L11a202

L11a201

L11a203

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Link Presentations

[edit Notes on L11a202's Link Presentations]

Planar diagram presentation	X8192 X2,9,3,10 X10,3,11,4 X6718 X18,11,19,12 X16,6,17,5 X4,18,5,17 X20,13,21,14 X22,15,7,16 X12,19,13,20 X14,21,15,22
Gauss code	{1, -2, 3, -7, 6, -4}, {4, -1, 2, -3, 5, -10, 8, -11, 9, -6, 7, -5, 10, -8, 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{t(1)^2 t(2)^6-t(1) t(2)^6-t(1)^2 t(2)^5+t(1) t(2)^5-t(2)^5+t(1)^2 t(2)^4-t(1) t(2)^4+t(2)^4-t(1)^2 t(2)^3+t(1) t(2)^3-t(2)^3+t(1)^2 t(2)^2-t(1) t(2)^2+t(2)^2-t(1)^2 t(2)+t(1) t(2)-t(2)-t(1)+1}{t(1) t(2)^3} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{1}{q^{3/2}}+\frac{1}{q^{5/2}}-\frac{3}{q^{7/2}}+\frac{3}{q^{9/2}}-\frac{5}{q^{11/2}}+\frac{5}{q^{13/2}}-\frac{5}{q^{15/2}}+\frac{5}{q^{17/2}}-\frac{4}{q^{19/2}}+\frac{3}{q^{21/2}}-\frac{2}{q^{23/2}}+\frac{1}{q^{25/2}} }[/math] (db)
Signature	-7 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^7 a^9-6 z^5 a^9-11 z^3 a^9-7 z a^9-2 a^9 z^{-1} +z^9 a^7+8 z^7 a^7+23 z^5 a^7+30 z^3 a^7+19 z a^7+5 a^7 z^{-1} -z^7 a^5-7 z^5 a^5-16 z^3 a^5-13 z a^5-3 a^5 z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{16} z^2+2 a^{15} z^3+3 a^{14} z^4-2 a^{14} z^2+4 a^{13} z^5-6 a^{13} z^3+5 a^{12} z^6-13 a^{12} z^4+6 a^{12} z^2-a^{12}+5 a^{11} z^7-17 a^{11} z^5+13 a^{11} z^3-3 a^{11} z+4 a^{10} z^8-15 a^{10} z^6+11 a^{10} z^4+a^{10} z^2+3 a^9 z^9-14 a^9 z^7+18 a^9 z^5-9 a^9 z^3+6 a^9 z-2 a^9 z^{-1} +a^8 z^{10}-2 a^8 z^8-11 a^8 z^6+31 a^8 z^4-22 a^8 z^2+5 a^8+4 a^7 z^9-27 a^7 z^7+62 a^7 z^5-59 a^7 z^3+25 a^7 z-5 a^7 z^{-1} +a^6 z^{10}-6 a^6 z^8+9 a^6 z^6+4 a^6 z^4-14 a^6 z^2+5 a^6+a^5 z^9-8 a^5 z^7+23 a^5 z^5-29 a^5 z^3+16 a^5 z-3 a^5 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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 χ -2                       1 1 -4                         0 -6                   3 1   2 -8                 1 1     0 -10               4 2       2 -12             2 2         0 -14           3 3           0 -16         2 2             0 -18       2 3               -1 -20     1 2                 1 -22   1 2                   -1 -24   1                     1 -26 1                       -1

Integral Khovanov Homology (db, data source)

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