L11n435

L11n434

L11n436

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Link Presentations

[edit Notes on L11n435's Link Presentations]

Planar diagram presentation	X8192 X14,4,15,3 X12,14,7,13 X2738 X22,10,13,9 X6,22,1,21 X20,16,21,15 X16,5,17,6 X11,19,12,18 X17,11,18,10 X4,19,5,20
Gauss code	{1, -4, 2, -11, 8, -6}, {4, -1, 5, 10, -9, -3}, {3, -2, 7, -8, -10, 9, 11, -7, 6, -5}

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)^2 t(3)^3+t(1) t(3)^3+t(1)^2 t(2) t(3)^3-2 t(1) t(2) t(3)^3+t(2) t(3)^3+2 t(1)^2 t(3)^2-2 t(1) t(2)^2 t(3)^2+t(2)^2 t(3)^2-2 t(1) t(3)^2-2 t(1)^2 t(2) t(3)^2+5 t(1) t(2) t(3)^2-2 t(2) t(3)^2-t(1)^2 t(3)+2 t(1) t(2)^2 t(3)-2 t(2)^2 t(3)+2 t(1) t(3)+2 t(1)^2 t(2) t(3)-5 t(1) t(2) t(3)+2 t(2) t(3)-t(1) t(2)^2+t(2)^2-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^8+4 q^7-8 q^6+12 q^5-14 q^4+16 q^3-13 q^2+11 q+3 q^{-1} -6 }[/math] (db)
Signature	2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^6 a^{-4} -4 z^4 a^{-2} +3 z^4 a^{-4} -z^4 a^{-6} -10 z^2 a^{-2} +6 z^2 a^{-4} -z^2 a^{-6} +3 z^2-8 a^{-2} +5 a^{-4} - a^{-6} +4-2 a^{-2} z^{-2} + a^{-4} z^{-2} + z^{-2} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^5 a^{-9} -z^3 a^{-9} +4 z^6 a^{-8} -6 z^4 a^{-8} +z^2 a^{-8} +7 z^7 a^{-7} -12 z^5 a^{-7} +4 z^3 a^{-7} -z a^{-7} +7 z^8 a^{-6} -12 z^6 a^{-6} +7 z^4 a^{-6} -4 z^2 a^{-6} + a^{-6} +3 z^9 a^{-5} +4 z^7 a^{-5} -17 z^5 a^{-5} +14 z^3 a^{-5} -3 z a^{-5} +13 z^8 a^{-4} -34 z^6 a^{-4} +41 z^4 a^{-4} -24 z^2 a^{-4} - a^{-4} z^{-2} +8 a^{-4} +3 z^9 a^{-3} -7 z^5 a^{-3} +12 z^3 a^{-3} -8 z a^{-3} +2 a^{-3} z^{-1} +6 z^8 a^{-2} -18 z^6 a^{-2} +34 z^4 a^{-2} -31 z^2 a^{-2} -2 a^{-2} z^{-2} +12 a^{-2} +3 z^7 a^{-1} -3 z^5 a^{-1} +3 z^3 a^{-1} -6 z a^{-1} +2 a^{-1} z^{-1} +6 z^4-12 z^2- z^{-2} +6 }[/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   \ -2 -1 0 1 2 3 4 5 6 7 χ 17                   1 -1 15                 3   3 13               5 1   -4 11             7 3     4 9           8 6       -2 7         8 6         2 5       6 9           3 3     5 7             -2 1   2 7               5 -1 1 4                 -3 -3 3                   3

Integral Khovanov Homology (db, data source)

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