L11n374

L11n373

L11n375

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Link Presentations

[edit Notes on L11n374's Link Presentations]

Planar diagram presentation	X6172 X2,9,3,10 X12,3,13,4 X10,5,11,6 X16,11,5,12 X4,15,1,16 X13,20,14,21 X7,19,8,18 X17,9,18,8 X19,22,20,17 X21,14,22,15
Gauss code	{1, -2, 3, -6}, {-9, 8, -10, 7, -11, 10}, {4, -1, -8, 9, 2, -4, 5, -3, -7, 11, 6, -5}

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{(v-1) (w-1) \left(2 u w^2-u w+w-2\right)}{\sqrt{u} \sqrt{v} w^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ - q^{-9} +3 q^{-8} -5 q^{-7} +8 q^{-6} -8 q^{-5} +8 q^{-4} -6 q^{-3} +6 q^{-2} -2 q^{-1} +1 }[/math] (db)
Signature	-4 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a^{10}+z^4 a^8+4 z^2 a^8+3 a^8-z^6 a^6-4 z^4 a^6-5 z^2 a^6+a^6 z^{-2} -2 a^6-z^6 a^4-3 z^4 a^4-2 z^2 a^4-2 a^4 z^{-2} -3 a^4+z^4 a^2+3 z^2 a^2+a^2 z^{-2} +3 a^2 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^3 a^{11}-2 z a^{11}+3 z^4 a^{10}-4 z^2 a^{10}+a^{10}+2 z^7 a^9-7 z^5 a^9+14 z^3 a^9-6 z a^9+3 z^8 a^8-12 z^6 a^8+22 z^4 a^8-12 z^2 a^8+3 a^8+z^9 a^7+2 z^7 a^7-14 z^5 a^7+20 z^3 a^7-6 z a^7+5 z^8 a^6-15 z^6 a^6+14 z^4 a^6-5 z^2 a^6+a^6 z^{-2} -a^6+z^9 a^5+2 z^7 a^5-12 z^5 a^5+7 z^3 a^5+2 z a^5-2 a^5 z^{-1} +2 z^8 a^4-2 z^6 a^4-9 z^4 a^4+9 z^2 a^4+2 a^4 z^{-2} -6 a^4+2 z^7 a^3-5 z^5 a^3+4 z a^3-2 a^3 z^{-1} +z^6 a^2-4 z^4 a^2+6 z^2 a^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 3     0 -7           5 3       2 -9       1 4 3         0 -11       5 5           0 -13     2 5             3 -15   1 3               -2 -17   2                 2 -19 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}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/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=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/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.

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