L11a438

L11a437

L11a439

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Link Presentations

[edit Notes on L11a438's Link Presentations]

Planar diagram presentation	X6172 X2,16,3,15 X10,4,11,3 X14,6,15,5 X22,12,13,11 X12,14,5,13 X4,21,1,22 X20,17,21,18 X16,7,17,8 X8,20,9,19 X18,10,19,9
Gauss code	{1, -2, 3, -7}, {4, -1, 9, -10, 11, -3, 5, -6}, {6, -4, 2, -9, 8, -11, 10, -8, 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{(t(1)-1) (t(2)-1)^2 (t(3)-1)^2 (t(2) t(3)+1)}{\sqrt{t(1)} t(2)^{3/2} t(3)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^8+4 q^7-8 q^6+14 q^5-17 q^4+21 q^3-20 q^2+18 q-12+8 q^{-1} -4 q^{-2} + q^{-3} }[/math] (db)
Signature	2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^8 a^{-2} -5 z^6 a^{-2} +2 z^6 a^{-4} +z^6-9 z^4 a^{-2} +7 z^4 a^{-4} -z^4 a^{-6} +3 z^4-6 z^2 a^{-2} +6 z^2 a^{-4} -2 z^2 a^{-6} +2 z^2- a^{-4} +1+ a^{-2} z^{-2} -2 a^{-4} z^{-2} + a^{-6} z^{-2} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^5 a^{-9} -z^3 a^{-9} +4 z^6 a^{-8} -6 z^4 a^{-8} +2 z^2 a^{-8} +7 z^7 a^{-7} -10 z^5 a^{-7} +2 z^3 a^{-7} +8 z^8 a^{-6} -11 z^6 a^{-6} +3 z^4 a^{-6} +z^2 a^{-6} + a^{-6} z^{-2} -3 a^{-6} +6 z^9 a^{-5} -5 z^7 a^{-5} -z^5 a^{-5} -z^3 a^{-5} +4 z a^{-5} -2 a^{-5} z^{-1} +2 z^{10} a^{-4} +11 z^8 a^{-4} -35 z^6 a^{-4} +37 z^4 a^{-4} -11 z^2 a^{-4} +2 a^{-4} z^{-2} -4 a^{-4} +12 z^9 a^{-3} -28 z^7 a^{-3} +24 z^5 a^{-3} -7 z^3 a^{-3} +4 z a^{-3} -2 a^{-3} z^{-1} +2 z^{10} a^{-2} +10 z^8 a^{-2} +a^2 z^6-41 z^6 a^{-2} -2 a^2 z^4+47 z^4 a^{-2} -16 z^2 a^{-2} + a^{-2} z^{-2} - a^{-2} +6 z^9 a^{-1} +4 a z^7-12 z^7 a^{-1} -10 a z^5+4 z^5 a^{-1} +4 a z^3+z^3 a^{-1} +7 z^8-20 z^6+17 z^4-6 z^2+1 }[/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                   5 1   -4 11                 9 3     6 9               10 7       -3 7             11 7         4 5           9 10           1 3         9 11             -2 1       5 11               6 -1     3 7                 -4 -3   1 5                   4 -5   3                     -3 -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{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\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^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\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^{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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