L11a43

L11a42

L11a44

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Link Presentations

[edit Notes on L11a43's Link Presentations]

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

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{2 (t(1)-1) (t(2)-1) \left(t(2)^2-3 t(2)+1\right)}{\sqrt{t(1)} t(2)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -3 q^{9/2}+\frac{2}{q^{9/2}}+5 q^{7/2}-\frac{5}{q^{7/2}}-9 q^{5/2}+\frac{8}{q^{5/2}}+11 q^{3/2}-\frac{10}{q^{3/2}}+q^{11/2}-\frac{1}{q^{11/2}}-13 \sqrt{q}+\frac{12}{\sqrt{q}} }[/math] (db)
Signature	1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^5 z+a^5 z^{-1} +z a^{-5} -2 a^3 z^3-2 z^3 a^{-3} -3 a^3 z-2 a^3 z^{-1} -2 z a^{-3} - a^{-3} z^{-1} +a z^5+z^5 a^{-1} +a z^3+z^3 a^{-1} +a z+a z^{-1} +2 z a^{-1} + a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -a^2 z^{10}-z^{10}-2 a^3 z^9-5 a z^9-3 z^9 a^{-1} -2 a^4 z^8-2 a^2 z^8-4 z^8 a^{-2} -4 z^8-a^5 z^7+4 a^3 z^7+12 a z^7+3 z^7 a^{-1} -4 z^7 a^{-3} +8 a^4 z^6+16 a^2 z^6+3 z^6 a^{-2} -4 z^6 a^{-4} +15 z^6+5 a^5 z^5+8 a^3 z^5-3 z^5 a^{-5} -9 a^4 z^4-18 a^2 z^4+z^4 a^{-2} +3 z^4 a^{-4} -z^4 a^{-6} -12 z^4-8 a^5 z^3-20 a^3 z^3-13 a z^3+2 z^3 a^{-1} +7 z^3 a^{-3} +4 z^3 a^{-5} +4 a^4 z^2+8 a^2 z^2+z^2 a^{-6} +5 z^2+5 a^5 z+13 a^3 z+8 a z-3 z a^{-1} -5 z a^{-3} -2 z a^{-5} -a^4-3 a^2- a^{-2} -2-a^5 z^{-1} -2 a^3 z^{-1} -a z^{-1} + a^{-1} z^{-1} + a^{-3} z^{-1} }[/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 12                       1 -1 10                     2   2 8                   3 1   -2 6                 6 2     4 4               5 3       -2 2             8 6         2 0           6 7           1 -2         4 6             -2 -4       4 6               2 -6     1 4                 -3 -8   1 4                   3 -10   1                     -1 -12 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=-6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/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}^{4}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\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^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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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