L11a474

L11a473

L11a475

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Link Presentations

[edit Notes on L11a474's Link Presentations]

Planar diagram presentation	X6172 X10,4,11,3 X12,8,13,7 X20,14,21,13 X8,19,9,20 X18,9,19,10 X22,16,17,15 X16,18,5,17 X14,22,15,21 X2536 X4,12,1,11
Gauss code	{1, -10, 2, -11}, {8, -6, 5, -4, 9, -7}, {10, -1, 3, -5, 6, -2, 11, -3, 4, -9, 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(3)-1) \left(t(2)^2 t(3)^3-t(2) t(3)^3-t(2)^2 t(3)^2+t(2) t(3)^2-t(3)^2+t(2)^2 t(3)-t(2) t(3)+t(3)+t(2)-1\right)}{\sqrt{t(1)} t(2) t(3)^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^9+3 q^8-6 q^7+9 q^6-11 q^5+13 q^4-11 q^3+11 q^2-7 q+5-2 q^{-1} + q^{-2} }[/math] (db)
Signature	4 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^6 a^{-6} -4 z^4 a^{-6} -5 z^2 a^{-6} -3 a^{-6} +z^8 a^{-4} +6 z^6 a^{-4} +14 z^4 a^{-4} +17 z^2 a^{-4} + a^{-4} z^{-2} +9 a^{-4} -2 z^6 a^{-2} -10 z^4 a^{-2} -16 z^2 a^{-2} -2 a^{-2} z^{-2} -10 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} +7 z^9 a^{-3} +5 z^9 a^{-5} +2 z^8 a^{-2} +10 z^8 a^{-4} +9 z^8 a^{-6} +z^8-10 z^7 a^{-1} -28 z^7 a^{-3} -8 z^7 a^{-5} +10 z^7 a^{-7} -29 z^6 a^{-2} -54 z^6 a^{-4} -22 z^6 a^{-6} +9 z^6 a^{-8} -6 z^6+15 z^5 a^{-1} +27 z^5 a^{-3} -13 z^5 a^{-5} -19 z^5 a^{-7} +6 z^5 a^{-9} +61 z^4 a^{-2} +76 z^4 a^{-4} +12 z^4 a^{-6} -13 z^4 a^{-8} +3 z^4 a^{-10} +13 z^4-5 z^3 a^{-1} +3 z^3 a^{-3} +19 z^3 a^{-5} +6 z^3 a^{-7} -4 z^3 a^{-9} +z^3 a^{-11} -47 z^2 a^{-2} -44 z^2 a^{-4} -5 z^2 a^{-6} +5 z^2 a^{-8} -13 z^2-4 z a^{-1} -9 z a^{-3} -6 z a^{-5} +z a^{-9} +15 a^{-2} +13 a^{-4} +2 a^{-6} - 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                   4 1   -3 13                 5 2     3 11               6 4       -2 9             7 5         2 7           6 8           2 5         5 5             0 3       4 8               4 1     1 3                 -2 -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}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\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^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/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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