L11a210

L11a209

L11a211

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Link Presentations

[edit Notes on L11a210's Link Presentations]

Planar diagram presentation	X8192 X20,9,21,10 X6718 X22,15,7,16 X4,14,5,13 X16,6,17,5 X12,18,13,17 X10,4,11,3 X18,12,19,11 X14,21,15,22 X2,20,3,19
Gauss code	{1, -11, 8, -5, 6, -3}, {3, -1, 2, -8, 9, -7, 5, -10, 4, -6, 7, -9, 11, -2, 10, -4}

A Braid Representative	{{{braid_table}}}
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)^2 t(2)^4-3 t(1) t(2)^4+t(2)^4-5 t(1)^2 t(2)^3+9 t(1) t(2)^3-4 t(2)^3+6 t(1)^2 t(2)^2-11 t(1) t(2)^2+6 t(2)^2-4 t(1)^2 t(2)+9 t(1) t(2)-5 t(2)+t(1)^2-3 t(1)+2}{t(1) t(2)^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^{11/2}+4 q^{9/2}-9 q^{7/2}+14 q^{5/2}-20 q^{3/2}+22 \sqrt{q}-\frac{23}{\sqrt{q}}+\frac{20}{q^{3/2}}-\frac{15}{q^{5/2}}+\frac{9}{q^{7/2}}-\frac{4}{q^{9/2}}+\frac{1}{q^{11/2}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a z^7+z^7 a^{-1} -a^3 z^5+3 a z^5+3 z^5 a^{-1} -z^5 a^{-3} -2 a^3 z^3+3 a z^3+3 z^3 a^{-1} -2 z^3 a^{-3} -a^3 z+a z+2 z a^{-1} -z a^{-3} + a^{-1} z^{-1} - a^{-3} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -3 z^{10} a^{-2} -3 z^{10}-10 a z^9-16 z^9 a^{-1} -6 z^9 a^{-3} -15 a^2 z^8-5 z^8 a^{-2} -4 z^8 a^{-4} -16 z^8-14 a^3 z^7+8 a z^7+40 z^7 a^{-1} +17 z^7 a^{-3} -z^7 a^{-5} -9 a^4 z^6+24 a^2 z^6+37 z^6 a^{-2} +13 z^6 a^{-4} +57 z^6-4 a^5 z^5+19 a^3 z^5+17 a z^5-20 z^5 a^{-1} -11 z^5 a^{-3} +3 z^5 a^{-5} -a^6 z^4+7 a^4 z^4-11 a^2 z^4-38 z^4 a^{-2} -13 z^4 a^{-4} -44 z^4+a^5 z^3-10 a^3 z^3-15 a z^3-2 z^3 a^{-1} -z^3 a^{-3} -3 z^3 a^{-5} -2 a^4 z^2+2 a^2 z^2+10 z^2 a^{-2} +4 z^2 a^{-4} +10 z^2+2 a^3 z+3 a z+z a^{-5} - a^{-2} + a^{-1} z^{-1} + a^{-3} 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   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 6 χ 12                       1 1 10                     3   -3 8                   6 1   5 6                 8 3     -5 4               12 6       6 2             11 9         -2 0           12 11           1 -2         9 12             3 -4       6 11               -5 -6     3 9                 6 -8   1 6                   -5 -10   3                     3 -12 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=-5 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=6 }[/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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