10 3

10_2

10_4

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10 3 Quick Notes

10 3 Further Notes and Views

Knot presentations

Planar diagram presentation	X1627 X11,16,12,17 X5,13,6,12 X3,15,4,14 X13,5,14,4 X15,3,16,2 X7,20,8,1 X9,18,10,19 X17,10,18,11 X19,8,20,9
Gauss code	-1, 6, -4, 5, -3, 1, -7, 10, -8, 9, -2, 3, -5, 4, -6, 2, -9, 8, -10, 7
Dowker-Thistlethwaite code	6 14 12 20 18 16 4 2 10 8
Conway Notation	[64]

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 1 Bridge index 2 Super bridge index Missing Nakanishi index 1 Maximal Thurston-Bennequin number [-7][-5] Hyperbolic Volume 5.7321 A-Polynomial See Data:10 3/A-polynomial

[edit Notes for 10 3's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus [math]\displaystyle{ 0 }[/math] Topological 4 genus [math]\displaystyle{ 0 }[/math] Concordance genus [math]\displaystyle{ 0 }[/math] Rasmussen s-Invariant 0

[edit Notes for 10 3's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ -6 t+13-6 t^{-1} }[/math]
Conway polynomial	[math]\displaystyle{ 1-6 z^2 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 25, 0 }
Jones polynomial	[math]\displaystyle{ q^4-q^3+2 q^2-3 q+4-4 q^{-1} +3 q^{-2} -3 q^{-3} +2 q^{-4} - q^{-5} + q^{-6} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ a^6-z^2 a^4-2 z^2 a^2-a^2-2 z^2-z^2 a^{-2} + a^{-4} }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ a^3 z^9+a z^9+a^4 z^8+2 a^2 z^8+z^8+a^5 z^7-6 a^3 z^7-6 a z^7+z^7 a^{-1} +a^6 z^6-4 a^4 z^6-10 a^2 z^6+z^6 a^{-2} -4 z^6-4 a^5 z^5+15 a^3 z^5+15 a z^5-3 z^5 a^{-1} +z^5 a^{-3} -5 a^6 z^4+4 a^4 z^4+18 a^2 z^4-2 z^4 a^{-2} +z^4 a^{-4} +6 z^4+3 a^5 z^3-18 a^3 z^3-15 a z^3+4 z^3 a^{-1} -2 z^3 a^{-3} +6 a^6 z^2-2 a^4 z^2-12 a^2 z^2+z^2 a^{-2} -3 z^2 a^{-4} +6 a^3 z+6 a z-a^6+a^2+ a^{-4} }[/math]
The A2 invariant	[math]\displaystyle{ q^{20}+q^{18}+q^{14}-q^{10}-q^6-q^4+ q^{-2} - q^{-4} + q^{-8} + q^{-12} + q^{-14} }[/math]
The G2 invariant	Data:10 3/QuantumInvariant/G2/1,0

Further Quantum Invariants

Further quantum knot invariants for 10_3.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ q^{13}+q^9-q^7-q^3+ q^{-1} - q^{-3} + q^{-5} + q^{-9} }[/math]
2	[math]\displaystyle{ q^{38}+q^{32}-q^{30}-2 q^{28}+q^{26}-2 q^{22}+q^{20}+q^{18}-q^{16}+2 q^{12}+q^6+2 q^{-2} - q^{-4} - q^{-6} +2 q^{-8} - q^{-10} - q^{-12} - q^{-16} + q^{-20} + q^{-26} }[/math]
3	[math]\displaystyle{ q^{75}-q^{65}-q^{63}+q^{59}-q^{57}-2 q^{55}+3 q^{51}+2 q^{49}-2 q^{47}-2 q^{45}+q^{43}+3 q^{41}-q^{37}-q^{35}+q^{33}+2 q^{31}+q^{29}-3 q^{27}-q^{25}+2 q^{23}+2 q^{21}-2 q^{19}-2 q^{17}+q^{15}-q^{11}-q^9+q^7+q^3-q- q^{-1} +2 q^{-3} +2 q^{-5} -2 q^{-9} +3 q^{-13} + q^{-15} -2 q^{-19} +2 q^{-23} +2 q^{-25} - q^{-27} -2 q^{-29} + q^{-33} -2 q^{-37} - q^{-39} + q^{-43} + q^{-51} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ q^{20}+q^{18}+q^{14}-q^{10}-q^6-q^4+ q^{-2} - q^{-4} + q^{-8} + q^{-12} + q^{-14} }[/math]
2,0	[math]\displaystyle{ q^{52}+q^{50}+q^{48}+q^{44}+q^{42}-q^{40}-3 q^{38}-2 q^{36}-2 q^{30}-q^{28}+q^{26}+q^{24}-q^{20}+2 q^{18}+2 q^{16}+2 q^{10}+q^8+q^6+q^4+1+ q^{-2} + q^{-4} -2 q^{-6} -2 q^{-8} +2 q^{-10} -3 q^{-14} -2 q^{-16} - q^{-22} + q^{-26} + q^{-28} + q^{-32} + q^{-34} + q^{-36} }[/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 3"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ -6 t+13-6 t^{-1} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ 1-6 z^2 }[/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]=

{ 25, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ a^3 z^9+a z^9+a^4 z^8+2 a^2 z^8+z^8+a^5 z^7-6 a^3 z^7-6 a z^7+z^7 a^{-1} +a^6 z^6-4 a^4 z^6-10 a^2 z^6+z^6 a^{-2} -4 z^6-4 a^5 z^5+15 a^3 z^5+15 a z^5-3 z^5 a^{-1} +z^5 a^{-3} -5 a^6 z^4+4 a^4 z^4+18 a^2 z^4-2 z^4 a^{-2} +z^4 a^{-4} +6 z^4+3 a^5 z^3-18 a^3 z^3-15 a z^3+4 z^3 a^{-1} -2 z^3 a^{-3} +6 a^6 z^2-2 a^4 z^2-12 a^2 z^2+z^2 a^{-2} -3 z^2 a^{-4} +6 a^3 z+6 a z-a^6+a^2+ a^{-4} }[/math]

Vassiliev invariants

V2 and V3:

(-6, 3)

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{ -24 }[/math] [math]\displaystyle{ 24 }[/math] [math]\displaystyle{ 288 }[/math] [math]\displaystyle{ 372 }[/math] [math]\displaystyle{ 148 }[/math] [math]\displaystyle{ -576 }[/math] [math]\displaystyle{ -912 }[/math] [math]\displaystyle{ -192 }[/math] [math]\displaystyle{ -200 }[/math] [math]\displaystyle{ -2304 }[/math] [math]\displaystyle{ 288 }[/math] [math]\displaystyle{ -8928 }[/math] [math]\displaystyle{ -3552 }[/math] [math]\displaystyle{ -\frac{36071}{5} }[/math] [math]\displaystyle{ \frac{21692}{15} }[/math] [math]\displaystyle{ -\frac{99844}{15} }[/math] [math]\displaystyle{ \frac{2551}{3} }[/math] [math]\displaystyle{ -\frac{7751}{5} }[/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 10 3. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \	-6	-5	-4	-3	-2	-1	0	1	2	3	4	χ
9											1	1
7												0
5									2	1		1
3								1				-1
1							3	2				1
-1						2	2					0
-3					1	2						-1
-5				2	2							0
-7				1								-1
-9		1	2									1
-11												0
-13	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, 3]]
Out[2]=	10
In[3]:=	PD[Knot[10, 3]]
Out[3]=	PD[X[1, 6, 2, 7], X[11, 16, 12, 17], X[5, 13, 6, 12], X[3, 15, 4, 14], X[13, 5, 14, 4], X[15, 3, 16, 2], X[7, 20, 8, 1], X[9, 18, 10, 19], X[17, 10, 18, 11], X[19, 8, 20, 9]]
In[4]:=	GaussCode[Knot[10, 3]]
Out[4]=	GaussCode[-1, 6, -4, 5, -3, 1, -7, 10, -8, 9, -2, 3, -5, 4, -6, 2, -9, 8, -10, 7]
In[5]:=	BR[Knot[10, 3]]
Out[5]=	BR[6, {-1, -1, -2, 1, -2, -3, 2, 4, -3, 4, 5, -4, 5}]
In[6]:=	alex = Alexander[Knot[10, 3]][t]
Out[6]=	6 13 - - - 6 t t
In[7]:=	Conway[Knot[10, 3]][z]
Out[7]=	2 1 - 6 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 3]}
In[9]:=	{KnotDet[Knot[10, 3]], KnotSignature[Knot[10, 3]]}
Out[9]=	{25, 0}
In[10]:=	J=Jones[Knot[10, 3]][q]
Out[10]=	-6 -5 2 3 3 4 2 3 4 4 + q - q + -- - -- + -- - - - 3 q + 2 q - 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[10, 3]}
In[12]:=	A2Invariant[Knot[10, 3]][q]
Out[12]=	-20 -18 -14 -10 -6 -4 2 4 8 12 14 q + q + q - q - q - q + q - q + q + q + q
In[13]:=	Kauffman[Knot[10, 3]][a, z]
Out[13]=	2 2 -4 2 6 3 3 z z 2 2 4 2 a + a - a + 6 a z + 6 a z - ---- + -- - 12 a z - 2 a z + 4 2 a a 3 3 4 6 2 2 z 4 z 3 3 3 5 3 4 z 6 a z - ---- + ---- - 15 a z - 18 a z + 3 a z + 6 z + -- - 3 a 4 a a 4 5 5 2 z 2 4 4 4 6 4 z 3 z 5 ---- + 18 a z + 4 a z - 5 a z + -- - ---- + 15 a z + 2 3 a a a 6 7 3 5 5 5 6 z 2 6 4 6 6 6 z 15 a z - 4 a z - 4 z + -- - 10 a z - 4 a z + a z + -- - 2 a a 7 3 7 5 7 8 2 8 4 8 9 3 9 6 a z - 6 a z + a z + z + 2 a z + a z + a z + a z
In[14]:=	{Vassiliev[2][Knot[10, 3]], Vassiliev[3][Knot[10, 3]]}
Out[14]=	{0, 3}
In[15]:=	Kh[Knot[10, 3]][q, t]
Out[15]=	2 1 1 2 1 2 2 1 - + 3 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + q 13 6 9 5 9 4 7 3 5 3 5 2 3 2 q t q t q t q t q t q t q t 2 2 3 5 2 5 3 9 4 ---- + --- + 2 q t + q t + 2 q t + q t + q t 3 q t q t