L11a444

L11a443

L11a445

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Link Presentations

[edit Notes on L11a444's Link Presentations]

Planar diagram presentation	X6172 X14,3,15,4 X20,12,21,11 X16,8,17,7 X18,10,19,9 X10,20,11,19 X22,16,13,15 X12,18,5,17 X8,22,9,21 X2536 X4,13,1,14
Gauss code	{1, -10, 2, -11}, {10, -1, 4, -9, 5, -6, 3, -8}, {11, -2, 7, -4, 8, -5, 6, -3, 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)^2 t(2)^3-2 t(3)^2 t(2)^3-t(1) t(3) t(2)^3+t(3) t(2)^3+t(1) t(3)^3 t(2)^2-2 t(3)^3 t(2)^2-4 t(1) t(3)^2 t(2)^2+5 t(3)^2 t(2)^2-t(1) t(2)^2+4 t(1) t(3) t(2)^2-4 t(3) t(2)^2+t(2)^2-t(1) t(3)^3 t(2)+t(3)^3 t(2)+4 t(1) t(3)^2 t(2)-4 t(3)^2 t(2)+2 t(1) t(2)-5 t(1) t(3) t(2)+4 t(3) t(2)-t(2)-t(1) t(3)^2+t(3)^2+2 t(1) t(3)-t(3)}{\sqrt{t(1)} t(2)^{3/2} t(3)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^8+4 q^7-8 q^6+13 q^5-15 q^4+18 q^3+ q^{-3} -17 q^2-2 q^{-2} +14 q+6 q^{-1} -9 }[/math] (db)
Signature	2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^6 a^{-2} +z^6 a^{-4} +z^4 a^{-2} +2 z^4 a^{-4} -z^4 a^{-6} -2 z^4+a^2 z^2-3 z^2 a^{-2} +2 z^2 a^{-4} -z^2 a^{-6} -4 z^2+2 a^2-5 a^{-2} +3 a^{-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} +2 z^2 a^{-8} +7 z^7 a^{-7} -11 z^5 a^{-7} +3 z^3 a^{-7} +7 z^8 a^{-6} -9 z^6 a^{-6} +z^4 a^{-6} +2 z^2 a^{-6} -2 a^{-6} +4 z^9 a^{-5} +z^7 a^{-5} -8 z^5 a^{-5} +3 z^3 a^{-5} +z a^{-5} +z^{10} a^{-4} +9 z^8 a^{-4} -18 z^6 a^{-4} +12 z^4 a^{-4} -5 z^2 a^{-4} - a^{-4} z^{-2} +3 a^{-4} +6 z^9 a^{-3} -6 z^7 a^{-3} -5 z^5 a^{-3} +13 z^3 a^{-3} -8 z a^{-3} +2 a^{-3} z^{-1} +z^{10} a^{-2} +5 z^8 a^{-2} +a^2 z^6-13 z^6 a^{-2} -4 a^2 z^4+15 z^4 a^{-2} +5 a^2 z^2-15 z^2 a^{-2} -2 a^{-2} z^{-2} -2 a^2+9 a^{-2} +2 z^9 a^{-1} +2 a z^7+2 z^7 a^{-1} -5 a z^5-14 z^5 a^{-1} +2 a z^3+16 z^3 a^{-1} +a z-8 z a^{-1} +2 a^{-1} z^{-1} +3 z^8-7 z^6+6 z^4-5 z^2- z^{-2} +3 }[/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 17                       1 -1 15                     3   3 13                   5 1   -4 11                 8 3     5 9               9 7       -2 7             9 6         3 5           8 9           1 3         6 9             -3 1       4 9               5 -1     2 5                 -3 -3     4                   4 -5 1 2                     -1 -7 1                       1

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=-4 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=1 }[/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}^{9}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/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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