L11a453

L11a452

L11a454

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Link Presentations

[edit Notes on L11a453's Link Presentations]

Planar diagram presentation	X6172 X14,4,15,3 X18,11,19,12 X16,8,17,7 X20,17,21,18 X12,19,5,20 X8,22,9,21 X22,10,13,9 X10,14,11,13 X2536 X4,16,1,15
Gauss code	{1, -10, 2, -11}, {10, -1, 4, -7, 8, -9, 3, -6}, {9, -2, 11, -4, 5, -3, 6, -5, 7, -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{(v w-v+1) (v w-w+1) (u v w-u w+u-v w+v-1)}{\sqrt{u} v^{3/2} w^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^7-3 q^6+7 q^5-12 q^4+16 q^3-16 q^2+18 q-14+11 q^{-1} -6 q^{-2} +3 q^{-3} - q^{-4} }[/math] (db)
Signature	2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^6 a^{-4} +4 z^4 a^{-4} +6 z^2 a^{-4} + a^{-4} z^{-2} +3 a^{-4} -z^8 a^{-2} -6 z^6 a^{-2} -a^2 z^4-15 z^4 a^{-2} -3 a^2 z^2-18 z^2 a^{-2} -2 a^{-2} z^{-2} -2 a^2-9 a^{-2} +2 z^6+9 z^4+14 z^2+ z^{-2} +8 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^4 a^{-8} -z^2 a^{-8} +3 z^5 a^{-7} -2 z^3 a^{-7} +6 z^6 a^{-6} -6 z^4 a^{-6} +4 z^2 a^{-6} - a^{-6} +9 z^7 a^{-5} -14 z^5 a^{-5} +11 z^3 a^{-5} -z a^{-5} +9 z^8 a^{-4} -15 z^6 a^{-4} +11 z^4 a^{-4} -6 z^2 a^{-4} - a^{-4} z^{-2} +3 a^{-4} +5 z^9 a^{-3} +a^3 z^7+z^7 a^{-3} -4 a^3 z^5-22 z^5 a^{-3} +5 a^3 z^3+22 z^3 a^{-3} -2 a^3 z-8 z a^{-3} +2 a^{-3} z^{-1} +z^{10} a^{-2} +3 a^2 z^8+15 z^8 a^{-2} -12 a^2 z^6-51 z^6 a^{-2} +16 a^2 z^4+58 z^4 a^{-2} -9 a^2 z^2-37 z^2 a^{-2} -2 a^{-2} z^{-2} +3 a^2+11 a^{-2} +3 a z^9+8 z^9 a^{-1} -6 a z^7-15 z^7 a^{-1} -6 a z^5-7 z^5 a^{-1} +16 a z^3+20 z^3 a^{-1} -7 a z-12 z a^{-1} +2 a^{-1} z^{-1} +z^{10}+9 z^8-42 z^6+56 z^4-35 z^2- 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   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 6 χ 15                       1 1 13                     2   -2 11                   5 1   4 9                 8 3     -5 7               8 4       4 5             8 8         0 3           10 8           2 1         7 11             4 -1       4 7               -3 -3     2 7                 5 -5   1 4                   -3 -7   2                     2 -9 1                       -1

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