L11a515

L11a514

L11a516

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Link Presentations

[edit Notes on L11a515's Link Presentations]

Planar diagram presentation	X8192 X14,4,15,3 X20,11,21,12 X18,10,19,9 X22,19,13,20 X10,14,11,13 X12,21,7,22 X16,6,17,5 X2738 X4,16,5,15 X6,18,1,17
Gauss code	{1, -9, 2, -10, 8, -11}, {9, -1, 4, -6, 3, -7}, {6, -2, 10, -8, 11, -4, 5, -3, 7, -5}

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{u^2 v^2 w^3-u^2 v^2 w^2-2 u^2 v w^3+4 u^2 v w^2-u^2 v w+u^2 w^3-2 u^2 w^2+u^2 w-u v^2 w^3+2 u v^2 w^2-u v^2 w+u v w^3-4 u v w^2+4 u v w-u v+u w^2-2 u w+u-v^2 w^2+2 v^2 w-v^2+v w^2-4 v w+2 v+w-1}{u v w^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^7-3 q^6+6 q^5-9 q^4- q^{-4} +13 q^3+3 q^{-3} -13 q^2-5 q^{-2} +14 q+9 q^{-1} -11 }[/math] (db)
Signature	2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^6 a^{-4} +4 z^4 a^{-4} +5 z^2 a^{-4} + a^{-4} z^{-2} +3 a^{-4} -z^8 a^{-2} -6 z^6 a^{-2} -a^2 z^4-14 z^4 a^{-2} -3 a^2 z^2-16 z^2 a^{-2} -2 a^{-2} z^{-2} -a^2-8 a^{-2} +2 z^6+9 z^4+12 z^2+ z^{-2} +6 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^4 a^{-8} -z^2 a^{-8} +3 z^5 a^{-7} -3 z^3 a^{-7} +5 z^6 a^{-6} -6 z^4 a^{-6} +3 z^2 a^{-6} - a^{-6} +6 z^7 a^{-5} -8 z^5 a^{-5} +4 z^3 a^{-5} +6 z^8 a^{-4} -11 z^6 a^{-4} +10 z^4 a^{-4} -6 z^2 a^{-4} - a^{-4} z^{-2} +4 a^{-4} +4 z^9 a^{-3} +a^3 z^7-5 z^7 a^{-3} -4 a^3 z^5-3 z^5 a^{-3} +4 a^3 z^3+8 z^3 a^{-3} -a^3 z-6 z a^{-3} +2 a^{-3} z^{-1} +z^{10} a^{-2} +3 a^2 z^8+9 z^8 a^{-2} -13 a^2 z^6-40 z^6 a^{-2} +17 a^2 z^4+56 z^4 a^{-2} -9 a^2 z^2-37 z^2 a^{-2} -2 a^{-2} z^{-2} +2 a^2+12 a^{-2} +3 a z^9+7 z^9 a^{-1} -9 a z^7-21 z^7 a^{-1} +2 a z^5+14 z^5 a^{-1} +7 a z^3+4 z^3 a^{-1} -3 a z-8 z a^{-1} +2 a^{-1} z^{-1} +z^{10}+6 z^8-37 z^6+56 z^4-36 z^2- z^{-2} +10 }[/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                   4 1   3 9                 6 3     -3 7               7 3       4 5             7 7         0 3           7 6           1 1         5 8             3 -1       4 6               -2 -3     2 6                 4 -5   1 3                   -2 -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}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/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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