L11n341

L11n340

L11n342

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Link Presentations

[edit Notes on L11n341's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X18,12,19,11 X14,8,15,7 X8,14,9,13 X19,22,20,13 X15,20,16,21 X21,16,22,17 X12,18,5,17 X2536 X4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, 4, -5, 11, -2, 3, -9}, {5, -4, -7, 8, 9, -3, -6, 7, -8, 6}

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)-1) \left(t(1) t(3)^3-t(1) t(2) t(3)^3+2 t(2) t(3)^3+2 t(1) t(2) t(3)^2-2 t(2) t(3)^2-2 t(1) t(2) t(3)+2 t(2) t(3)+t(2)^2+2 t(1) t(2)-t(2)\right)}{\sqrt{t(1)} t(2)^{3/2} t(3)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -2 q^{-6} +5 q^{-5} -8 q^{-4} +q^3+11 q^{-3} -2 q^2-10 q^{-2} +6 q+11 q^{-1} -8 }[/math] (db)
Signature	-2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a^6 z^{-2} -a^6-z^4 a^4+z^2 a^4+4 a^4 z^{-2} +6 a^4+z^6 a^2+2 z^4 a^2-2 z^2 a^2-5 a^2 z^{-2} -7 a^2-2 z^4-4 z^2+2 z^{-2} +z^2 a^{-2} +2 a^{-2} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^3 z^9+a z^9+4 a^4 z^8+7 a^2 z^8+3 z^8+4 a^5 z^7+8 a^3 z^7+6 a z^7+2 z^7 a^{-1} +a^6 z^6-9 a^4 z^6-19 a^2 z^6+z^6 a^{-2} -8 z^6-9 a^5 z^5-29 a^3 z^5-25 a z^5-5 z^5 a^{-1} +4 a^6 z^4+19 a^4 z^4+27 a^2 z^4-4 z^4 a^{-2} +8 z^4+3 a^7 z^3+20 a^5 z^3+43 a^3 z^3+28 a z^3+2 z^3 a^{-1} -4 a^6 z^2-24 a^4 z^2-34 a^2 z^2+5 z^2 a^{-2} -9 z^2-3 a^7 z-17 a^5 z-33 a^3 z-18 a z+z a^{-1} +3 a^6+16 a^4+21 a^2-2 a^{-2} +7+a^7 z^{-1} +5 a^5 z^{-1} +9 a^3 z^{-1} +5 a z^{-1} -a^6 z^{-2} -4 a^4 z^{-2} -5 a^2 z^{-2} -2 z^{-2} }[/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 7                   1 1 5                 2 1 -1 3               4     4 1             4 2     -2 -1           7 4       3 -3         5 6         1 -5       6 5           1 -7     2 5             3 -9   3 6               -3 -11   3                 3 -13 2                   -2

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=-1 }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/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=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=1 }[/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}^{2}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=4 }[/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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