L11a10

L11a9

L11a11

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Link Presentations

[edit Notes on L11a10's Link Presentations]

Planar diagram presentation	X6172 X18,7,19,8 X4,19,1,20 X12,6,13,5 X8493 X16,10,17,9 X22,14,5,13 X10,16,11,15 X14,22,15,21 X20,12,21,11 X2,18,3,17
Gauss code	{1, -11, 5, -3}, {4, -1, 2, -5, 6, -8, 10, -4, 7, -9, 8, -6, 11, -2, 3, -10, 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{(t(1)-1) (t(2)-1) \left(4 t(2)^2-7 t(2)+4\right)}{\sqrt{t(1)} t(2)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^{17/2}+4 q^{15/2}-7 q^{13/2}+12 q^{11/2}-16 q^{9/2}+18 q^{7/2}-20 q^{5/2}+16 q^{3/2}-13 \sqrt{q}+\frac{8}{\sqrt{q}}-\frac{4}{q^{3/2}}+\frac{1}{q^{5/2}} }[/math] (db)
Signature	1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^3 a^{-7} + a^{-7} z^{-1} +z^5 a^{-5} -2 z a^{-5} -2 a^{-5} z^{-1} +2 z^5 a^{-3} +3 z^3 a^{-3} +2 z a^{-3} +z^5 a^{-1} -a z^3+ a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^7 a^{-9} -3 z^5 a^{-9} +2 z^3 a^{-9} +4 z^8 a^{-8} -15 z^6 a^{-8} +16 z^4 a^{-8} -3 z^2 a^{-8} -2 a^{-8} +5 z^9 a^{-7} -16 z^7 a^{-7} +13 z^5 a^{-7} -2 z^3 a^{-7} + a^{-7} z^{-1} +2 z^{10} a^{-6} +5 z^8 a^{-6} -32 z^6 a^{-6} +32 z^4 a^{-6} -3 z^2 a^{-6} -5 a^{-6} +11 z^9 a^{-5} -27 z^7 a^{-5} +11 z^5 a^{-5} +5 z^3 a^{-5} -3 z a^{-5} +2 a^{-5} z^{-1} +2 z^{10} a^{-4} +11 z^8 a^{-4} -37 z^6 a^{-4} +25 z^4 a^{-4} -z^2 a^{-4} -3 a^{-4} +6 z^9 a^{-3} +z^7 a^{-3} -24 z^5 a^{-3} +19 z^3 a^{-3} -4 z a^{-3} +10 z^8 a^{-2} -12 z^6 a^{-2} +a^2 z^4+z^4 a^{-2} + a^{-2} +11 z^7 a^{-1} +4 a z^5-15 z^5 a^{-1} -2 a z^3+8 z^3 a^{-1} -z a^{-1} - a^{-1} z^{-1} +8 z^6-7 z^4+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   \ -3 -2 -1 0 1 2 3 4 5 6 7 8 χ 18                       1 1 16                     3   -3 14                   4 1   3 12                 8 3     -5 10               8 4       4 8             10 8         -2 6           10 8           2 4         6 10             4 2       7 10               -3 0     3 8                 5 -2   1 5                   -4 -4   3                     3 -6 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=-3 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=7 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=8 }[/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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