L11a449

L11a448

L11a450

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Link Presentations

[edit Notes on L11a449's Link Presentations]

Planar diagram presentation	X6172 X14,4,15,3 X22,18,13,17 X16,8,17,7 X12,14,5,13 X8,21,9,22 X20,11,21,12 X18,9,19,10 X10,19,11,20 X2536 X4,16,1,15
Gauss code	{1, -10, 2, -11}, {10, -1, 4, -6, 8, -9, 7, -5}, {5, -2, 11, -4, 3, -8, 9, -7, 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{t(1) t(3)^3 t(2)^3-t(3)^3 t(2)^3-t(1) t(3)^2 t(2)^3+2 t(3)^2 t(2)^3-t(3) t(2)^3-2 t(1) t(3)^3 t(2)^2+t(3)^3 t(2)^2+4 t(1) t(3)^2 t(2)^2-4 t(3)^2 t(2)^2-2 t(1) t(3) t(2)^2+5 t(3) t(2)^2-t(2)^2+t(1) t(3)^3 t(2)-5 t(1) t(3)^2 t(2)+2 t(3)^2 t(2)-t(1) t(2)+4 t(1) t(3) t(2)-4 t(3) t(2)+2 t(2)+t(1) t(3)^2+t(1)-2 t(1) t(3)+t(3)-1}{\sqrt{t(1)} t(2)^{3/2} t(3)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^5+3 q^4-6 q^3+10 q^2-13 q+17-15 q^{-1} +14 q^{-2} -10 q^{-3} +7 q^{-4} -3 q^{-5} + q^{-6} }[/math] (db)
Signature	0 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^8-2 a^2 z^6-z^6 a^{-2} +6 z^6+a^4 z^4-9 a^2 z^4-4 z^4 a^{-2} +14 z^4+3 a^4 z^2-14 a^2 z^2-5 z^2 a^{-2} +15 z^2+3 a^4-9 a^2-2 a^{-2} +8+a^4 z^{-2} -2 a^2 z^{-2} + z^{-2} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^6 z^6-3 a^6 z^4+3 a^6 z^2-a^6+3 a^5 z^7-8 a^5 z^5+z^5 a^{-5} +5 a^5 z^3-2 z^3 a^{-5} -a^5 z+4 a^4 z^8-8 a^4 z^6+3 z^6 a^{-4} +a^4 z^4-6 z^4 a^{-4} -2 a^4 z^2+z^2 a^{-4} -a^4 z^{-2} +3 a^4+3 a^3 z^9-a^3 z^7+5 z^7 a^{-3} -13 a^3 z^5-11 z^5 a^{-3} +14 a^3 z^3+7 z^3 a^{-3} -8 a^3 z-2 z a^{-3} +2 a^3 z^{-1} +a^2 z^{10}+8 a^2 z^8+6 z^8 a^{-2} -30 a^2 z^6-16 z^6 a^{-2} +39 a^2 z^4+21 z^4 a^{-2} -28 a^2 z^2-13 z^2 a^{-2} -2 a^2 z^{-2} +11 a^2+3 a^{-2} +7 a z^9+4 z^9 a^{-1} -14 a z^7-5 z^7 a^{-1} +3 a z^5-4 z^5 a^{-1} +16 a z^3+16 z^3 a^{-1} -12 a z-7 z a^{-1} +2 a z^{-1} +z^{10}+10 z^8-40 z^6+62 z^4-37 z^2- z^{-2} +11 }[/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 11                       1 -1 9                     2   2 7                   4 1   -3 5                 6 2     4 3               7 4       -3 1             10 6         4 -1           8 10           2 -3         6 7             -1 -5       4 8               4 -7     3 6                 -3 -9   1 5                   4 -11   2                     -2 -13 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=1 }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/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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