L11a448

L11a447

L11a449

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Link Presentations

[edit Notes on L11a448's Link Presentations]

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

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(3)^3 t(2)^3-t(3)^3 t(2)^3-t(1) t(3)^2 t(2)^3+2 t(3)^2 t(2)^3-t(3) t(2)^3-2 t(1) t(3)^3 t(2)^2+t(3)^3 t(2)^2+2 t(1) t(3)^2 t(2)^2-2 t(3)^2 t(2)^2-t(1) t(3) t(2)^2+2 t(3) t(2)^2-t(2)^2+t(1) t(3)^3 t(2)-2 t(1) t(3)^2 t(2)+t(3)^2 t(2)-t(1) t(2)+2 t(1) t(3) t(2)-2 t(3) t(2)+2 t(2)+t(1) t(3)^2+t(1)-2 t(1) t(3)+t(3)-1}{\sqrt{t(1)} t(2)^{3/2} t(3)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^9+3 q^8-5 q^7+7 q^6-9 q^5+11 q^4-9 q^3+9 q^2+ q^{-2} -6 q-2 q^{-1} +5 }[/math] (db)
Signature	4 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^6 a^{-6} -4 z^4 a^{-6} -4 z^2 a^{-6} -2 a^{-6} +z^8 a^{-4} +6 z^6 a^{-4} +13 z^4 a^{-4} +14 z^2 a^{-4} + a^{-4} z^{-2} +7 a^{-4} -2 z^6 a^{-2} -10 z^4 a^{-2} -15 z^2 a^{-2} -2 a^{-2} z^{-2} -9 a^{-2} +z^4+4 z^2+ z^{-2} +4 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^{10} a^{-2} +z^{10} a^{-4} +2 z^9 a^{-1} +6 z^9 a^{-3} +4 z^9 a^{-5} +z^8 a^{-2} +6 z^8 a^{-4} +6 z^8 a^{-6} +z^8-10 z^7 a^{-1} -25 z^7 a^{-3} -9 z^7 a^{-5} +6 z^7 a^{-7} -25 z^6 a^{-2} -39 z^6 a^{-4} -14 z^6 a^{-6} +6 z^6 a^{-8} -6 z^6+14 z^5 a^{-1} +24 z^5 a^{-3} -3 z^5 a^{-5} -8 z^5 a^{-7} +5 z^5 a^{-9} +54 z^4 a^{-2} +56 z^4 a^{-4} +5 z^4 a^{-6} -7 z^4 a^{-8} +3 z^4 a^{-10} +13 z^4-3 z^3 a^{-1} +4 z^3 a^{-3} +9 z^3 a^{-5} -3 z^3 a^{-7} -4 z^3 a^{-9} +z^3 a^{-11} -41 z^2 a^{-2} -31 z^2 a^{-4} +2 z^2 a^{-8} -z^2 a^{-10} -13 z^2-5 z a^{-1} -8 z a^{-3} -3 z a^{-5} +z a^{-7} +z a^{-9} +13 a^{-2} +9 a^{-4} - a^{-8} +6+2 a^{-1} z^{-1} +2 a^{-3} z^{-1} -2 a^{-2} z^{-2} - a^{-4} 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   \ -4 -3 -2 -1 0 1 2 3 4 5 6 7 χ 19                       1 -1 17                     2   2 15                   3 1   -2 13                 4 2     2 11               6 4       -2 9             5 3         2 7           4 6           2 5         5 5             0 3       4 7               3 1     1 2                 -1 -1   1 4                   3 -3   1                     -1 -5 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=5 }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/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}^{4}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=7 }[/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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