L11a333

L11a332

L11a334

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Link Presentations

[edit Notes on L11a333's Link Presentations]

Planar diagram presentation	X10,1,11,2 X12,4,13,3 X20,5,21,6 X16,7,17,8 X8,9,1,10 X18,12,19,11 X6,15,7,16 X4,14,5,13 X22,18,9,17 X2,19,3,20 X14,22,15,21
Gauss code	{1, -10, 2, -8, 3, -7, 4, -5}, {5, -1, 6, -2, 8, -11, 7, -4, 9, -6, 10, -3, 11, -9}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in [math]\displaystyle{ u }[/math], [math]\displaystyle{ v }[/math], [math]\displaystyle{ w }[/math], ...)	[math]\displaystyle{ -\frac{(t(1) t(2)+1) \left(t(1)^2 t(2)^4-t(1) t(2)^4-3 t(1)^2 t(2)^3+5 t(1) t(2)^3-t(2)^3+3 t(1)^2 t(2)^2-7 t(1) t(2)^2+3 t(2)^2-t(1)^2 t(2)+5 t(1) t(2)-3 t(2)-t(1)+1\right)}{t(1)^{3/2} t(2)^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{9/2}-4 q^{7/2}+9 q^{5/2}-14 q^{3/2}+19 \sqrt{q}-\frac{23}{\sqrt{q}}+\frac{22}{q^{3/2}}-\frac{20}{q^{5/2}}+\frac{14}{q^{7/2}}-\frac{9}{q^{9/2}}+\frac{4}{q^{11/2}}-\frac{1}{q^{13/2}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^3 z^7+4 a^3 z^5+5 a^3 z^3+3 a^3 z+2 a^3 z^{-1} -a z^9-6 a z^7+z^7 a^{-1} -13 a z^5+4 z^5 a^{-1} -13 a z^3+5 z^3 a^{-1} -8 a z+3 z a^{-1} -3 a z^{-1} + a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -3 a^2 z^{10}-3 z^{10}-8 a^3 z^9-15 a z^9-7 z^9 a^{-1} -10 a^4 z^8-9 a^2 z^8-7 z^8 a^{-2} -6 z^8-8 a^5 z^7+11 a^3 z^7+37 a z^7+14 z^7 a^{-1} -4 z^7 a^{-3} -4 a^6 z^6+18 a^4 z^6+32 a^2 z^6+16 z^6 a^{-2} -z^6 a^{-4} +27 z^6-a^7 z^5+13 a^5 z^5-5 a^3 z^5-37 a z^5-9 z^5 a^{-1} +9 z^5 a^{-3} +5 a^6 z^4-13 a^4 z^4-34 a^2 z^4-9 z^4 a^{-2} +2 z^4 a^{-4} -27 z^4+a^7 z^3-6 a^5 z^3+a^3 z^3+19 a z^3+7 z^3 a^{-1} -4 z^3 a^{-3} +4 a^4 z^2+13 a^2 z^2+3 z^2 a^{-2} -z^2 a^{-4} +13 z^2+2 a^5 z-4 a^3 z-9 a z-3 z a^{-1} -3 a^2- a^{-2} -3+2 a^3 z^{-1} +3 a z^{-1} + a^{-1} z^{-1} }[/math] (db)

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]).

\   r    \    j   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 10                       1 -1 8                     3   3 6                   6 1   -5 4                 8 3     5 2               11 6       -5 0             12 8         4 -2           11 12           1 -4         9 11             -2 -6       5 11               6 -8     4 9                 -5 -10   1 6                   5 -12   3                     -3 -14 1                       1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-2 }[/math] [math]\displaystyle{ i=0 }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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.

Modifying This Page

Read me first: Modifying Knot Pages See/edit the Link Page master template (intermediate). See/edit the Link_Splice_Base (expert). Back to the top.

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