L11a232

L11a231

L11a233

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Link Presentations

[edit Notes on L11a232's Link Presentations]

Planar diagram presentation	X8192 X12,4,13,3 X18,6,19,5 X22,16,7,15 X16,11,17,12 X20,17,21,18 X10,21,11,22 X14,10,15,9 X4,20,5,19 X2738 X6,14,1,13
Gauss code	{1, -10, 2, -9, 3, -11}, {10, -1, 8, -7, 5, -2, 11, -8, 4, -5, 6, -3, 9, -6, 7, -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{t(1)^2 t(2)^4-3 t(1) t(2)^4+2 t(2)^4-4 t(1)^2 t(2)^3+11 t(1) t(2)^3-6 t(2)^3+7 t(1)^2 t(2)^2-17 t(1) t(2)^2+7 t(2)^2-6 t(1)^2 t(2)+11 t(1) t(2)-4 t(2)+2 t(1)^2-3 t(1)+1}{t(1) t(2)^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -10 q^{9/2}+\frac{1}{q^{9/2}}+17 q^{7/2}-\frac{5}{q^{7/2}}-24 q^{5/2}+\frac{11}{q^{5/2}}+27 q^{3/2}-\frac{18}{q^{3/2}}-q^{13/2}+4 q^{11/2}-28 \sqrt{q}+\frac{24}{\sqrt{q}} }[/math] (db)
Signature	1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^3 a^{-5} -z a^{-5} - a^{-5} z^{-1} +2 z^5 a^{-3} -a^3 z^3+4 z^3 a^{-3} +5 z a^{-3} +2 a^{-3} z^{-1} -z^7 a^{-1} +2 a z^5-3 z^5 a^{-1} +3 a z^3-6 z^3 a^{-1} +2 a z-5 z a^{-1} +a z^{-1} -2 a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^5 a^{-7} -z^3 a^{-7} +4 z^6 a^{-6} -4 z^4 a^{-6} +z^2 a^{-6} +9 z^7 a^{-5} -13 z^5 a^{-5} +10 z^3 a^{-5} -4 z a^{-5} + a^{-5} z^{-1} +12 z^8 a^{-4} +a^4 z^6-17 z^6 a^{-4} -a^4 z^4+11 z^4 a^{-4} -3 z^2 a^{-4} +9 z^9 a^{-3} +5 a^3 z^7-9 a^3 z^5-24 z^5 a^{-3} +4 a^3 z^3+27 z^3 a^{-3} -12 z a^{-3} +2 a^{-3} z^{-1} +3 z^{10} a^{-2} +10 a^2 z^8+21 z^8 a^{-2} -21 a^2 z^6-53 z^6 a^{-2} +12 a^2 z^4+38 z^4 a^{-2} -a^2 z^2-10 z^2 a^{-2} + a^{-2} +9 a z^9+18 z^9 a^{-1} -9 a z^7-23 z^7 a^{-1} -12 a z^5-13 z^5 a^{-1} +13 a z^3+25 z^3 a^{-1} -4 a z-12 z a^{-1} +a z^{-1} +2 a^{-1} z^{-1} +3 z^{10}+19 z^8-54 z^6+36 z^4-7 z^2 }[/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 14                       1 1 12                     3   -3 10                   7 1   6 8                 10 3     -7 6               14 7       7 4             14 11         -3 2           14 13           1 0         11 15             4 -2       7 13               -6 -4     4 11                 7 -6   1 7                   -6 -8   4                     4 -10 1                       -1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=0 }[/math] [math]\displaystyle{ i=2 }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{15}\oplus{\mathbb Z}_2^{13} }[/math] [math]\displaystyle{ {\mathbb Z}^{14} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{14} }[/math] [math]\displaystyle{ {\mathbb Z}^{14} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{13} }[/math] [math]\displaystyle{ {\mathbb Z}^{14} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/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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