L11a506

L11a505

L11a507

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Link Presentations

[edit Notes on L11a506's Link Presentations]

Planar diagram presentation	X8192 X16,8,17,7 X14,6,15,5 X10,4,11,3 X4,14,5,13 X2,18,3,17 X18,9,19,10 X12,21,7,22 X22,11,13,12 X20,16,21,15 X6,19,1,20
Gauss code	{1, -6, 4, -5, 3, -11}, {2, -1, 7, -4, 9, -8}, {5, -3, 10, -2, 6, -7, 11, -10, 8, -9}

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(3)-1) \left(t(2)^2 t(1)^2+t(2)^2 t(3)^2 t(1)^2-2 t(2) t(3)^2 t(1)^2+t(3)^2 t(1)^2-t(2) t(1)^2-t(2)^2 t(3) t(1)^2+2 t(2) t(3) t(1)^2-t(3) t(1)^2-2 t(2)^2 t(1)-t(2)^2 t(3)^2 t(1)+3 t(2) t(3)^2 t(1)-2 t(3)^2 t(1)+3 t(2) t(1)+2 t(2)^2 t(3) t(1)-3 t(2) t(3) t(1)+2 t(3) t(1)-t(1)+t(2)^2-t(2) t(3)^2+t(3)^2-2 t(2)-t(2)^2 t(3)+2 t(2) t(3)-t(3)+1\right)}{t(1) t(2) t(3)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^7-4 q^6+9 q^5-16 q^4+22 q^3-24 q^2+26 q-21+17 q^{-1} -10 q^{-2} +5 q^{-3} - q^{-4} }[/math] (db)
Signature	2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^6 a^{-4} +3 z^4 a^{-4} +3 z^2 a^{-4} -z^8 a^{-2} -5 z^6 a^{-2} -a^2 z^4-10 z^4 a^{-2} -a^2 z^2-7 z^2 a^{-2} +a^2 z^{-2} + a^{-2} z^{-2} +a^2+ a^{-2} +2 z^6+6 z^4+4 z^2-2 z^{-2} -2 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^4 a^{-8} +4 z^5 a^{-7} -z^3 a^{-7} +9 z^6 a^{-6} -6 z^4 a^{-6} +z^2 a^{-6} +15 z^7 a^{-5} -20 z^5 a^{-5} +9 z^3 a^{-5} +18 z^8 a^{-4} -34 z^6 a^{-4} +24 z^4 a^{-4} -8 z^2 a^{-4} +13 z^9 a^{-3} +a^3 z^7-17 z^7 a^{-3} -2 a^3 z^5-8 z^5 a^{-3} +a^3 z^3+10 z^3 a^{-3} +4 z^{10} a^{-2} +5 a^2 z^8+19 z^8 a^{-2} -15 a^2 z^6-77 z^6 a^{-2} +13 a^2 z^4+71 z^4 a^{-2} -2 a^2 z^2-21 z^2 a^{-2} +a^2 z^{-2} + a^{-2} z^{-2} -2 a^2-2 a^{-2} +8 a z^9+21 z^9 a^{-1} -23 a z^7-56 z^7 a^{-1} +16 a z^5+34 z^5 a^{-1} -a z^3-2 z^3 a^{-1} +2 a z+2 z a^{-1} -2 a z^{-1} -2 a^{-1} z^{-1} +4 z^{10}+6 z^8-49 z^6+53 z^4-14 z^2+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   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 6 χ 15                       1 1 13                     3   -3 11                   6 1   5 9                 10 3     -7 7               12 6       6 5             13 11         -2 3           13 11           2 1         10 15             5 -1       7 11               -4 -3     4 11                 7 -5   1 6                   -5 -7   4                     4 -9 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=-5 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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}^{11}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{15}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{13} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{13} }[/math] [math]\displaystyle{ {\mathbb Z}^{13} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/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}^{3}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=6 }[/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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