L11n314

L11n313

L11n315

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Link Presentations

[edit Notes on L11n314's Link Presentations]

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

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-5 }[/math] [math]\displaystyle{ i=-3 }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-3 }[/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}^{5}\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^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/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}_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.

L11n313 L11n315