L11a394

L11a393

L11a395

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Link Presentations

[edit Notes on L11a394's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X22,16,9,15 X18,12,19,11 X14,20,15,19 X20,14,21,13 X12,22,13,21 X8,18,5,17 X16,8,17,7 X2536 X4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, 9, -8}, {11, -2, 4, -7, 6, -5, 3, -9, 8, -4, 5, -6, 7, -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) t(3)^3+2 t(1) t(2) t(3)^3-4 t(2) t(3)^3+2 t(3)^3+5 t(1) t(3)^2-3 t(1) t(2) t(3)^2+6 t(2) t(3)^2-3 t(3)^2-6 t(1) t(3)+3 t(1) t(2) t(3)-5 t(2) t(3)+3 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+3 q^7-7 q^6+11 q^5-16 q^4+18 q^3+ q^{-3} -16 q^2-2 q^{-2} +16 q+7 q^{-1} -10 }[/math] (db)
Signature	2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^4 a^{-6} -2 z^2 a^{-6} - a^{-6} z^{-2} -2 a^{-6} +z^6 a^{-4} +3 z^4 a^{-4} +6 z^2 a^{-4} +3 a^{-4} z^{-2} +6 a^{-4} +z^6 a^{-2} +z^4 a^{-2} +a^2 z^2-2 z^2 a^{-2} +a^2 z^{-2} -2 a^{-2} z^{-2} +2 a^2-3 a^{-2} -2 z^4-4 z^2- z^{-2} -3 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^5 a^{-9} -2 z^3 a^{-9} +3 z^6 a^{-8} -5 z^4 a^{-8} +6 z^7 a^{-7} -14 z^5 a^{-7} +14 z^3 a^{-7} -7 z a^{-7} +2 a^{-7} z^{-1} +7 z^8 a^{-6} -16 z^6 a^{-6} +17 z^4 a^{-6} -5 z^2 a^{-6} - a^{-6} z^{-2} + a^{-6} +4 z^9 a^{-5} +z^7 a^{-5} -21 z^5 a^{-5} +40 z^3 a^{-5} -27 z a^{-5} +8 a^{-5} z^{-1} +z^{10} a^{-4} +10 z^8 a^{-4} -25 z^6 a^{-4} +23 z^4 a^{-4} -9 z^2 a^{-4} -3 a^{-4} z^{-2} +5 a^{-4} +6 z^9 a^{-3} -2 z^7 a^{-3} -23 z^5 a^{-3} +43 z^3 a^{-3} -34 z a^{-3} +10 a^{-3} z^{-1} +z^{10} a^{-2} +6 z^8 a^{-2} +a^2 z^6-11 z^6 a^{-2} -4 a^2 z^4+2 z^4 a^{-2} +6 a^2 z^2-4 z^2 a^{-2} +a^2 z^{-2} -2 a^{-2} z^{-2} -4 a^2+4 a^{-2} +2 z^9 a^{-1} +2 a z^7+5 z^7 a^{-1} -4 a z^5-21 z^5 a^{-1} +19 z^3 a^{-1} +4 a z-10 z a^{-1} -2 a z^{-1} +2 a^{-1} z^{-1} +3 z^8-4 z^6-3 z^4+6 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   \ -4 -3 -2 -1 0 1 2 3 4 5 6 7 χ 17                       1 -1 15                     2   2 13                   5 1   -4 11                 6 2     4 9               10 5       -5 7             8 6         2 5           8 10           2 3         8 8             0 1       5 11               6 -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}^{11}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/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}^{10}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/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.

L11a393 L11a395