L11a135

L11a134

L11a136

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Link Presentations

[edit Notes on L11a135's Link Presentations]

Planar diagram presentation	X6172 X2,9,3,10 X14,3,15,4 X10,5,11,6 X20,11,21,12 X22,13,5,14 X12,21,13,22 X4,19,1,20 X18,16,19,15 X16,8,17,7 X8,18,9,17
Gauss code	{1, -2, 3, -8}, {4, -1, 10, -11, 2, -4, 5, -7, 6, -3, 9, -10, 11, -9, 8, -5, 7, -6}

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{2 t(1) t(2)^5-2 t(2)^5-6 t(1) t(2)^4+6 t(2)^4+10 t(1) t(2)^3-9 t(2)^3-9 t(1) t(2)^2+10 t(2)^2+6 t(1) t(2)-6 t(2)-2 t(1)+2}{\sqrt{t(1)} t(2)^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{3/2}-4 \sqrt{q}+\frac{8}{\sqrt{q}}-\frac{14}{q^{3/2}}+\frac{19}{q^{5/2}}-\frac{23}{q^{7/2}}+\frac{22}{q^{9/2}}-\frac{20}{q^{11/2}}+\frac{15}{q^{13/2}}-\frac{9}{q^{15/2}}+\frac{4}{q^{17/2}}-\frac{1}{q^{19/2}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^7 z^5+2 a^7 z^3+a^7 z-a^5 z^7-3 a^5 z^5-3 a^5 z^3-a^5 z+a^5 z^{-1} -a^3 z^7-3 a^3 z^5-3 a^3 z^3-2 a^3 z-a^3 z^{-1} +a z^5+2 a z^3 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^5 a^{11}+z^3 a^{11}-4 z^6 a^{10}+5 z^4 a^{10}-2 z^2 a^{10}-8 z^7 a^9+12 z^5 a^9-7 z^3 a^9+z a^9-9 z^8 a^8+10 z^6 a^8-z^4 a^8-z^2 a^8-6 z^9 a^7-2 z^7 a^7+14 z^5 a^7-6 z^3 a^7-2 z^{10} a^6-13 z^8 a^6+27 z^6 a^6-14 z^4 a^6+3 z^2 a^6-12 z^9 a^5+19 z^7 a^5-9 z^5 a^5+4 z^3 a^5+z a^5-a^5 z^{-1} -2 z^{10} a^4-11 z^8 a^4+32 z^6 a^4-24 z^4 a^4+4 z^2 a^4+a^4-6 z^9 a^3+9 z^7 a^3-4 z^3 a^3+2 z a^3-a^3 z^{-1} -7 z^8 a^2+18 z^6 a^2-14 z^4 a^2+2 z^2 a^2-4 z^7 a+10 z^5 a-6 z^3 a-z^6+2 z^4 }[/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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 χ 4                       1 -1 2                     3   3 0                   5 1   -4 -2                 9 3     6 -4               11 6       -5 -6             12 8         4 -8           10 11           1 -10         10 12             -2 -12       6 11               5 -14     3 9                 -6 -16   1 6                   5 -18   3                     -3 -20 1                       1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-4 }[/math] [math]\displaystyle{ i=-2 }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{12} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=3 }[/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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