L11a445

L11a444

L11a446

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Link Presentations

[edit Notes on L11a445's Link Presentations]

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

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-t(3)^2 t(2)^3-t(1) t(3) t(2)^3+2 t(3) t(2)^3-t(2)^3+t(1) t(3)^3 t(2)^2-t(3)^3 t(2)^2-5 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)^2-6 t(3) t(2)^2+2 t(2)^2-2 t(1) t(3)^3 t(2)+t(3)^3 t(2)+6 t(1) t(3)^2 t(2)-4 t(3)^2 t(2)+t(1) t(2)-5 t(1) t(3) t(2)+5 t(3) t(2)-t(2)+t(1) t(3)^3-2 t(1) t(3)^2+t(3)^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^7-9 q^6+14 q^5-17 q^4+21 q^3+ q^{-3} -19 q^2-3 q^{-2} +17 q+7 q^{-1} -11 }[/math] (db)
Signature	2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^4 a^{-6} -z^2 a^{-6} - a^{-6} +z^6 a^{-4} +2 z^4 a^{-4} +3 z^2 a^{-4} + a^{-4} z^{-2} +3 a^{-4} +z^6 a^{-2} +z^4 a^{-2} +a^2 z^2-z^2 a^{-2} -2 a^{-2} z^{-2} +a^2-3 a^{-2} -2 z^4-3 z^2+ z^{-2} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^5 a^{-9} -z^3 a^{-9} +4 z^6 a^{-8} -5 z^4 a^{-8} +z^2 a^{-8} +8 z^7 a^{-7} -13 z^5 a^{-7} +6 z^3 a^{-7} -2 z a^{-7} +9 z^8 a^{-6} -15 z^6 a^{-6} +11 z^4 a^{-6} -7 z^2 a^{-6} +3 a^{-6} +5 z^9 a^{-5} +3 z^7 a^{-5} -20 z^5 a^{-5} +20 z^3 a^{-5} -7 z a^{-5} +z^{10} a^{-4} +16 z^8 a^{-4} -42 z^6 a^{-4} +47 z^4 a^{-4} -31 z^2 a^{-4} - a^{-4} z^{-2} +11 a^{-4} +8 z^9 a^{-3} -6 z^7 a^{-3} -15 z^5 a^{-3} +24 z^3 a^{-3} -12 z a^{-3} +2 a^{-3} z^{-1} +z^{10} a^{-2} +11 z^8 a^{-2} +a^2 z^6-31 z^6 a^{-2} -3 a^2 z^4+35 z^4 a^{-2} +3 a^2 z^2-26 z^2 a^{-2} -2 a^{-2} z^{-2} -a^2+11 a^{-2} +3 z^9 a^{-1} +3 a z^7+2 z^7 a^{-1} -8 a z^5-17 z^5 a^{-1} +6 a z^3+17 z^3 a^{-1} -a z-8 z a^{-1} +2 a^{-1} z^{-1} +4 z^8-7 z^6+z^4- 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                   6 1   -5 11                 8 3     5 9               10 7       -3 7             11 7         4 5           8 10           2 3         9 11             -2 1       5 11               6 -1     2 6                 -4 -3   1 5                   4 -5   2                     -2 -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{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/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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