L11a512

L11a511

L11a513

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Link Presentations

[edit Notes on L11a512's Link Presentations]

Planar diagram presentation	X8192 X16,5,17,6 X22,15,13,16 X14,4,15,3 X4,14,5,13 X12,17,7,18 X10,19,11,20 X18,9,19,10 X20,11,21,12 X2738 X6,21,1,22
Gauss code	{1, -10, 4, -5, 2, -11}, {10, -1, 8, -7, 9, -6}, {5, -4, 3, -2, 6, -8, 7, -9, 11, -3}

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)^2 t(3)^3-t(2)^2 t(3)^3+t(1)^2 t(2) t(3)^3-2 t(1) t(2) t(3)^3+t(2) t(3)^3+t(1)^2 t(3)^2+t(1)^2 t(2)^2 t(3)^2-4 t(1) t(2)^2 t(3)^2+2 t(2)^2 t(3)^2-2 t(1) t(3)^2-3 t(1)^2 t(2) t(3)^2+7 t(1) t(2) t(3)^2-3 t(2) t(3)^2+t(3)^2-2 t(1)^2 t(3)-t(1)^2 t(2)^2 t(3)+2 t(1) t(2)^2 t(3)-t(2)^2 t(3)+4 t(1) t(3)+3 t(1)^2 t(2) t(3)-7 t(1) t(2) t(3)+3 t(2) t(3)-t(3)+t(1)^2-t(1)-t(1)^2 t(2)+2 t(1) t(2)-t(2)}{t(1) t(2) t(3)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ - q^{-11} +4 q^{-10} -8 q^{-9} +13 q^{-8} -17 q^{-7} +20 q^{-6} -18 q^{-5} +17 q^{-4} -11 q^{-3} +7 q^{-2} -3 q^{-1} +1 }[/math] (db)
Signature	-4 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^{10} \left(-z^2\right)-a^{10}+3 a^8 z^4+7 a^8 z^2+a^8 z^{-2} +4 a^8-2 a^6 z^6-7 a^6 z^4-10 a^6 z^2-2 a^6 z^{-2} -8 a^6-a^4 z^6-a^4 z^4+4 a^4 z^2+a^4 z^{-2} +5 a^4+a^2 z^4+2 a^2 z^2 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{13} z^5-a^{13} z^3+4 a^{12} z^6-6 a^{12} z^4+2 a^{12} z^2+7 a^{11} z^7-11 a^{11} z^5+5 a^{11} z^3-a^{11} z+7 a^{10} z^8-7 a^{10} z^6-a^{10} z^4+a^{10}+4 a^9 z^9+6 a^9 z^7-23 a^9 z^5+18 a^9 z^3-3 a^9 z+a^8 z^{10}+14 a^8 z^8-36 a^8 z^6+39 a^8 z^4-24 a^8 z^2-a^8 z^{-2} +8 a^8+8 a^7 z^9-9 a^7 z^7-6 a^7 z^5+14 a^7 z^3-8 a^7 z+2 a^7 z^{-1} +a^6 z^{10}+12 a^6 z^8-40 a^6 z^6+53 a^6 z^4-37 a^6 z^2-2 a^6 z^{-2} +12 a^6+4 a^5 z^9-5 a^5 z^7-3 a^5 z^5+7 a^5 z^3-6 a^5 z+2 a^5 z^{-1} +5 a^4 z^8-14 a^4 z^6+16 a^4 z^4-13 a^4 z^2-a^4 z^{-2} +6 a^4+3 a^3 z^7-8 a^3 z^5+5 a^3 z^3+a^2 z^6-3 a^2 z^4+2 a^2 z^2 }[/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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 1                       1 1 -1                     2   -2 -3                   5 1   4 -5                 7 3     -4 -7               10 4       6 -9             9 8         -1 -11           11 9           2 -13         7 10             3 -15       6 10               -4 -17     3 8                 5 -19   1 5                   -4 -21   3                     3 -23 1                       -1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-5 }[/math] [math]\displaystyle{ i=-3 }[/math] [math]\displaystyle{ r=-9 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=2 }[/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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