L11a105

L11a104

L11a106

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Link Presentations

[edit Notes on L11a105's Link Presentations]

Planar diagram presentation	X6172 X14,3,15,4 X16,8,17,7 X18,10,19,9 X8,18,9,17 X22,20,5,19 X20,11,21,12 X10,21,11,22 X12,16,13,15 X2536 X4,13,1,14
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -5, 4, -8, 7, -9, 11, -2, 9, -3, 5, -4, 6, -7, 8, -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(2)^5-2 t(2)^5-4 t(1) t(2)^4+6 t(2)^4+6 t(1) t(2)^3-6 t(2)^3-6 t(1) t(2)^2+6 t(2)^2+6 t(1) t(2)-4 t(2)-2 t(1)+1}{\sqrt{t(1)} t(2)^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{11/2}-4 q^{9/2}+7 q^{7/2}-11 q^{5/2}+15 q^{3/2}-16 \sqrt{q}+\frac{15}{\sqrt{q}}-\frac{13}{q^{3/2}}+\frac{9}{q^{5/2}}-\frac{6}{q^{7/2}}+\frac{2}{q^{9/2}}-\frac{1}{q^{11/2}} }[/math] (db)
Signature	1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^5 z+2 a^5 z^{-1} +z^5 a^{-3} -3 a^3 z^3+2 z^3 a^{-3} -8 a^3 z-4 a^3 z^{-1} -z^7 a^{-1} +3 a z^5-4 z^5 a^{-1} +10 a z^3-6 z^3 a^{-1} +9 a z-4 z a^{-1} +3 a z^{-1} - a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -a^2 z^{10}-z^{10}-2 a^3 z^9-6 a z^9-4 z^9 a^{-1} -2 a^4 z^8-4 a^2 z^8-7 z^8 a^{-2} -9 z^8-a^5 z^7+2 a^3 z^7+9 a z^7-2 z^7 a^{-1} -8 z^7 a^{-3} +7 a^4 z^6+18 a^2 z^6+5 z^6 a^{-2} -7 z^6 a^{-4} +23 z^6+5 a^5 z^5+13 a^3 z^5+13 a z^5+16 z^5 a^{-1} +7 z^5 a^{-3} -4 z^5 a^{-5} -6 a^4 z^4-12 a^2 z^4+7 z^4 a^{-2} +7 z^4 a^{-4} -z^4 a^{-6} -7 z^4-9 a^5 z^3-25 a^3 z^3-25 a z^3-11 z^3 a^{-1} +z^3 a^{-3} +3 z^3 a^{-5} -a^4 z^2-3 a^2 z^2-7 z^2 a^{-2} -2 z^2 a^{-4} -7 z^2+7 a^5 z+16 a^3 z+13 a z+3 z a^{-1} -z a^{-3} +2 a^4+3 a^2+ a^{-2} +3-2 a^5 z^{-1} -4 a^3 z^{-1} -3 a z^{-1} - a^{-1} 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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 12                       1 -1 10                     3   3 8                   4 1   -3 6                 7 3     4 4               8 4       -4 2             8 7         1 0           8 9           1 -2         5 7             -2 -4       4 8               4 -6     2 5                 -3 -8   1 5                   4 -10   1                     -1 -12 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=-6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=5 }[/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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