L11n308

L11n307

L11n309

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Link Presentations

[edit Notes on L11n308's Link Presentations]

Planar diagram presentation	X6172 X12,3,13,4 X7,17,8,16 X9,21,10,20 X15,9,16,8 X19,5,20,10 X13,19,14,18 X17,11,18,22 X21,15,22,14 X2536 X4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, -3, 5, -4, 6}, {11, -2, -7, 9, -5, 3, -8, 7, -6, 4, -9, 8}

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) t(3)^4-2 t(2)^2 t(3)^3+2 t(2) t(3)^3+t(3)^3+t(1) t(2)^2 t(3)^2+t(2)^2 t(3)^2-t(1) t(3)^2+t(1) t(2) t(3)^2-t(2) t(3)^2-t(3)^2-t(1) t(2)^2 t(3)+2 t(1) t(3)-2 t(1) t(2) t(3)+t(1) t(2)}{\sqrt{t(1)} t(2) t(3)^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^7+q^6-q^4+3 q^3+ q^{-3} -3 q^2-2 q^{-2} +5 q+4 q^{-1} -3 }[/math] (db)
Signature	2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^2 a^{-6} - a^{-6} z^{-2} -2 a^{-6} +z^4 a^{-4} +6 z^2 a^{-4} +4 a^{-4} z^{-2} +9 a^{-4} -2 z^4 a^{-2} +a^2 z^2-8 z^2 a^{-2} -5 a^{-2} z^{-2} +a^2-11 a^{-2} -z^4-z^2+2 z^{-2} +3 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^7 a^{-7} -6 z^5 a^{-7} +9 z^3 a^{-7} -4 z a^{-7} + a^{-7} z^{-1} +z^8 a^{-6} -7 z^6 a^{-6} +13 z^4 a^{-6} -9 z^2 a^{-6} - a^{-6} z^{-2} +3 a^{-6} +z^7 a^{-5} -10 z^5 a^{-5} +23 z^3 a^{-5} -19 z a^{-5} +5 a^{-5} z^{-1} +2 z^8 a^{-4} -15 z^6 a^{-4} +33 z^4 a^{-4} -29 z^2 a^{-4} -4 a^{-4} z^{-2} +15 a^{-4} +z^9 a^{-3} -3 z^7 a^{-3} -10 z^5 a^{-3} +37 z^3 a^{-3} -33 z a^{-3} +9 a^{-3} z^{-1} +4 z^8 a^{-2} +a^2 z^6-23 z^6 a^{-2} -4 a^2 z^4+43 z^4 a^{-2} +3 a^2 z^2-38 z^2 a^{-2} -5 a^{-2} z^{-2} -a^2+20 a^{-2} +z^9 a^{-1} +2 a z^7-z^7 a^{-1} -7 a z^5-13 z^5 a^{-1} +3 a z^3+26 z^3 a^{-1} -18 z a^{-1} +5 a^{-1} z^{-1} +3 z^8-14 z^6+19 z^4-15 z^2-2 z^{-2} +8 }[/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   \ -4 -3 -2 -1 0 1 2 3 4 5 6 7 χ 15                       1 -1 13                         0 11                 1 1 1   1 9               2 1       -1 7             2 1 1       2 5           4 5 1         0 3         2 1 1           2 1       2 5 1             2 -1     2 2 1               1 -3     2                   2 -5 1 2                     -1 -7 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{ i=3 }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=7 }[/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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