L11n448

L11n447

L11n449

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Link Presentations

[edit Notes on L11n448's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X13,20,14,21 X16,12,17,11 X19,12,20,13 X8,16,5,15 X14,8,15,7 X17,19,18,22 X21,9,22,18 X2536 X4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, 7, -6}, {-5, 3, -9, 8}, {11, -2, 4, 5, -3, -7, 6, -4, -8, 9}

A Braid Representative	{{{braid_table}}}
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)-1) (t(2)-1) (t(3)-1) (t(4)-1)}{\sqrt{t(1)} \sqrt{t(2)} \sqrt{t(3)} \sqrt{t(4)}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{9/2}-\frac{1}{q^{9/2}}-3 q^{7/2}+4 q^{5/2}-\frac{4}{q^{5/2}}-5 q^{3/2}+\frac{3}{q^{3/2}}+4 \sqrt{q}-\frac{7}{\sqrt{q}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^5 z^{-3} +a^5 z^{-1} -3 a^3 z^{-3} +z^3 a^{-3} -2 a^3 z-4 a^3 z^{-1} +z a^{-3} -z^5 a^{-1} +2 a z^3+3 a z^{-3} -3 z^3 a^{-1} - a^{-1} z^{-3} +4 a z+5 a z^{-1} -3 z a^{-1} -2 a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^5 z^3-a^5 z^{-3} -3 a^5 z+3 a^5 z^{-1} +z^6 a^{-4} -3 z^4 a^{-4} +3 a^4 z^2+z^2 a^{-4} +3 a^4 z^{-2} -6 a^4+3 z^7 a^{-3} +a^3 z^5-11 z^5 a^{-3} +8 z^3 a^{-3} -3 a^3 z^{-3} -5 a^3 z-3 z a^{-3} +6 a^3 z^{-1} +a^2 z^8+3 z^8 a^{-2} -5 a^2 z^6-11 z^6 a^{-2} +9 a^2 z^4+7 z^4 a^{-2} +a^2 z^2+z^2 a^{-2} +6 a^2 z^{-2} -11 a^2+a z^9+z^9 a^{-1} -3 a z^7-a z^5-13 z^5 a^{-1} +10 a z^3+19 z^3 a^{-1} -3 a z^{-3} - a^{-1} z^{-3} -7 a z-8 z a^{-1} +6 a z^{-1} +3 a^{-1} z^{-1} +4 z^8-17 z^6+19 z^4-2 z^2+3 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   \ -4 -3 -2 -1 0 1 2 3 4 5 χ 10                   1 -1 8                 2   2 6               2 1   -1 4             3 2     1 2         2 3 2       1 0       1 7 3         3 -2       3 7           4 -4     2 1             1 -6 1   3               4 -8 2 1                 1 -10 1                   1

Integral Khovanov Homology (db, data source)

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