L11n35

L11n34

L11n36

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Link Presentations

[edit Notes on L11n35's Link Presentations]

Planar diagram presentation	X6172 X20,7,21,8 X4,21,1,22 X9,14,10,15 X8493 X12,5,13,6 X22,13,5,14 X15,18,16,19 X11,17,12,16 X17,11,18,10 X2,20,3,19
Gauss code	{1, -11, 5, -3}, {6, -1, 2, -5, -4, 10, -9, -6, 7, 4, -8, 9, -10, 8, 11, -2, 3, -7}

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(1)-1) (t(2)-2) (t(2)-1) (2 t(2)-1)}{\sqrt{t(1)} t(2)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{10}{q^{9/2}}+\frac{12}{q^{7/2}}-\frac{12}{q^{5/2}}-2 q^{3/2}+\frac{11}{q^{3/2}}+\frac{1}{q^{15/2}}-\frac{4}{q^{13/2}}+\frac{7}{q^{11/2}}+4 \sqrt{q}-\frac{9}{\sqrt{q}} }[/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^3-8 a^3 z-3 a^3 z^{-1} +4 a z^3+7 a z+4 a z^{-1} -2 z a^{-1} -2 a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -2 a^5 z^9-2 a^3 z^9-5 a^6 z^8-10 a^4 z^8-5 a^2 z^8-4 a^7 z^7-5 a^5 z^7-5 a^3 z^7-4 a z^7-a^8 z^6+12 a^6 z^6+26 a^4 z^6+12 a^2 z^6-z^6+11 a^7 z^5+27 a^5 z^5+26 a^3 z^5+10 a z^5+2 a^8 z^4-4 a^6 z^4-17 a^4 z^4-13 a^2 z^4-2 z^4-7 a^7 z^3-22 a^5 z^3-33 a^3 z^3-21 a z^3-3 z^3 a^{-1} -a^8 z^2-a^2 z^2+a^7 z+6 a^5 z+15 a^3 z+15 a z+5 z a^{-1} +a^6+3 a^4+3 a^2+2-a^5 z^{-1} -3 a^3 z^{-1} -4 a z^{-1} -2 a^{-1} 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                   2 2 2                 2   -2 0               7 2   5 -2             6 4     -2 -4           6 5       1 -6         6 6         0 -8       4 6           -2 -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}^{4} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/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}^{5}\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^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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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