L11n124

L11n123

L11n125

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Link Presentations

[edit Notes on L11n124's Link Presentations]

Planar diagram presentation	X6172 X3,15,4,14 X9,22,10,5 X7,19,8,18 X17,9,18,8 X19,13,20,12 X11,21,12,20 X15,10,16,11 X21,16,22,17 X2536 X13,1,14,4
Gauss code	{1, -10, -2, 11}, {10, -1, -4, 5, -3, 8, -7, 6, -11, 2, -8, 9, -5, 4, -6, 7, -9, 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{(u-1) (v-1) \left(2 v^2-3 v+2\right)}{\sqrt{u} v^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ 3 q^{9/2}-6 q^{7/2}+\frac{1}{q^{7/2}}+8 q^{5/2}-\frac{4}{q^{5/2}}-9 q^{3/2}+\frac{6}{q^{3/2}}-q^{11/2}+9 \sqrt{q}-\frac{9}{\sqrt{q}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -2 z^5 a^{-1} +3 a z^3-7 z^3 a^{-1} +3 z^3 a^{-3} -a^3 z+5 a z-9 z a^{-1} +6 z a^{-3} -z a^{-5} +2 a z^{-1} -4 a^{-1} z^{-1} +3 a^{-3} z^{-1} - a^{-5} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -2 z^9 a^{-1} -2 z^9 a^{-3} -10 z^8 a^{-2} -3 z^8 a^{-4} -7 z^8-7 a z^7-5 z^7 a^{-1} +z^7 a^{-3} -z^7 a^{-5} -2 a^2 z^6+39 z^6 a^{-2} +12 z^6 a^{-4} +25 z^6+25 a z^5+42 z^5 a^{-1} +21 z^5 a^{-3} +4 z^5 a^{-5} +a^2 z^4-42 z^4 a^{-2} -14 z^4 a^{-4} -27 z^4-4 a^3 z^3-32 a z^3-56 z^3 a^{-1} -34 z^3 a^{-3} -6 z^3 a^{-5} -a^4 z^2+17 z^2 a^{-2} +6 z^2 a^{-4} +12 z^2+3 a^3 z+15 a z+26 z a^{-1} +18 z a^{-3} +4 z a^{-5} -a^2-3 a^{-2} - a^{-4} -2-2 a z^{-1} -4 a^{-1} z^{-1} -3 a^{-3} z^{-1} - a^{-5} 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   \ -3 -2 -1 0 1 2 3 4 5 6 χ 12                   1 1 10                 2   -2 8               4 1   3 6             4 2     -2 4           5 4       1 2       1 5 4         0 0       5 5           0 -2     3 6             3 -4   1 3               -2 -6   3                 3 -8 1                   -1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-2 }[/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}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/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=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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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