L11n294

L11n293

L11n295

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Link Presentations

[edit Notes on L11n294's Link Presentations]

Planar diagram presentation	X6172 X12,4,13,3 X20,16,21,15 X14,8,15,7 X10,22,5,21 X11,19,12,18 X9,17,10,16 X17,11,18,22 X19,9,20,8 X2536 X4,14,1,13
Gauss code	{1, -10, 2, -11}, {10, -1, 4, 9, -7, -5}, {-6, -2, 11, -4, 3, 7, -8, 6, -9, -3, 5, 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(1) t(2)^2 t(3)^4-t(2)^2 t(3)^4-t(1) t(2) t(3)^4+t(2)^2 t(3)^3+t(1) t(2) t(3)^2-t(2) t(3)^2-t(1) t(3)+t(1)+t(2)-1}{\sqrt{t(1)} t(2) t(3)^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^8+q^7-q^6+3 q^5-q^4+3 q^3-q^2+q }[/math] (db)
Signature	6 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^8 a^{-6} +z^6 a^{-4} -7 z^6 a^{-6} +z^6 a^{-8} +6 z^4 a^{-4} -17 z^4 a^{-6} +6 z^4 a^{-8} +11 z^2 a^{-4} -21 z^2 a^{-6} +10 z^2 a^{-8} -z^2 a^{-10} +8 a^{-4} -15 a^{-6} +8 a^{-8} - a^{-10} +2 a^{-4} z^{-2} -5 a^{-6} z^{-2} +4 a^{-8} z^{-2} - a^{-10} z^{-2} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -2 z a^{-11} + a^{-11} z^{-1} -3 z^2 a^{-10} - a^{-10} z^{-2} +2 a^{-10} +2 z^7 a^{-9} -12 z^5 a^{-9} +19 z^3 a^{-9} -13 z a^{-9} +5 a^{-9} z^{-1} +3 z^8 a^{-8} -19 z^6 a^{-8} +36 z^4 a^{-8} -29 z^2 a^{-8} -4 a^{-8} z^{-2} +13 a^{-8} +z^9 a^{-7} -3 z^7 a^{-7} -9 z^5 a^{-7} +31 z^3 a^{-7} -27 z a^{-7} +9 a^{-7} z^{-1} +4 z^8 a^{-6} -26 z^6 a^{-6} +53 z^4 a^{-6} -45 z^2 a^{-6} -5 a^{-6} z^{-2} +20 a^{-6} +z^9 a^{-5} -5 z^7 a^{-5} +3 z^5 a^{-5} +12 z^3 a^{-5} -16 z a^{-5} +5 a^{-5} z^{-1} +z^8 a^{-4} -7 z^6 a^{-4} +17 z^4 a^{-4} -19 z^2 a^{-4} -2 a^{-4} z^{-2} +10 a^{-4} }[/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   \ -2 -1 0 1 2 3 4 5 6 χ 17               2 1 -1 15             1 2 1 0 13           2 2     0 11         1 1 2     2 9       1 3         2 7     2 1 1         2 5   1 3             2 3                   0 1 1                 1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=3 }[/math] [math]\displaystyle{ i=5 }[/math] [math]\displaystyle{ i=7 }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=3 }[/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}^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\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}^{2} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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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