L11a234

L11a233

L11a235

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Link Presentations

[edit Notes on L11a234's Link Presentations]

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

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{t(1)^2 t(2)^4-3 t(1) t(2)^4+2 t(2)^4-3 t(1)^2 t(2)^3+8 t(1) t(2)^3-4 t(2)^3+4 t(1)^2 t(2)^2-9 t(1) t(2)^2+4 t(2)^2-4 t(1)^2 t(2)+8 t(1) t(2)-3 t(2)+2 t(1)^2-3 t(1)+1}{t(1) t(2)^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{17/2}-3 q^{15/2}+7 q^{13/2}-12 q^{11/2}+16 q^{9/2}-19 q^{7/2}+18 q^{5/2}-17 q^{3/2}+12 \sqrt{q}-\frac{8}{\sqrt{q}}+\frac{4}{q^{3/2}}-\frac{1}{q^{5/2}} }[/math] (db)
Signature	3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^3 a^{-7} +2 z a^{-7} + a^{-7} z^{-1} -2 z^5 a^{-5} -6 z^3 a^{-5} -7 z a^{-5} -3 a^{-5} z^{-1} +z^7 a^{-3} +4 z^5 a^{-3} +8 z^3 a^{-3} +8 z a^{-3} +2 a^{-3} z^{-1} -2 z^5 a^{-1} +a z^3-5 z^3 a^{-1} +a z-3 z a^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^4 a^{-10} -z^2 a^{-10} +3 z^5 a^{-9} -2 z^3 a^{-9} +6 z^6 a^{-8} -6 z^4 a^{-8} +4 z^2 a^{-8} - a^{-8} +9 z^7 a^{-7} -14 z^5 a^{-7} +13 z^3 a^{-7} -5 z a^{-7} + a^{-7} z^{-1} +9 z^8 a^{-6} -12 z^6 a^{-6} +z^4 a^{-6} +7 z^2 a^{-6} -3 a^{-6} +6 z^9 a^{-5} -z^7 a^{-5} -23 z^5 a^{-5} +27 z^3 a^{-5} -13 z a^{-5} +3 a^{-5} z^{-1} +2 z^{10} a^{-4} +11 z^8 a^{-4} -38 z^6 a^{-4} +26 z^4 a^{-4} -2 z^2 a^{-4} -3 a^{-4} +11 z^9 a^{-3} -25 z^7 a^{-3} +3 z^5 a^{-3} +15 z^3 a^{-3} -10 z a^{-3} +2 a^{-3} z^{-1} +2 z^{10} a^{-2} +6 z^8 a^{-2} -34 z^6 a^{-2} +32 z^4 a^{-2} -7 z^2 a^{-2} +5 z^9 a^{-1} +a z^7-14 z^7 a^{-1} -3 a z^5+6 z^5 a^{-1} +3 a z^3+6 z^3 a^{-1} -a z-3 z a^{-1} +4 z^8-14 z^6+14 z^4-3 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   \ -4 -3 -2 -1 0 1 2 3 4 5 6 7 χ 18                       1 -1 16                     2   2 14                   5 1   -4 12                 7 2     5 10               9 5       -4 8             10 7         3 6           9 10           1 4         8 9             -1 2       5 10               5 0     3 7                 -4 -2   1 5                   4 -4   3                     -3 -6 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=-4 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=7 }[/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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