L11a242

L11a241

L11a243

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Link Presentations

[edit Notes on L11a242's Link Presentations]

Planar diagram presentation	X8192 X12,4,13,3 X22,10,7,9 X14,12,15,11 X10,16,11,15 X18,6,19,5 X20,18,21,17 X16,22,17,21 X2738 X4,14,5,13 X6,20,1,19
Gauss code	{1, -9, 2, -10, 6, -11}, {9, -1, 3, -5, 4, -2, 10, -4, 5, -8, 7, -6, 11, -7, 8, -3}

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{4 t(2)^2 t(1)^2-7 t(2) t(1)^2+3 t(1)^2-7 t(2)^2 t(1)+13 t(2) t(1)-7 t(1)+3 t(2)^2-7 t(2)+4}{t(1) t(2)} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{21/2}-4 q^{19/2}+8 q^{17/2}-12 q^{15/2}+16 q^{13/2}-18 q^{11/2}+17 q^{9/2}-15 q^{7/2}+10 q^{5/2}-6 q^{3/2}+2 \sqrt{q}-\frac{1}{\sqrt{q}} }[/math] (db)
Signature	3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^3 a^{-9} -z^5 a^{-7} +z a^{-7} -2 z^5 a^{-5} -3 z^3 a^{-5} -z a^{-5} -z^5 a^{-3} -z^3 a^{-3} -z a^{-3} - a^{-3} z^{-1} +z^3 a^{-1} +2 z a^{-1} + a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^{10} a^{-6} -z^{10} a^{-8} -3 z^9 a^{-5} -7 z^9 a^{-7} -4 z^9 a^{-9} -4 z^8 a^{-4} -9 z^8 a^{-6} -11 z^8 a^{-8} -6 z^8 a^{-10} -3 z^7 a^{-3} -z^7 a^{-5} +7 z^7 a^{-7} +z^7 a^{-9} -4 z^7 a^{-11} -2 z^6 a^{-2} +5 z^6 a^{-4} +23 z^6 a^{-6} +31 z^6 a^{-8} +14 z^6 a^{-10} -z^6 a^{-12} -z^5 a^{-1} +2 z^5 a^{-3} +7 z^5 a^{-5} +10 z^5 a^{-7} +16 z^5 a^{-9} +10 z^5 a^{-11} +3 z^4 a^{-2} -6 z^4 a^{-4} -23 z^4 a^{-6} -23 z^4 a^{-8} -7 z^4 a^{-10} +2 z^4 a^{-12} +3 z^3 a^{-1} +3 z^3 a^{-3} -9 z^3 a^{-5} -16 z^3 a^{-7} -13 z^3 a^{-9} -6 z^3 a^{-11} +4 z^2 a^{-4} +7 z^2 a^{-6} +5 z^2 a^{-8} +z^2 a^{-10} -z^2 a^{-12} -3 z a^{-1} -3 z a^{-3} +3 z a^{-5} +5 z a^{-7} +3 z a^{-9} +z a^{-11} - 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   \ -2 -1 0 1 2 3 4 5 6 7 8 9 χ 22                       1 -1 20                     3   3 18                   5 1   -4 16                 7 3     4 14               9 5       -4 12             9 7         2 10           8 9           1 8         7 9             -2 6       4 9               5 4     2 6                 -4 2   1 5                   4 0   1                     -1 -2 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=4 }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=7 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=8 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=9 }[/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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