L11a385

L11a384

L11a386

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Link Presentations

[edit Notes on L11a385's Link Presentations]

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

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) t(3)^3+2 t(1) t(2) t(3)^3-4 t(2) t(3)^3+2 t(3)^3+6 t(1) t(3)^2-4 t(1) t(2) t(3)^2+7 t(2) t(3)^2-4 t(3)^2-7 t(1) t(3)+4 t(1) t(2) t(3)-6 t(2) t(3)+4 t(3)+4 t(1)-2 t(1) t(2)+2 t(2)-2}{\sqrt{t(1)} \sqrt{t(2)} t(3)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^8+4 q^7-10 q^6+14 q^5-18 q^4+21 q^3+ q^{-3} -19 q^2-2 q^{-2} +17 q+7 q^{-1} -10 }[/math] (db)
Signature	2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^6 a^{-2} +z^6 a^{-4} +z^4 a^{-2} +2 z^4 a^{-4} -z^4 a^{-6} -2 z^4+a^2 z^2-z^2 a^{-2} +4 z^2 a^{-4} -z^2 a^{-6} -4 z^2+2 a^2-3 a^{-2} +6 a^{-4} -2 a^{-6} -3+a^2 z^{-2} -2 a^{-2} z^{-2} +3 a^{-4} z^{-2} - a^{-6} z^{-2} - z^{-2} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^{10} a^{-2} +z^{10} a^{-4} +2 z^9 a^{-1} +7 z^9 a^{-3} +5 z^9 a^{-5} +8 z^8 a^{-2} +15 z^8 a^{-4} +10 z^8 a^{-6} +3 z^8+2 a z^7+6 z^7 a^{-1} +z^7 a^{-3} +6 z^7 a^{-5} +9 z^7 a^{-7} +a^2 z^6-15 z^6 a^{-2} -31 z^6 a^{-4} -17 z^6 a^{-6} +4 z^6 a^{-8} -4 z^6-4 a z^5-23 z^5 a^{-1} -33 z^5 a^{-3} -31 z^5 a^{-5} -16 z^5 a^{-7} +z^5 a^{-9} -4 a^2 z^4+6 z^4 a^{-2} +18 z^4 a^{-4} +10 z^4 a^{-6} -4 z^4 a^{-8} -2 z^4+23 z^3 a^{-1} +50 z^3 a^{-3} +39 z^3 a^{-5} +11 z^3 a^{-7} -z^3 a^{-9} +6 a^2 z^2-3 z^2 a^{-2} -4 z^2 a^{-4} -2 z^2 a^{-6} +5 z^2+4 a z-10 z a^{-1} -34 z a^{-3} -27 z a^{-5} -7 z a^{-7} -4 a^2+4 a^{-2} +5 a^{-4} + a^{-6} -3-2 a z^{-1} +2 a^{-1} z^{-1} +10 a^{-3} z^{-1} +8 a^{-5} z^{-1} +2 a^{-7} z^{-1} +a^2 z^{-2} -2 a^{-2} z^{-2} -3 a^{-4} z^{-2} - a^{-6} 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 17                       1 -1 15                     3   3 13                   7 1   -6 11                 7 3     4 9               11 7       -4 7             10 7         3 5           9 11           2 3         8 10             -2 1       5 12               7 -1     2 5                 -3 -3     5                   5 -5 1 2                     -1 -7 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=-4 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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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