L11a480

L11a479

L11a481

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Link Presentations

[edit Notes on L11a480's Link Presentations]

Planar diagram presentation	X6172 X10,4,11,3 X18,7,19,8 X14,21,15,22 X20,10,21,9 X8,13,9,14 X22,15,17,16 X16,17,5,18 X12,20,13,19 X2536 X4,12,1,11
Gauss code	{1, -10, 2, -11}, {8, -3, 9, -5, 4, -7}, {10, -1, 3, -6, 5, -2, 11, -9, 6, -4, 7, -8}

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)-1) (t(2)-1) (t(3)-1)^3 (t(2) t(3)+1)}{\sqrt{t(1)} t(2) t(3)^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^3-4 q^2+9 q-13+19 q^{-1} -20 q^{-2} +21 q^{-3} -17 q^{-4} +13 q^{-5} -7 q^{-6} +3 q^{-7} - q^{-8} }[/math] (db)
Signature	-2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a^6 z^4-3 a^6 z^2-2 a^6+2 a^4 z^6+8 a^4 z^4+11 a^4 z^2+a^4 z^{-2} +6 a^4-a^2 z^8-5 a^2 z^6-10 a^2 z^4-11 a^2 z^2-2 a^2 z^{-2} -7 a^2+z^6+3 z^4+3 z^2+ z^{-2} +3 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ 2 a^4 z^{10}+2 a^2 z^{10}+5 a^5 z^9+12 a^3 z^9+7 a z^9+6 a^6 z^8+10 a^4 z^8+12 a^2 z^8+8 z^8+5 a^7 z^7-2 a^5 z^7-26 a^3 z^7-15 a z^7+4 z^7 a^{-1} +3 a^8 z^6-6 a^6 z^6-32 a^4 z^6-47 a^2 z^6+z^6 a^{-2} -23 z^6+a^9 z^5-6 a^7 z^5-5 a^5 z^5+18 a^3 z^5+7 a z^5-9 z^5 a^{-1} -5 a^8 z^4+a^6 z^4+39 a^4 z^4+56 a^2 z^4-2 z^4 a^{-2} +21 z^4-2 a^9 z^3+a^7 z^3+6 a^5 z^3+a^3 z^3+a z^3+3 z^3 a^{-1} +3 a^8 z^2-23 a^4 z^2-30 a^2 z^2-10 z^2+a^9 z+a^7 z-3 a^5 z-6 a^3 z-3 a z-a^8+7 a^4+9 a^2+4+2 a^3 z^{-1} +2 a z^{-1} -a^4 z^{-2} -2 a^2 z^{-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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 7                       1 1 5                     3   -3 3                   6 1   5 1                 7 3     -4 -1               12 6       6 -3             10 9         -1 -5           11 10           1 -7         8 12             4 -9       5 9               -4 -11     2 8                 6 -13   1 5                   -4 -15   2                     2 -17 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{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=4 }[/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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