L11a442

L11a441

L11a443

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Link Presentations

[edit Notes on L11a442's Link Presentations]

Planar diagram presentation	X6172 X14,3,15,4 X22,16,13,15 X8,18,9,17 X16,8,17,7 X18,10,19,9 X20,12,21,11 X10,20,11,19 X12,22,5,21 X2536 X4,13,1,14
Gauss code	{1, -10, 2, -11}, {10, -1, 5, -4, 6, -8, 7, -9}, {11, -2, 3, -5, 4, -6, 8, -7, 9, -3}

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 v^3 w^3-u v^3 w^2-u v^2 w^3+2 u v^2 w^2-u v^2 w-u v w^2+2 u v w-u v-u w+2 u-2 v^3 w^3+v^3 w^2+v^2 w^3-2 v^2 w^2+v^2 w+v w^2-2 v w+v+w-1}{\sqrt{u} v^{3/2} w^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^9+3 q^8-4 q^7+6 q^6-7 q^5+8 q^4-7 q^3+6 q^2-4 q+4- q^{-1} + q^{-2} }[/math] (db)
Signature	4 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^6 a^{-6} -4 z^4 a^{-6} -3 z^2 a^{-6} +z^8 a^{-4} +6 z^6 a^{-4} +12 z^4 a^{-4} +11 z^2 a^{-4} + a^{-4} z^{-2} +5 a^{-4} -2 z^6 a^{-2} -11 z^4 a^{-2} -18 z^2 a^{-2} -2 a^{-2} z^{-2} -11 a^{-2} +z^4+5 z^2+ z^{-2} +6 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^{10} a^{-2} +z^{10} a^{-4} +z^9 a^{-1} +4 z^9 a^{-3} +3 z^9 a^{-5} -3 z^8 a^{-2} +4 z^8 a^{-6} +z^8-4 z^7 a^{-1} -18 z^7 a^{-3} -10 z^7 a^{-5} +4 z^7 a^{-7} -5 z^6 a^{-2} -13 z^6 a^{-4} -11 z^6 a^{-6} +4 z^6 a^{-8} -7 z^6+18 z^5 a^{-3} +8 z^5 a^{-5} -6 z^5 a^{-7} +4 z^5 a^{-9} +22 z^4 a^{-2} +19 z^4 a^{-4} +8 z^4 a^{-6} -3 z^4 a^{-8} +3 z^4 a^{-10} +17 z^4+12 z^3 a^{-1} +6 z^3 a^{-3} -2 z^3 a^{-5} -3 z^3 a^{-9} +z^3 a^{-11} -25 z^2 a^{-2} -10 z^2 a^{-4} -3 z^2 a^{-6} -2 z^2 a^{-8} -2 z^2 a^{-10} -18 z^2-11 z a^{-1} -11 z a^{-3} +13 a^{-2} +5 a^{-4} + a^{-8} +8+2 a^{-1} z^{-1} +2 a^{-3} z^{-1} -2 a^{-2} z^{-2} - a^{-4} z^{-2} - 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   \ -4 -3 -2 -1 0 1 2 3 4 5 6 7 χ 19                       1 -1 17                     2   2 15                   2 1   -1 13                 4 2     2 11               5 4       -1 9             3 2         1 7           4 5           1 5         2 3             -1 3       3 5               2 1     1 1                 0 -1     3                   3 -3 1 1                     0 -5 1                       1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=3 }[/math] [math]\displaystyle{ i=5 }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=7 }[/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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