{47}-q^{45}-2 q^{43}-3 q^{41}-2 q^{39}+q^{37}+3 q^{35}+3 q^{33}+q^{31}-2 q^{29}-4 q^{27}-2 q^{25}+q^{23}+3 q^{21}+4 q^{19}+q^{17}-2 q^{15}-3 q^{13}-q^{11}+2 q^9+3 q^7+2 q^5-q^3-3 q-2 q^{-1} + q^{-3} +3 q^{-5} +2 q^{-7} -2 q^{-11} - q^{-13} + q^{-15} +2 q^{-17} + q^{-19} - q^{-21} - q^{-23} + q^{-27} - q^{-31} - q^{-33} + q^{-37} + q^{-39} + q^{-55} +2 q^{-57} + q^{-59} -2 q^{-61} -4 q^{-63} -4 q^{-65} +5 q^{-69} +6 q^{-71} + q^{-73} -5 q^{-75} -6 q^{-77} -3 q^{-79} +3 q^{-81} +6 q^{-83} +5 q^{-85} -3 q^{-89} -3 q^{-91} -2 q^{-93} + q^{-97} + q^{-99} }[/math]

A2 Invariants.

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

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{20}+q^{16}+q^{14}+q^{12}-q^4- q^{-4} + q^{-12} + q^{-14} + q^{-16} + q^{-20} }[/math]
1,0,0	[math]\displaystyle{ q^{13}+q^{11}+2 q^9+q^7-q^3-2 q-2 q^{-1} - q^{-3} + q^{-7} +2 q^{-9} + q^{-11} + q^{-13} }[/math]

A4 Invariants.

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

B2 Invariants.

Weight	Invariant
0,1	[math]\displaystyle{ q^{20}+q^{16}+q^{14}+q^{12}-q^4-2- q^{-4} + q^{-12} + q^{-14} + q^{-16} + q^{-20} }[/math]
1,0	[math]\displaystyle{ q^{34}+q^{26}+q^{18}-q^{14}+1- q^{-14} + q^{-18} + q^{-26} + q^{-34} }[/math]

D4 Invariants.

Weight	Invariant
1,0,0,0	[math]\displaystyle{ q^{26}+q^{22}+q^{20}+2 q^{18}+q^{16}+q^{14}+q^{12}-q^6-q^4-2 q^2-2 q^{-2} - q^{-4} - q^{-6} + q^{-12} + q^{-14} + q^{-16} +2 q^{-18} + q^{-20} + q^{-22} + q^{-26} }[/math]

G2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ q^{46}+q^{42}+2 q^{32}+q^{26}+q^{24}+q^{22}+q^{20}-q^{18}+q^{16}+q^{14}-q^{12}+q^{10}-q^8-q^4-2 q^2-1- q^{-2} - q^{-4} -2 q^{-6} - q^{-8} - q^{-10} + q^{-12} - q^{-14} - q^{-16} + q^{-20} + q^{-22} + q^{-24} + q^{-26} +3 q^{-30} + q^{-34} + q^{-36} + q^{-40} + q^{-46} - q^{-50} - q^{-54} + q^{-56} - q^{-60} + q^{-62} }[/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 42"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ -t^2+2 t-1+2 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]=

{ 7, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ a z^7+z^7 a^{-1} +a^2 z^6+z^6 a^{-2} +2 z^6-5 a z^5-5 z^5 a^{-1} -5 a^2 z^4-5 z^4 a^{-2} -10 z^4+6 a z^3+6 z^3 a^{-1} +6 a^2 z^2+6 z^2 a^{-2} +12 z^2-2 a z-2 z a^{-1} -2 a^2-2 a^{-2} -3 }[/math]

Vassiliev invariants

V2 and V3:

(-2, 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{ -8 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ \frac{164}{3} }[/math] [math]\displaystyle{ \frac{76}{3} }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ -\frac{256}{3} }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ -\frac{1312}{3} }[/math] [math]\displaystyle{ -\frac{608}{3} }[/math] [math]\displaystyle{ -\frac{6271}{15} }[/math] [math]\displaystyle{ \frac{1484}{15} }[/math] [math]\displaystyle{ -\frac{19564}{45} }[/math] [math]\displaystyle{ \frac{607}{9} }[/math] [math]\displaystyle{ -\frac{1471}{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 42. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \	-4	-3	-2	-1	0	1	2	χ
7							1	1
5								0
3					1	1		0
1				1	1			0
-1				1	1			0
-3		1	1					0
-5								0
-7	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, 42]]
Out[2]=	9
In[3]:=	PD[Knot[9, 42]]
Out[3]=	PD[X[1, 4, 2, 5], X[5, 10, 6, 11], X[3, 9, 4, 8], X[9, 3, 10, 2], X[16, 12, 17, 11], X[14, 7, 15, 8], X[6, 15, 7, 16], X[18, 14, 1, 13], X[12, 18, 13, 17]]
In[4]:=	GaussCode[Knot[9, 42]]
Out[4]=	GaussCode[-1, 4, -3, 1, -2, -7, 6, 3, -4, 2, 5, -9, 8, -6, 7, -5, 9, -8]
In[5]:=	BR[Knot[9, 42]]
Out[5]=	BR[4, {1, 1, 1, -2, -1, -1, 3, -2, 3}]
In[6]:=	alex = Alexander[Knot[9, 42]][t]
Out[6]=	-2 2 2 -1 - t + - + 2 t - t t
In[7]:=	Conway[Knot[9, 42]][z]
Out[7]=	2 4 1 - 2 z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[9, 42]}
In[9]:=	{KnotDet[Knot[9, 42]], KnotSignature[Knot[9, 42]]}
Out[9]=	{7, 2}
In[10]:=	J=Jones[Knot[9, 42]][q]
Out[10]=	-3 -2 1 2 3 -1 + q - q + - + q - q + q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[9, 42]}
In[12]:=	A2Invariant[Knot[9, 42]][q]
Out[12]=	-10 -8 -6 -2 2 6 8 10 -1 + q + q + q - q - q + q + q + q
In[13]:=	Kauffman[Knot[9, 42]][a, z]
Out[13]=	2 3 2 2 2 z 2 6 z 2 2 6 z 3 -3 - -- - 2 a - --- - 2 a z + 12 z + ---- + 6 a z + ---- + 6 a z - 2 a 2 a a a 4 5 6 7 4 5 z 2 4 5 z 5 6 z 2 6 z 7 10 z - ---- - 5 a z - ---- - 5 a z + 2 z + -- + a z + -- + a z 2 a 2 a a a
In[14]:=	{Vassiliev[2][Knot[9, 42]], Vassiliev[3][Knot[9, 42]]}
Out[14]=	{0, 0}
In[15]:=	Kh[Knot[9, 42]][q, t]
Out[15]=	1 3 1 1 1 1 q 3 7 2 - + q + q + ----- + ----- + ----- + --- + - + q t + q t q 7 4 3 3 3 2 q t t q t q t q t