L11n246

L11n245

L11n247

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Link Presentations

[edit Notes on L11n246's Link Presentations]

Planar diagram presentation	X12,1,13,2 X3849 X5,14,6,15 X7,18,8,19 X9,21,10,20 X10,11,1,12 X13,6,14,7 X17,4,18,5 X15,11,16,22 X19,3,20,2 X21,17,22,16
Gauss code	{1, 10, -2, 8, -3, 7, -4, 2, -5, -6}, {6, -1, -7, 3, -9, 11, -8, 4, -10, 5, -11, 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+v) \left(u^2 v^2-2 u^2 v+u^2-2 u v^2+5 u v-2 u+v^2-2 v+1\right)}{u^{3/2} v^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^{3/2}+4 \sqrt{q}-\frac{8}{\sqrt{q}}+\frac{10}{q^{3/2}}-\frac{12}{q^{5/2}}+\frac{11}{q^{7/2}}-\frac{10}{q^{9/2}}+\frac{7}{q^{11/2}}-\frac{4}{q^{13/2}}+\frac{1}{q^{15/2}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^7 (-z)+3 a^5 z^3+4 a^5 z+a^5 z^{-1} -2 a^3 z^5-6 a^3 z^3-7 a^3 z-a^3 z^{-1} +3 a z^3+3 a z-z a^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -2 a^5 z^9-2 a^3 z^9-5 a^6 z^8-9 a^4 z^8-4 a^2 z^8-4 a^7 z^7-4 a^5 z^7-2 a^3 z^7-2 a z^7-a^8 z^6+12 a^6 z^6+23 a^4 z^6+10 a^2 z^6+11 a^7 z^5+24 a^5 z^5+15 a^3 z^5+2 a z^5+2 a^8 z^4-3 a^6 z^4-15 a^4 z^4-14 a^2 z^4-4 z^4-7 a^7 z^3-20 a^5 z^3-19 a^3 z^3-7 a z^3-z^3 a^{-1} -a^8 z^2-a^6 z^2+2 a^4 z^2+4 a^2 z^2+2 z^2+a^7 z+7 a^5 z+8 a^3 z+3 a z+z a^{-1} +a^4-a^5 z^{-1} -a^3 z^{-1} }[/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 4                   1 1 2                 3   -3 0               5 1   4 -2             6 4     -2 -4           6 4       2 -6         5 6         1 -8       5 6           -1 -10     3 6             3 -12   1 4               -3 -14   3                 3 -16 1                   -1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-2 }[/math] [math]\displaystyle{ i=0 }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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}^{6}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\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^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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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