L11a518

L11a517

L11a519

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Link Presentations

[edit Notes on L11a518's Link Presentations]

Planar diagram presentation	X8192 X14,4,15,3 X22,9,13,10 X20,11,21,12 X12,19,7,20 X10,21,11,22 X18,14,19,13 X16,6,17,5 X2738 X4,16,5,15 X6,18,1,17
Gauss code	{1, -9, 2, -10, 8, -11}, {9, -1, 3, -6, 4, -5}, {7, -2, 10, -8, 11, -7, 5, -4, 6, -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{-2 t(1)^2 t(3)^3+t(1) t(3)^3+t(1)^2 t(2) t(3)^3-t(1) t(2) t(3)^3+t(1)^2 t(3)^2+t(1)^2 t(2)^2 t(3)^2-t(1) t(2)^2 t(3)^2-2 t(1) t(3)^2-2 t(1)^2 t(2) t(3)^2+2 t(1) t(2) t(3)^2-t(2) t(3)^2+t(3)^2-t(1)^2 t(2)^2 t(3)+2 t(1) t(2)^2 t(3)-t(2)^2 t(3)+t(1) t(3)+t(1)^2 t(2) t(3)-2 t(1) t(2) t(3)+2 t(2) t(3)-t(3)-t(1) t(2)^2+2 t(2)^2+t(1) t(2)-t(2)}{t(1) t(2) t(3)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^6+3 q^5+ q^{-5} -5 q^4- q^{-4} +8 q^3+4 q^{-3} -9 q^2-5 q^{-2} +10 q+8 q^{-1} -9 }[/math] (db)
Signature	2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^6 a^{-2} +z^6-2 a^2 z^4+3 z^4 a^{-2} -z^4 a^{-4} +3 z^4+a^4 z^2-7 a^2 z^2+2 z^2 a^{-2} -2 z^2 a^{-4} +2 z^2+3 a^4-6 a^2+ a^{-2} +2+a^4 z^{-2} -2 a^2 z^{-2} + z^{-2} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^3 a^{-7} +3 z^4 a^{-6} -z^2 a^{-6} +5 z^5 a^{-5} -3 z^3 a^{-5} +a^4 z^8-7 a^4 z^6+7 z^6 a^{-4} +17 a^4 z^4-10 z^4 a^{-4} -17 a^4 z^2+4 z^2 a^{-4} -a^4 z^{-2} +7 a^4+a^3 z^9-4 a^3 z^7+7 z^7 a^{-3} +a^3 z^5-13 z^5 a^{-3} +10 a^3 z^3+5 z^3 a^{-3} -9 a^3 z+2 a^3 z^{-1} +a^2 z^{10}-3 a^2 z^8+5 z^8 a^{-2} -3 a^2 z^6-8 z^6 a^{-2} +17 a^2 z^4-5 z^4 a^{-2} -20 a^2 z^2+7 z^2 a^{-2} -2 a^2 z^{-2} +11 a^2-2 a^{-2} +4 a z^9+3 z^9 a^{-1} -16 a z^7-5 z^7 a^{-1} +15 a z^5-4 z^5 a^{-1} +4 a z^3+3 z^3 a^{-1} -9 a z+2 a z^{-1} +z^{10}+z^8-11 z^6+8 z^4-z^2- z^{-2} +3 }[/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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 13                       1 -1 11                     2   2 9                   3 1   -2 7                 5 2     3 5               5 4       -1 3             5 4         1 1           6 7           1 -1         2 3             -1 -3       3 6               3 -5     1 2                 -1 -7     3                   3 -9 1 1                     0 -11 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=-6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=5 }[/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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