L11a519

L11a518

L11a520

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Link Presentations

[edit Notes on L11a519's Link Presentations]

Planar diagram presentation	X8192 X16,6,17,5 X22,20,13,19 X18,10,19,9 X10,21,11,22 X20,11,21,12 X2,13,3,14 X14,3,15,4 X4758 X12,16,7,15 X6,18,1,17
Gauss code	{1, -7, 8, -9, 2, -11}, {9, -1, 4, -5, 6, -10}, {7, -8, 10, -2, 11, -4, 3, -6, 5, -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(2)-1)^2 (t(3)-1) (t(3) t(1)-t(1)+1) (t(1) t(3)-t(3)+1)}{t(1) t(2) t(3)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{-6} -q^5-4 q^{-5} +4 q^4+9 q^{-4} -9 q^3-14 q^{-3} +15 q^2+21 q^{-2} -20 q-22 q^{-1} +24 }[/math] (db)
Signature	0 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^4 z^4+2 a^4 z^2+a^4 z^{-2} +a^4-2 a^2 z^6-z^6 a^{-2} -7 a^2 z^4-3 z^4 a^{-2} -8 a^2 z^2-3 z^2 a^{-2} -2 a^2 z^{-2} -4 a^2- a^{-2} +z^8+5 z^6+10 z^4+9 z^2+ z^{-2} +4 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ 2 a^2 z^{10}+2 z^{10}+6 a^3 z^9+13 a z^9+7 z^9 a^{-1} +7 a^4 z^8+14 a^2 z^8+10 z^8 a^{-2} +17 z^8+4 a^5 z^7-6 a^3 z^7-20 a z^7-2 z^7 a^{-1} +8 z^7 a^{-3} +a^6 z^6-16 a^4 z^6-44 a^2 z^6-16 z^6 a^{-2} +4 z^6 a^{-4} -47 z^6-9 a^5 z^5-7 a^3 z^5+3 a z^5-12 z^5 a^{-1} -12 z^5 a^{-3} +z^5 a^{-5} -2 a^6 z^4+12 a^4 z^4+45 a^2 z^4+12 z^4 a^{-2} -5 z^4 a^{-4} +48 z^4+5 a^5 z^3+6 a^3 z^3+6 a z^3+12 z^3 a^{-1} +6 z^3 a^{-3} -z^3 a^{-5} +a^6 z^2-6 a^4 z^2-25 a^2 z^2-6 z^2 a^{-2} +z^2 a^{-4} -25 z^2-2 a^3 z-4 a z-3 z a^{-1} -z a^{-3} +3 a^4+7 a^2+2 a^{-2} +7+2 a^3 z^{-1} +2 a z^{-1} -a^4 z^{-2} -2 a^2 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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 11                       1 -1 9                     3   3 7                   6 1   -5 5                 9 3     6 3               11 6       -5 1             13 9         4 -1           11 13           2 -3         10 11             -1 -5       6 13               7 -7     3 8                 -5 -9   1 6                   5 -11   3                     -3 -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}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{13} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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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