L11n233

L11n232

L11n234

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Link Presentations

[edit Notes on L11n233's Link Presentations]

Planar diagram presentation	X10,1,11,2 X2,11,3,12 X12,3,13,4 X9,18,10,19 X6,13,7,14 X14,7,15,8 X8,15,1,16 X17,22,18,9 X21,16,22,17 X19,4,20,5 X5,20,6,21
Gauss code	{1, -2, 3, 10, -11, -5, 6, -7}, {-4, -1, 2, -3, 5, -6, 7, 9, -8, 4, -10, 11, -9, 8}

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^3 v^5+u^2 v^4-3 u^2 v^3+2 u^2 v^2-u^2 v-u v^4+2 u v^3-3 u v^2+u v-1}{u^{3/2} v^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{1}{q^{9/2}}-\frac{1}{q^{13/2}}-\frac{1}{q^{15/2}}+\frac{3}{q^{17/2}}-\frac{4}{q^{19/2}}+\frac{5}{q^{21/2}}-\frac{5}{q^{23/2}}+\frac{4}{q^{25/2}}-\frac{3}{q^{27/2}}+\frac{1}{q^{29/2}} }[/math] (db)
Signature	-7 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -2 a^{13} z^3-4 a^{13} z-a^{13} z^{-1} +a^{11} z^7+9 a^{11} z^5+23 a^{11} z^3+19 a^{11} z+3 a^{11} z^{-1} -a^9 z^9-9 a^9 z^7-28 a^9 z^5-37 a^9 z^3-19 a^9 z-2 a^9 z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{18} z^4-a^{18} z^2+3 a^{17} z^5-5 a^{17} z^3+a^{17} z+3 a^{16} z^6-4 a^{16} z^4+a^{15} z^7+3 a^{15} z^5-7 a^{15} z^3+3 a^{15} z+3 a^{14} z^6-2 a^{14} z^4-a^{14} z^2+a^{14}+4 a^{13} z^5-6 a^{13} z^3+5 a^{13} z-a^{13} z^{-1} +a^{12} z^8-9 a^{12} z^6+26 a^{12} z^4-21 a^{12} z^2+3 a^{12}+a^{11} z^9-10 a^{11} z^7+32 a^{11} z^5-41 a^{11} z^3+22 a^{11} z-3 a^{11} z^{-1} +a^{10} z^8-9 a^{10} z^6+23 a^{10} z^4-19 a^{10} z^2+3 a^{10}+a^9 z^9-9 a^9 z^7+28 a^9 z^5-37 a^9 z^3+19 a^9 z-2 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   \ -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -8                       1 1 -10                       1 1 -12                   1     1 -14               2         2 -16             2 1 1       -2 -18           3 2           1 -20         2 2 1           -1 -22       3 3               0 -24     2 3                 1 -26   1 2                   -1 -28   2                     2 -30 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=-11 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-10 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-9 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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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