L11n316

L11n315

L11n317

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Link Presentations

[edit Notes on L11n316's Link Presentations]

Planar diagram presentation	X6172 X12,4,13,3 X7,17,8,16 X20,10,21,9 X18,12,19,11 X22,20,11,19 X15,9,16,8 X10,22,5,21 X17,14,18,15 X2536 X4,14,1,13
Gauss code	{1, -10, 2, -11}, {10, -1, -3, 7, 4, -8}, {5, -2, 11, 9, -7, 3, -9, -5, 6, -4, 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{2 u v^2 w-u v^2+u v w^2-3 u v w+2 u v-u w^2+u w-v^2 w+v^2-2 v w^2+3 v w-v+w^2-2 w}{\sqrt{u} v w} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^9-3 q^8+5 q^7-6 q^6+8 q^5-7 q^4+7 q^3-4 q^2+3 q }[/math] (db)
Signature	2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^2 a^{-8} -z^4 a^{-6} + a^{-6} z^{-2} +2 a^{-6} -2 z^4 a^{-4} -5 z^2 a^{-4} -2 a^{-4} z^{-2} -6 a^{-4} +3 z^2 a^{-2} + a^{-2} z^{-2} +4 a^{-2} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^9 a^{-5} +z^9 a^{-7} +z^8 a^{-4} +4 z^8 a^{-6} +3 z^8 a^{-8} -2 z^7 a^{-5} +z^7 a^{-7} +3 z^7 a^{-9} -z^6 a^{-4} -11 z^6 a^{-6} -9 z^6 a^{-8} +z^6 a^{-10} +3 z^5 a^{-3} +5 z^5 a^{-5} -8 z^5 a^{-7} -10 z^5 a^{-9} -z^4 a^{-4} +8 z^4 a^{-6} +6 z^4 a^{-8} -3 z^4 a^{-10} -4 z^3 a^{-3} -8 z^3 a^{-5} +3 z^3 a^{-7} +7 z^3 a^{-9} +5 z^2 a^{-2} +10 z^2 a^{-4} -3 z^2 a^{-8} +2 z^2 a^{-10} +6 z a^{-3} +6 z a^{-5} -5 a^{-2} -8 a^{-4} -3 a^{-6} + a^{-8} -2 a^{-3} z^{-1} -2 a^{-5} z^{-1} + a^{-2} z^{-2} +2 a^{-4} z^{-2} + a^{-6} z^{-2} }[/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               2   -2 15             3 1   2 13           4 3     -1 11         4 2       2 9       3 4         1 7     4 4           0 5   1 4             3 3 2 3               -1 1 3                 3

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