L11a98

L11a97

L11a99

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Link Presentations

[edit Notes on L11a98's Link Presentations]

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

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)^5-5 t(1) t(2)^4+7 t(2)^4+8 t(1) t(2)^3-9 t(2)^3-9 t(1) t(2)^2+8 t(2)^2+7 t(1) t(2)-5 t(2)-2 t(1)+2}{\sqrt{t(1)} t(2)^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{11/2}-4 q^{9/2}+10 q^{7/2}-15 q^{5/2}+19 q^{3/2}-22 \sqrt{q}+\frac{20}{\sqrt{q}}-\frac{18}{q^{3/2}}+\frac{12}{q^{5/2}}-\frac{7}{q^{7/2}}+\frac{3}{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^5-4 a z^5-3 z^5 a^{-1} +z^5 a^{-3} +3 a^3 z^3-7 a z^3-3 z^3 a^{-1} +2 z^3 a^{-3} +3 a^3 z-5 a z-2 z a^{-1} +2 z a^{-3} +a^3 z^{-1} -2 a^{-1} z^{-1} + a^{-3} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^4 a^{-6} +a^5 z^7-4 a^5 z^5+4 z^5 a^{-5} +5 a^5 z^3-2 a^5 z+3 a^4 z^8-11 a^4 z^6+10 z^6 a^{-4} +13 a^4 z^4-9 z^4 a^{-4} -5 a^4 z^2+5 z^2 a^{-4} -2 a^{-4} +4 a^3 z^9-11 a^3 z^7+15 z^7 a^{-3} +7 a^3 z^5-21 z^5 a^{-3} +10 z^3 a^{-3} +2 a^3 z-2 z a^{-3} -a^3 z^{-1} + a^{-3} z^{-1} +2 a^2 z^{10}+5 a^2 z^8+14 z^8 a^{-2} -30 a^2 z^6-18 z^6 a^{-2} +34 a^2 z^4-2 z^4 a^{-2} -12 a^2 z^2+10 z^2 a^{-2} +a^2-5 a^{-2} +12 a z^9+8 z^9 a^{-1} -28 a z^7-z^7 a^{-1} +14 a z^5-22 z^5 a^{-1} -2 a z^3+13 z^3 a^{-1} +2 a z-4 z a^{-1} +2 a^{-1} z^{-1} +2 z^{10}+16 z^8-47 z^6+29 z^4-2 z^2-3 }[/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                   7 1   -6 6                 8 3     5 4               11 7       -4 2             11 8         3 0           10 12           2 -2         8 10             -2 -4       4 10               6 -6     3 8                 -5 -8   1 5                   4 -10   2                     -2 -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}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/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=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\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^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/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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