L11n382

L11n381

L11n383

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Link Presentations

[edit Notes on L11n382's Link Presentations]

Planar diagram presentation	X6172 X14,7,15,8 X4,15,1,16 X5,10,6,11 X8493 X22,18,19,17 X20,12,21,11 X12,20,13,19 X18,22,5,21 X9,16,10,17 X2,14,3,13
Gauss code	{1, -11, 5, -3}, {8, -7, 9, -6}, {-4, -1, 2, -5, -10, 4, 7, -8, 11, -2, 3, 10, 6, -9}

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

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