L11n405

L11n404

L11n406

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Link Presentations

[edit Notes on L11n405's Link Presentations]

Planar diagram presentation	X6172 X14,7,15,8 X16,9,17,10 X8,15,9,16 X4,17,1,18 X22,12,19,11 X10,4,11,3 X5,21,6,20 X21,5,22,18 X12,20,13,19 X2,14,3,13
Gauss code	{1, -11, 7, -5}, {10, 8, -9, -6}, {-8, -1, 2, -4, 3, -7, 6, -10, 11, -2, 4, -3, 5, 9}

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) (w-1)^2 \left(w^2+1\right) (v w+1)}{\sqrt{u} \sqrt{v} w^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ - q^{-5} +3 q^4+3 q^{-4} -4 q^3-6 q^{-3} +8 q^2+8 q^{-2} -9 q-10 q^{-1} +12 }[/math] (db)
Signature	0 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^8-a^2 z^6-z^6 a^{-2} +6 z^6-4 a^2 z^4-5 z^4 a^{-2} +13 z^4-5 a^2 z^2-10 z^2 a^{-2} +z^2 a^{-4} +14 z^2-3 a^2-11 a^{-2} +3 a^{-4} +11-a^2 z^{-2} -5 a^{-2} z^{-2} +2 a^{-4} z^{-2} +4 z^{-2} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^5 z^5-2 a^5 z^3+a^5 z+3 a^4 z^6-6 a^4 z^4+6 z^4 a^{-4} +a^4 z^2-17 z^2 a^{-4} -2 a^{-4} z^{-2} +10 a^{-4} +4 a^3 z^7+3 z^7 a^{-3} -7 a^3 z^5-9 z^5 a^{-3} +a^3 z^3+14 z^3 a^{-3} -2 a^3 z-15 z a^{-3} +a^3 z^{-1} +5 a^{-3} z^{-1} +4 a^2 z^8+5 z^8 a^{-2} -9 a^2 z^6-22 z^6 a^{-2} +10 a^2 z^4+46 z^4 a^{-2} -9 a^2 z^2-45 z^2 a^{-2} -a^2 z^{-2} -5 a^{-2} z^{-2} +4 a^2+22 a^{-2} +2 a z^9+2 z^9 a^{-1} -z^7 a^{-1} -11 a z^5-12 z^5 a^{-1} +23 a z^3+34 z^3 a^{-1} -17 a z-29 z a^{-1} +5 a z^{-1} +9 a^{-1} z^{-1} +9 z^8-34 z^6+56 z^4-38 z^2-4 z^{-2} +17 }[/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   \ -5 -4 -3 -2 -1 0 1 2 3 4 χ 9                   3 3 7                 3 2 -1 5               5 1   4 3             4 3     -1 1           8 5       3 -1         4 6         2 -3       4 6           -2 -5     2 4             2 -7   1 4               -3 -9   2                 2 -11 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{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\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^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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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