L10n112

L10n111

L10n113

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Link Presentations

[edit Notes on L10n112's Link Presentations]

Planar diagram presentation	X6172 X2536 X11,19,12,18 X3,11,4,10 X9,1,10,4 X7,15,8,14 X13,5,14,8 X15,17,16,20 X19,13,20,16 X17,9,18,12
Gauss code	{1, -2, -4, 5}, {2, -1, -6, 7}, {-5, 4, -3, 10}, {-7, 6, -8, 9}, {-10, 3, -9, 8}

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)+t(1) t(3) t(2)-t(3) t(2)+t(1) t(4) t(2)-t(1) t(3) t(4) t(2)+t(3) t(4) t(2)+2 t(1) t(5) t(2)-t(1) t(3) t(5) t(2)+t(3) t(5) t(2)-t(1) t(4) t(5) t(2)-t(5) t(2)+t(3)-t(1) t(4)+t(1) t(3) t(4)-2 t(3) t(4)+t(4)-t(1) t(5)-t(3) t(5)+t(1) t(4) t(5)+t(3) t(4) t(5)-t(4) t(5)+t(5)}{\sqrt{t(1)} \sqrt{t(2)} \sqrt{t(3)} \sqrt{t(4)} \sqrt{t(5)}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^9-q^8+6 q^7-5 q^6+11 q^5-5 q^4+10 q^3-5 q^2+4 q }[/math] (db)
Signature	2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^{-10} z^{-4} + a^{-10} z^{-2} -4 a^{-8} z^{-4} -7 a^{-8} z^{-2} -4 a^{-8} +6 a^{-6} z^{-4} +6 z^2 a^{-6} +15 a^{-6} z^{-2} +14 a^{-6} -3 z^4 a^{-4} -4 a^{-4} z^{-4} -10 z^2 a^{-4} -13 a^{-4} z^{-2} -16 a^{-4} + a^{-2} z^{-4} +4 z^2 a^{-2} +4 a^{-2} z^{-2} +6 a^{-2} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^8 a^{-6} +z^8 a^{-8} +5 z^7 a^{-5} +6 z^7 a^{-7} +z^7 a^{-9} +10 z^6 a^{-4} +12 z^6 a^{-6} +3 z^6 a^{-8} +z^6 a^{-10} +6 z^5 a^{-3} -6 z^5 a^{-7} -25 z^4 a^{-4} -39 z^4 a^{-6} -19 z^4 a^{-8} -5 z^4 a^{-10} -10 z^3 a^{-3} -30 z^3 a^{-5} -30 z^3 a^{-7} -10 z^3 a^{-9} +10 z^2 a^{-2} +30 z^2 a^{-4} +40 z^2 a^{-6} +30 z^2 a^{-8} +10 z^2 a^{-10} +20 z a^{-3} +55 z a^{-5} +55 z a^{-7} +20 z a^{-9} -10 a^{-2} -25 a^{-4} -31 a^{-6} -25 a^{-8} -10 a^{-10} -15 a^{-3} z^{-1} -41 a^{-5} z^{-1} -41 a^{-7} z^{-1} -15 a^{-9} z^{-1} +5 a^{-2} z^{-2} +14 a^{-4} z^{-2} +18 a^{-6} z^{-2} +14 a^{-8} z^{-2} +5 a^{-10} z^{-2} +4 a^{-3} z^{-3} +12 a^{-5} z^{-3} +12 a^{-7} z^{-3} +4 a^{-9} z^{-3} - a^{-2} z^{-4} -4 a^{-4} z^{-4} -6 a^{-6} z^{-4} -4 a^{-8} z^{-4} - a^{-10} z^{-4} }[/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   \ 0 1 2 3 4 5 6 7 8 χ 19                 1 1 17               1 1 0 15             5     5 13             1     1 11         11 5       6 9       4 10         6 7     6 1           5 5 1   4             5 3 5 6               -1 1 4                 4

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{ i=5 }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=7 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=8 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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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