L11a40

L11a39

L11a41

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Link Presentations

[edit Notes on L11a40's Link Presentations]

Planar diagram presentation	X6172 X10,4,11,3 X12,10,13,9 X18,15,19,16 X16,7,17,8 X8,17,9,18 X22,20,5,19 X20,14,21,13 X14,22,15,21 X2536 X4,12,1,11
Gauss code	{1, -10, 2, -11}, {10, -1, 5, -6, 3, -2, 11, -3, 8, -9, 4, -5, 6, -4, 7, -8, 9, -7}

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{(u-1) (v-1)^5}{\sqrt{u} v^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^{13/2}+4 q^{11/2}-8 q^{9/2}+14 q^{7/2}-19 q^{5/2}+20 q^{3/2}-21 \sqrt{q}+\frac{17}{\sqrt{q}}-\frac{13}{q^{3/2}}+\frac{7}{q^{5/2}}-\frac{3}{q^{7/2}}+\frac{1}{q^{9/2}} }[/math] (db)
Signature	1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^7 a^{-1} +2 a z^5-4 z^5 a^{-1} +2 z^5 a^{-3} -a^3 z^3+6 a z^3-9 z^3 a^{-1} +5 z^3 a^{-3} -z^3 a^{-5} -2 a^3 z+8 a z-10 z a^{-1} +5 z a^{-3} -z a^{-5} -a^3 z^{-1} +4 a z^{-1} -4 a^{-1} z^{-1} + a^{-3} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^5 a^{-7} -z^3 a^{-7} +4 z^6 a^{-6} -6 z^4 a^{-6} +3 z^2 a^{-6} +7 z^7 a^{-5} -10 z^5 a^{-5} +6 z^3 a^{-5} -2 z a^{-5} +7 z^8 a^{-4} +a^4 z^6-3 z^6 a^{-4} -3 a^4 z^4-10 z^4 a^{-4} +3 a^4 z^2+9 z^2 a^{-4} -a^4- a^{-4} +4 z^9 a^{-3} +3 a^3 z^7+10 z^7 a^{-3} -8 a^3 z^5-31 z^5 a^{-3} +8 a^3 z^3+26 z^3 a^{-3} -4 a^3 z-10 z a^{-3} +a^3 z^{-1} + a^{-3} z^{-1} +z^{10} a^{-2} +4 a^2 z^8+14 z^8 a^{-2} -5 a^2 z^6-21 z^6 a^{-2} -5 a^2 z^4-3 z^4 a^{-2} +10 a^2 z^2+13 z^2 a^{-2} -4 a^2-4 a^{-2} +3 a z^9+7 z^9 a^{-1} +5 a z^7+5 z^7 a^{-1} -26 a z^5-38 z^5 a^{-1} +29 a z^3+40 z^3 a^{-1} -15 a z-19 z a^{-1} +4 a z^{-1} +4 a^{-1} z^{-1} +z^{10}+11 z^8-20 z^6-z^4+14 z^2-7 }[/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 14                       1 1 12                     3   -3 10                   5 1   4 8                 9 3     -6 6               10 5       5 4             10 9         -1 2           11 10           1 0         8 12             4 -2       5 9               -4 -4     2 8                 6 -6   1 5                   -4 -8   2                     2 -10 1                       -1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=0 }[/math] [math]\displaystyle{ i=2 }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/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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