L11n254

L11n253

L11n255

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Link Presentations

[edit Notes on L11n254's Link Presentations]

Planar diagram presentation	X12,1,13,2 X2,13,3,14 X14,3,15,4 X16,5,17,6 X22,7,11,8 X9,18,10,19 X17,20,18,21 X19,10,20,1 X8,11,9,12 X4,15,5,16 X6,21,7,22
Gauss code	{1, -2, 3, -10, 4, -11, 5, -9, -6, 8}, {9, -1, 2, -3, 10, -4, -7, 6, -8, 7, 11, -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{u^4 v^4+u^3 v^2+u^2 v^3-u^2 v^2+u^2 v+u v^2+1}{u^2 v^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{1}{q^{9/2}}-\frac{1}{q^{13/2}}+\frac{1}{q^{17/2}}-\frac{1}{q^{19/2}}+\frac{2}{q^{21/2}}-\frac{2}{q^{23/2}}+\frac{1}{q^{25/2}}-\frac{1}{q^{27/2}} }[/math] (db)
Signature	-7 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^3 a^{13}-2 z a^{13}+z^7 a^{11}+8 z^5 a^{11}+19 z^3 a^{11}+14 z a^{11}+a^{11} z^{-1} -z^9 a^9-9 z^7 a^9-28 z^5 a^9-36 z^3 a^9-17 z a^9-a^9 z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{17} z^3-2 a^{17} z+a^{16} z^4-a^{16} z^2+a^{15} z^5-2 a^{15} z^3+2 a^{15} z+a^{14} z^4+2 a^{13} z+a^{12} z^8-8 a^{12} z^6+19 a^{12} z^4-13 a^{12} z^2+a^{11} z^9-9 a^{11} z^7+27 a^{11} z^5-33 a^{11} z^3+15 a^{11} z-a^{11} z^{-1} +a^{10} z^8-8 a^{10} z^6+19 a^{10} z^4-14 a^{10} z^2+a^{10}+a^9 z^9-9 a^9 z^7+28 a^9 z^5-36 a^9 z^3+17 a^9 z-a^9 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   \ -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -8                     1 1 -10                     1 1 -12                 1     1 -14             1         1 -16           1 1 1       -1 -18         1 1           0 -20       1 1 1           -1 -22     1 1               0 -24     1                 1 -26 1 1                   0 -28 1                     1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-10 }[/math] [math]\displaystyle{ i=-8 }[/math] [math]\displaystyle{ i=-6 }[/math] [math]\displaystyle{ r=-10 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-9 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/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=-1 }[/math] [math]\displaystyle{ r=0 }[/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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