L11a157

L11a156

L11a158

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Link Presentations

[edit Notes on L11a157's Link Presentations]

Planar diagram presentation	X8192 X10,4,11,3 X22,10,7,9 X2738 X18,13,19,14 X6,12,1,11 X20,18,21,17 X16,22,17,21 X4,16,5,15 X14,6,15,5 X12,19,13,20
Gauss code	{1, -4, 2, -9, 10, -6}, {4, -1, 3, -2, 6, -11, 5, -10, 9, -8, 7, -5, 11, -7, 8, -3}

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(2)^4-t(2)^4+2 t(1)^2 t(2)^3-8 t(1) t(2)^3+5 t(2)^3-6 t(1)^2 t(2)^2+15 t(1) t(2)^2-6 t(2)^2+5 t(1)^2 t(2)-8 t(1) t(2)+2 t(2)-t(1)^2+t(1)}{t(1) t(2)^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{15/2}-4 q^{13/2}+8 q^{11/2}-13 q^{9/2}+17 q^{7/2}-20 q^{5/2}+19 q^{3/2}-17 \sqrt{q}+\frac{12}{\sqrt{q}}-\frac{7}{q^{3/2}}+\frac{3}{q^{5/2}}-\frac{1}{q^{7/2}} }[/math] (db)
Signature	1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z a^{-7} -3 z^3 a^{-5} -3 z a^{-5} - a^{-5} z^{-1} +2 z^5 a^{-3} +4 z^3 a^{-3} +a^3 z+6 z a^{-3} +2 a^{-3} z^{-1} +z^5 a^{-1} -2 a z^3-2 z^3 a^{-1} -4 z a^{-1} +a z^{-1} -2 a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^{10} a^{-2} -z^{10} a^{-4} -4 z^9 a^{-1} -8 z^9 a^{-3} -4 z^9 a^{-5} -14 z^8 a^{-2} -14 z^8 a^{-4} -6 z^8 a^{-6} -6 z^8-5 a z^7-3 z^7 a^{-1} +4 z^7 a^{-3} -2 z^7 a^{-5} -4 z^7 a^{-7} -3 a^2 z^6+36 z^6 a^{-2} +39 z^6 a^{-4} +13 z^6 a^{-6} -z^6 a^{-8} +8 z^6-a^3 z^5+7 a z^5+18 z^5 a^{-1} +25 z^5 a^{-3} +25 z^5 a^{-5} +10 z^5 a^{-7} +5 a^2 z^4-34 z^4 a^{-2} -31 z^4 a^{-4} -6 z^4 a^{-6} +2 z^4 a^{-8} -6 z^4+2 a^3 z^3-4 a z^3-24 z^3 a^{-1} -35 z^3 a^{-3} -24 z^3 a^{-5} -7 z^3 a^{-7} -2 a^2 z^2+11 z^2 a^{-2} +8 z^2 a^{-4} +z^2 a^{-6} -z^2 a^{-8} +3 z^2-a^3 z+3 a z+13 z a^{-1} +15 z a^{-3} +8 z a^{-5} +2 z a^{-7} - a^{-2} -a z^{-1} -2 a^{-1} z^{-1} -2 a^{-3} z^{-1} - a^{-5} z^{-1} }[/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 16                       1 -1 14                     3   3 12                   5 1   -4 10                 8 3     5 8               9 5       -4 6             11 8         3 4           9 10           1 2         8 10             -2 0       5 10               5 -2     2 7                 -5 -4   1 5                   4 -6   2                     -2 -8 1                       1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=0 }[/math] [math]\displaystyle{ i=2 }[/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}^{7}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{8} }[/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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