L11n269

L11n268

L11n270

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Link Presentations

[edit Notes on L11n269's Link Presentations]

Planar diagram presentation	X6172 X3,11,4,10 X7,17,8,16 X15,5,16,8 X18,11,19,12 X22,17,9,18 X20,13,21,14 X12,19,13,20 X14,21,15,22 X2536 X9,1,10,4
Gauss code	{1, -10, -2, 11}, {10, -1, -3, 4}, {-11, 2, 5, -8, 7, -9, -4, 3, 6, -5, 8, -7, 9, -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(3)^5-t(1) t(3)^4-t(2) t(3)^4+t(1) t(3)^3+t(2) t(3)^3-t(1) t(3)^2-t(2) t(3)^2+t(1) t(3)+t(2) t(3)-t(1) t(2)}{\sqrt{t(1)} \sqrt{t(2)} t(3)^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ - q^{-8} + q^{-7} -2 q^{-6} +2 q^{-5} -3 q^{-4} +3 q^{-3} +q^2+2 q^{-1} +1 }[/math] (db)
Signature	-2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^6 \left(-z^4\right)-4 a^6 z^2-2 a^6 z^{-2} -5 a^6+a^4 z^6+7 a^4 z^4+18 a^4 z^2+7 a^4 z^{-2} +18 a^4-a^2 z^6-8 a^2 z^4-20 a^2 z^2-8 a^2 z^{-2} -20 a^2+z^4+5 z^2+3 z^{-2} +7 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^9 z^5-4 a^9 z^3+3 a^9 z-a^9 z^{-1} +a^8 z^6-3 a^8 z^4+a^8+a^7 z^7-3 a^7 z^5+3 a^7 z-a^7 z^{-1} +a^6 z^8-5 a^6 z^6+9 a^6 z^4-9 a^6 z^2-2 a^6 z^{-2} +7 a^6+2 a^5 z^7-12 a^5 z^5+26 a^5 z^3-21 a^5 z+7 a^5 z^{-1} +a^4 z^8-8 a^4 z^6+24 a^4 z^4-32 a^4 z^2-7 a^4 z^{-2} +22 a^4+2 a^3 z^7-17 a^3 z^5+46 a^3 z^3-45 a^3 z+15 a^3 z^{-1} +a^2 z^8-10 a^2 z^6+33 a^2 z^4-46 a^2 z^2-8 a^2 z^{-2} +28 a^2+a z^7-9 a z^5+24 a z^3-24 a z+8 a z^{-1} +z^8-8 z^6+21 z^4-23 z^2-3 z^{-2} +13 }[/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 3 4 χ 5                       1 1 3                       1 1 1                   1     1 -1               3         3 -3             1 4 1       2 -5           3             3 -7         1 2 1           0 -9       2 3               -1 -11       1 1               0 -13   1 2                   -1 -15                         0 -17 1                       -1

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