L11a59

L11a58

L11a60

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Link Presentations

[edit Notes on L11a59's Link Presentations]

Planar diagram presentation	X6172 X12,3,13,4 X14,8,15,7 X20,16,21,15 X18,9,19,10 X8,19,9,20 X22,18,5,17 X16,22,17,21 X10,14,11,13 X2536 X4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -6, 5, -9, 11, -2, 9, -3, 4, -8, 7, -5, 6, -4, 8, -7}

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) t(2)^5-2 t(2)^5-4 t(1) t(2)^4+6 t(2)^4+7 t(1) t(2)^3-7 t(2)^3-7 t(1) t(2)^2+7 t(2)^2+6 t(1) t(2)-4 t(2)-2 t(1)+1}{\sqrt{t(1)} t(2)^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{11/2}-4 q^{9/2}+8 q^{7/2}-13 q^{5/2}+16 q^{3/2}-17 \sqrt{q}+\frac{17}{\sqrt{q}}-\frac{14}{q^{3/2}}+\frac{9}{q^{5/2}}-\frac{6}{q^{7/2}}+\frac{2}{q^{9/2}}-\frac{1}{q^{11/2}} }[/math] (db)
Signature	1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^5 z+2 a^5 z^{-1} +z^5 a^{-3} -3 a^3 z^3+2 z^3 a^{-3} -8 a^3 z+z a^{-3} -4 a^3 z^{-1} -z^7 a^{-1} +3 a z^5-4 z^5 a^{-1} +10 a z^3-7 z^3 a^{-1} +10 a z-6 z a^{-1} +3 a z^{-1} - a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -a^2 z^{10}-z^{10}-2 a^3 z^9-7 a z^9-5 z^9 a^{-1} -2 a^4 z^8-5 a^2 z^8-10 z^8 a^{-2} -13 z^8-a^5 z^7+2 a^3 z^7+12 a z^7-2 z^7 a^{-1} -11 z^7 a^{-3} +7 a^4 z^6+22 a^2 z^6+15 z^6 a^{-2} -8 z^6 a^{-4} +38 z^6+5 a^5 z^5+12 a^3 z^5+9 a z^5+21 z^5 a^{-1} +15 z^5 a^{-3} -4 z^5 a^{-5} -6 a^4 z^4-19 a^2 z^4-7 z^4 a^{-2} +7 z^4 a^{-4} -z^4 a^{-6} -28 z^4-9 a^5 z^3-23 a^3 z^3-22 a z^3-16 z^3 a^{-1} -6 z^3 a^{-3} +2 z^3 a^{-5} -a^4 z^2+a^2 z^2+z^2 a^{-2} -z^2 a^{-4} +4 z^2+7 a^5 z+15 a^3 z+12 a z+4 z a^{-1} +2 a^4+3 a^2+ a^{-2} +3-2 a^5 z^{-1} -4 a^3 z^{-1} -3 a z^{-1} - a^{-1} 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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 12                       1 -1 10                     3   3 8                   5 1   -4 6                 8 3     5 4               8 5       -3 2             9 8         1 0           9 9           0 -2         5 8             -3 -4       4 9               5 -6     2 5                 -3 -8   1 5                   4 -10   1                     -1 -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=0 }[/math] [math]\displaystyle{ i=2 }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\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^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=5 }[/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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