10 153: Difference between revisions

Revision as of 19:14, 28 August 2005

Visit 10 153's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 10 153's page at the original Knot Atlas!

10_153 is not [math]\displaystyle{ k }[/math]-colourable for any [math]\displaystyle{ k }[/math]. See The Determinant and the Signature.

Knot presentations

Planar diagram presentation	X₄₂₅₁ X₈₄₉₃ X_12,6,13,5 X_13,18,14,19 X_9,16,10,17 X_17,10,18,11 X_15,20,16,1 X_19,14,20,15 X_6,12,7,11 X₂₈₃₇
Gauss code	1, -10, 2, -1, 3, -9, 10, -2, -5, 6, 9, -3, -4, 8, -7, 5, -6, 4, -8, 7
Dowker-Thistlethwaite code	4 8 12 2 -16 6 -18 -20 -10 -14
Conway Notation	[(3,2)-(21,2)]

Three dimensional invariants

Symmetry type	Chiral
Unknotting number	2
3-genus	3
Bridge index	3
Super bridge index	Missing
Nakanishi index	1
Maximal Thurston-Bennequin number	[-6][-4]
Hyperbolic Volume	7.37434
A-Polynomial	See Data:10 153/A-polynomial

[edit Notes for 10 153'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 10 153's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ t^3-t^2-t+3- t^{-1} - t^{-2} + t^{-3} }[/math]
Conway polynomial	[math]\displaystyle{ z^6+5 z^4+4 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 1, 0 }
Jones polynomial	[math]\displaystyle{ -q^4+q^3-q^2+q+1+ q^{-2} - q^{-3} + q^{-4} - q^{-5} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ z^6-z^4 a^{-2} +6 z^4-a^4 z^2-a^2 z^2-4 z^2 a^{-2} +10 z^2-a^4-a^2-3 a^{-2} +6 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ z^8 a^{-2} +z^8+a z^7+2 z^7 a^{-1} +z^7 a^{-3} +a^4 z^6-6 z^6 a^{-2} -7 z^6+a^5 z^5+a^3 z^5-7 a z^5-13 z^5 a^{-1} -6 z^5 a^{-3} -4 a^4 z^4+10 z^4 a^{-2} +14 z^4-4 a^5 z^3-4 a^3 z^3+12 a z^3+22 z^3 a^{-1} +10 z^3 a^{-3} +3 a^4 z^2-2 a^2 z^2-7 z^2 a^{-2} -12 z^2+3 a^5 z+2 a^3 z-6 a z-10 z a^{-1} -5 z a^{-3} -a^4+a^2+3 a^{-2} +6 }[/math]
The A2 invariant	[math]\displaystyle{ -q^{16}-q^{12}-q^{10}+2 q^4+2 q^2+3+2 q^{-2} - q^{-8} - q^{-10} - q^{-12} }[/math]
The G2 invariant	[math]\displaystyle{ q^{80}+q^{76}-q^{74}+q^{70}-2 q^{68}+q^{64}-3 q^{62}+q^{60}-q^{58}-4 q^{56}+5 q^{54}-5 q^{52}+q^{50}+q^{48}-5 q^{46}+5 q^{44}-3 q^{42}-2 q^{40}+2 q^{38}-4 q^{36}+q^{34}+3 q^{32}-5 q^{30}+3 q^{28}-q^{24}+2 q^{22}-2 q^{20}+2 q^{18}+4 q^{14}-2 q^{12}+3 q^{10}+4 q^8-q^6+5 q^4-q^2+2+6 q^{-2} - q^{-4} +2 q^{-6} +4 q^{-8} -2 q^{-10} +6 q^{-12} - q^{-14} -3 q^{-16} +4 q^{-18} -4 q^{-20} +2 q^{-22} -3 q^{-26} + q^{-28} - q^{-30} -3 q^{-32} -2 q^{-36} - q^{-38} -3 q^{-42} - q^{-48} - q^{-52} + q^{-56} + q^{-60} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 10_153.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ -q^{11}+q^3+q+2 q^{-1} - q^{-9} }[/math]
2	[math]\displaystyle{ q^{32}-q^{28}+q^{24}-3 q^{20}-q^{18}+2 q^{16}-2 q^{14}-q^{12}+2 q^{10}+q^6+q^4+2 q^2+1+ q^{-2} +2 q^{-4} -2 q^{-8} + q^{-10} + q^{-12} -2 q^{-14} - q^{-16} - q^{-24} + q^{-28} }[/math]
3	[math]\displaystyle{ -q^{63}+q^{59}+q^{57}-2 q^{53}-q^{51}+q^{49}+4 q^{47}+3 q^{45}-2 q^{43}-5 q^{41}-2 q^{39}+5 q^{37}+4 q^{35}-4 q^{33}-7 q^{31}-q^{29}+5 q^{27}+q^{25}-4 q^{23}-2 q^{21}+q^{19}+q^{17}+q^{15}+q^9+q^7-q^5+q^3+5 q+4 q^{-1} -2 q^{-3} -3 q^{-5} +6 q^{-7} +5 q^{-9} -3 q^{-11} -7 q^{-13} - q^{-15} +5 q^{-17} +2 q^{-19} -3 q^{-21} -5 q^{-23} -2 q^{-25} +3 q^{-27} +3 q^{-29} -3 q^{-33} -2 q^{-35} + q^{-37} + q^{-39} + q^{-41} - q^{-47} + q^{-51} + q^{-53} - q^{-57} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ -q^{16}-q^{12}-q^{10}+2 q^4+2 q^2+3+2 q^{-2} - q^{-8} - q^{-10} - q^{-12} }[/math]
1,1	[math]\displaystyle{ q^{44}+2 q^{40}-2 q^{38}+2 q^{34}-4 q^{32}+6 q^{30}-8 q^{28}+4 q^{26}-2 q^{24}-4 q^{22}+2 q^{20}-12 q^{18}+4 q^{16}-6 q^{14}+3 q^{12}+6 q^8+4 q^6+5 q^4+8 q^2+10 q^{-2} -2 q^{-4} +4 q^{-6} -4 q^{-10} +4 q^{-12} -6 q^{-14} -2 q^{-16} -2 q^{-18} -4 q^{-20} +2 q^{-22} -4 q^{-24} +2 q^{-32} + q^{-36} }[/math]
2,0	[math]\displaystyle{ q^{42}+q^{36}+q^{34}+q^{32}-q^{28}-2 q^{26}-5 q^{24}-2 q^{22}-3 q^{20}-3 q^{18}-q^{16}+q^{12}+3 q^8+3 q^6+6 q^4+5 q^2+8+4 q^{-2} +2 q^{-4} + q^{-6} -2 q^{-8} -2 q^{-10} -2 q^{-12} -2 q^{-14} -2 q^{-16} -3 q^{-18} -2 q^{-20} - q^{-22} - q^{-24} + q^{-30} + q^{-32} + q^{-34} }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{34}+q^{30}-q^{24}-3 q^{22}-q^{20}-3 q^{18}-3 q^{16}-q^{14}-2 q^{12}-2 q^{10}+q^8+3 q^6+5 q^4+8 q^2+10+7 q^{-2} +4 q^{-4} - q^{-6} -2 q^{-8} -7 q^{-10} -5 q^{-12} -3 q^{-14} -3 q^{-16} + q^{-20} + q^{-22} + q^{-26} }[/math]
1,0,0	[math]\displaystyle{ -q^{21}-q^{17}-q^{15}-q^{13}-q^{11}+2 q^5+3 q^3+4 q+3 q^{-1} +3 q^{-3} - q^{-7} - q^{-9} -2 q^{-11} - q^{-13} - q^{-15} }[/math]

A4 Invariants.

Weight	Invariant
0,1,0,0	[math]\displaystyle{ q^{44}+q^{40}+2 q^{38}+q^{36}+q^{34}+q^{32}-q^{30}-3 q^{28}-3 q^{26}-5 q^{24}-6 q^{22}-7 q^{20}-6 q^{18}-7 q^{16}-8 q^{14}-2 q^{12}+q^{10}+4 q^8+13 q^6+19 q^4+20 q^2+20+17 q^{-2} +8 q^{-4} -8 q^{-8} -11 q^{-10} -15 q^{-12} -13 q^{-14} -8 q^{-16} -5 q^{-18} -2 q^{-20} +2 q^{-22} +3 q^{-24} +2 q^{-26} +2 q^{-28} + q^{-30} + q^{-32} }[/math]
1,0,0,0	[math]\displaystyle{ -q^{26}-q^{22}-q^{20}-q^{18}-q^{16}-q^{14}-q^{12}+2 q^6+3 q^4+5 q^2+4+4 q^{-2} +3 q^{-4} - q^{-8} -2 q^{-10} -2 q^{-12} -2 q^{-14} - q^{-16} - q^{-18} }[/math]

B2 Invariants.

Weight	Invariant
0,1	[math]\displaystyle{ -q^{34}-q^{30}-q^{24}+q^{22}-q^{20}+q^{18}-q^{16}+q^{14}+q^8-q^6+3 q^4+2+ q^{-2} +2 q^{-4} + q^{-6} + q^{-10} - q^{-12} + q^{-14} - q^{-16} - q^{-20} - q^{-22} - q^{-26} }[/math]
1,0	[math]\displaystyle{ q^{56}+q^{48}-q^{44}+q^{40}-q^{38}-2 q^{36}-q^{34}-q^{30}-3 q^{28}-2 q^{26}-2 q^{20}-q^{18}+q^{14}+q^{12}+q^{10}+2 q^8+4 q^6+3 q^4+4 q^2+3+4 q^{-2} +3 q^{-4} +3 q^{-6} - q^{-8} - q^{-14} -3 q^{-16} -3 q^{-18} - q^{-20} - q^{-22} -3 q^{-24} -2 q^{-26} + q^{-36} + q^{-44} }[/math]

D4 Invariants.

Weight	Invariant
1,0,0,0	[math]\displaystyle{ q^{46}+q^{42}+q^{38}-2 q^{34}-q^{32}-3 q^{30}-q^{28}-3 q^{26}-q^{24}-3 q^{22}-q^{20}-q^{18}-2 q^{16}-2 q^{14}-2 q^{12}+q^{10}+q^8+6 q^6+6 q^4+10 q^2+10+10 q^{-2} +6 q^{-4} +4 q^{-6} -4 q^{-10} -5 q^{-12} -7 q^{-14} -5 q^{-16} -6 q^{-18} -2 q^{-20} -2 q^{-22} + q^{-26} + q^{-28} + q^{-30} + q^{-34} }[/math]

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{80}+q^{76}-q^{74}+q^{70}-2 q^{68}+q^{64}-3 q^{62}+q^{60}-q^{58}-4 q^{56}+5 q^{54}-5 q^{52}+q^{50}+q^{48}-5 q^{46}+5 q^{44}-3 q^{42}-2 q^{40}+2 q^{38}-4 q^{36}+q^{34}+3 q^{32}-5 q^{30}+3 q^{28}-q^{24}+2 q^{22}-2 q^{20}+2 q^{18}+4 q^{14}-2 q^{12}+3 q^{10}+4 q^8-q^6+5 q^4-q^2+2+6 q^{-2} - q^{-4} +2 q^{-6} +4 q^{-8} -2 q^{-10} +6 q^{-12} - q^{-14} -3 q^{-16} +4 q^{-18} -4 q^{-20} +2 q^{-22} -3 q^{-26} + q^{-28} - q^{-30} -3 q^{-32} -2 q^{-36} - q^{-38} -3 q^{-42} - q^{-48} - q^{-52} + q^{-56} + q^{-60} }[/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.

In[3]:= K = Knot["10 153"];

In[4]:= Alexander[K][t]

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

Out[4]= [math]\displaystyle{ t^3-t^2-t+3- t^{-1} - t^{-2} + t^{-3} }[/math]

In[5]:= Conway[K][z]

Out[5]= [math]\displaystyle{ z^6+5 z^4+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]= { 1, 0 }

In[8]:= Jones[K][q]

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

Out[8]= [math]\displaystyle{ -q^4+q^3-q^2+q+1+ q^{-2} - q^{-3} + 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^6-z^4 a^{-2} +6 z^4-a^4 z^2-a^2 z^2-4 z^2 a^{-2} +10 z^2-a^4-a^2-3 a^{-2} +6 }[/math]

In[10]:= Kauffman[K][a, z]

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

Out[10]= [math]\displaystyle{ z^8 a^{-2} +z^8+a z^7+2 z^7 a^{-1} +z^7 a^{-3} +a^4 z^6-6 z^6 a^{-2} -7 z^6+a^5 z^5+a^3 z^5-7 a z^5-13 z^5 a^{-1} -6 z^5 a^{-3} -4 a^4 z^4+10 z^4 a^{-2} +14 z^4-4 a^5 z^3-4 a^3 z^3+12 a z^3+22 z^3 a^{-1} +10 z^3 a^{-3} +3 a^4 z^2-2 a^2 z^2-7 z^2 a^{-2} -12 z^2+3 a^5 z+2 a^3 z-6 a z-10 z a^{-1} -5 z a^{-3} -a^4+a^2+3 a^{-2} +6 }[/math]

Vassiliev invariants

V₂ and V₃:

(4, -1)

V_2,1 through V_6,9:

V_2,1

V_3,1

V_4,1

V_4,2

V_4,3

V_5,1

V_5,2

V_5,3

V_5,4

V_6,1

V_6,2

V_6,3

V_6,4

V_6,5

V_6,6

V_6,7

V_6,8

V_6,9

[math]\displaystyle{ 16 }[/math]

[math]\displaystyle{ -8 }[/math]

[math]\displaystyle{ 128 }[/math]

[math]\displaystyle{ \frac{488}{3} }[/math]

[math]\displaystyle{ \frac{64}{3} }[/math]

[math]\displaystyle{ -128 }[/math]

[math]\displaystyle{ -\frac{560}{3} }[/math]

[math]\displaystyle{ -\frac{128}{3} }[/math]

[math]\displaystyle{ -8 }[/math]

[math]\displaystyle{ \frac{2048}{3} }[/math]

[math]\displaystyle{ 32 }[/math]

[math]\displaystyle{ \frac{7808}{3} }[/math]

[math]\displaystyle{ \frac{1024}{3} }[/math]

[math]\displaystyle{ \frac{41222}{15} }[/math]

[math]\displaystyle{ \frac{464}{5} }[/math]

[math]\displaystyle{ \frac{44528}{45} }[/math]

[math]\displaystyle{ \frac{442}{9} }[/math]

[math]\displaystyle{ \frac{1622}{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]0 is the signature of 10 153. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

-5

-4

-3

-2

-1

0

1

2

3

4

5

χ

9

1

-1

7

0

5

1

0

3

1

0

1

2

-1

1

3

1

-3

1

-5

1

0

-7

1

0

-9

0

-11

1

-1

Integral Khovanov Homology

(db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math]	[math]\displaystyle{ i=-3 }[/math]	[math]\displaystyle{ i=-1 }[/math]	[math]\displaystyle{ i=1 }[/math]
[math]\displaystyle{ r=-5 }[/math]		[math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-4 }[/math]		[math]\displaystyle{ {\mathbb Z}_2 }[/math]	[math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-3 }[/math]		[math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-2 }[/math]		[math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math]	[math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-1 }[/math]	[math]\displaystyle{ {\mathbb Z} }[/math]	[math]\displaystyle{ {\mathbb Z}_2 }[/math]	[math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=0 }[/math]	[math]\displaystyle{ {\mathbb Z}_2 }[/math]	[math]\displaystyle{ {\mathbb Z}^{3} }[/math]	[math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=1 }[/math]	[math]\displaystyle{ {\mathbb Z} }[/math]	[math]\displaystyle{ {\mathbb Z}_2 }[/math]	[math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=2 }[/math]	[math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math]	[math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=3 }[/math]	[math]\displaystyle{ {\mathbb Z}_2 }[/math]	[math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=4 }[/math]	[math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=5 }[/math]	[math]\displaystyle{ {\mathbb Z}_2 }[/math]	[math]\displaystyle{ {\mathbb Z} }[/math]

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, 153]]
Out[2]=	10
In[3]:=	PD[Knot[10, 153]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 4, 9, 3], X[12, 6, 13, 5], X[13, 18, 14, 19], X[9, 16, 10, 17], X[17, 10, 18, 11], X[15, 20, 16, 1], X[19, 14, 20, 15], X[6, 12, 7, 11], X[2, 8, 3, 7]]
In[4]:=	GaussCode[Knot[10, 153]]
Out[4]=	GaussCode[1, -10, 2, -1, 3, -9, 10, -2, -5, 6, 9, -3, -4, 8, -7, 5, -6, 4, -8, 7]
In[5]:=	BR[Knot[10, 153]]
Out[5]=	BR[4, {-1, -1, -1, -2, -1, -1, 3, 2, 2, 2, 3}]
In[6]:=	alex = Alexander[Knot[10, 153]][t]
Out[6]=	-3 -2 1 2 3 3 + t - t - - - t - t + t t
In[7]:=	Conway[Knot[10, 153]][z]
Out[7]=	2 4 6 1 + 4 z + 5 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 153]}
In[9]:=	{KnotDet[Knot[10, 153]], KnotSignature[Knot[10, 153]]}
Out[9]=	{1, 0}
In[10]:=	J=Jones[Knot[10, 153]][q]
Out[10]=	-5 -4 -3 -2 2 3 4 1 - q + q - q + q + q - q + q - q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] \|\| (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[10, 153]}
In[12]:=	A2Invariant[Knot[10, 153]][q]
Out[12]=	-16 -12 -10 2 2 2 8 10 12 3 - q - q - q + -- + -- + 2 q - q - q - q 4 2 q q
In[13]:=	Kauffman[Knot[10, 153]][a, z]
Out[13]=	3 2 4 5 z 10 z 3 5 2 6 + -- + a - a - --- - ---- - 6 a z + 2 a z + 3 a z - 12 z - 2 3 a a a 2 3 3 7 z 2 2 4 2 10 z 22 z 3 3 3 ---- - 2 a z + 3 a z + ----- + ----- + 12 a z - 4 a z - 2 3 a a a 4 5 5 5 3 4 10 z 4 4 6 z 13 z 5 3 5 4 a z + 14 z + ----- - 4 a z - ---- - ----- - 7 a z + a z + 2 3 a a a 6 7 7 8 5 5 6 6 z 4 6 z 2 z 7 8 z a z - 7 z - ---- + a z + -- + ---- + a z + z + -- 2 3 a 2 a a a
In[14]:=	{Vassiliev[2][Knot[10, 153]], Vassiliev[3][Knot[10, 153]]}
Out[14]=	{0, -1}
In[15]:=	Kh[Knot[10, 153]][q, t]
Out[15]=	3 1 1 1 1 1 1 1 t - + q + ------ + ----- + ----- + ----- + ----- + ---- + --- + - + q 11 5 7 4 7 3 5 2 3 2 5 q t q q t q t q t q t q t q t 3 2 3 2 5 3 5 4 9 5 q t + q t + q t + q t + q t + q t

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+<!--  -->
+<!-- -->
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 <span id="top"></span>
+<!-- -->
 <!-- this relies on transclusion for next and previous links -->
 {{Knot Navigation Links|ext=gif}}
+{{Rolfsen Knot Page Header|n=10|k=153|KnotilusURL=http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/1,-10,2,-1,3,-9,10,-2,-5,6,9,-3,-4,8,-7,5,-6,4,-8,7/goTop.html}}
-{| align=left
-|- valign=top
-|[[Image:{{PAGENAME}}.gif]]
-|{{Rolfsen Knot Site Links|n=10|k=153|KnotilusURL=http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/1,-10,2,-1,3,-9,10,-2,-5,6,9,-3,-4,8,-7,5,-6,4,-8,7/goTop.html}}
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 {{Vassiliev Invariants}}
-===[[Khovanov Homology]]===
+{{Khovanov Homology|table=<table border=1>
-The coefficients of the monomials <math>t^rq^j</math> are shown, along with their alternating sums <math>\chi</math> (fixed <math>j</math>, alternation over <math>r</math>). The squares with <font class=HLYellow>yellow</font> highlighting are those on the "critical diagonals", where <math>j-2r=s+1</math> or <math>j-2r=s+1</math>, where <math>s=</math>{{Data:{{PAGENAME}}/Signature}} is the signature of {{PAGENAME}}. Nonzero entries off the critical diagonals (if any exist) are highlighted in <font class=HLRed>red</font>.
-<center><table border=1>
 <tr align=center>
 <td width=13.3333%><table cellpadding=0 cellspacing=0>
@@ Line 47: / Line 40: @@
 <tr align=center><td>-9</td><td bgcolor=yellow>&nbsp;</td><td bgcolor=yellow>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>0</td></tr>
 <tr align=center><td>-11</td><td bgcolor=yellow>1</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>-1</td></tr>
-</table></center>
+</table>}}
 {{Computer Talk Header}}
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   q  t + q t  + q  t  + q  t  + q  t  + q  t</nowiki></pre></td></tr>
 </table>
+ [[Category:Knot Page]]

10 153: Difference between revisions

Revision as of 19:14, 28 August 2005

Contents

Knot presentations

Three dimensional invariants

Four dimensional invariants

Polynomial invariants

Vassiliev invariants

Khovanov Homology

Computer Talk

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