L10a110

L10a109

L10a111

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Link Presentations

[edit Notes on L10a110's Link Presentations]

Planar diagram presentation	X12,1,13,2 X8493 X18,14,19,13 X20,16,11,15 X14,20,15,19 X6,17,7,18 X16,7,17,8 X10,6,1,5 X4,10,5,9 X2,11,3,12
Gauss code	{1, -10, 2, -9, 8, -6, 7, -2, 9, -8}, {10, -1, 3, -5, 4, -7, 6, -3, 5, -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(2)^3 t(1)^3-2 t(2)^2 t(1)^3+2 t(2) t(1)^3-2 t(1)^3-2 t(2)^3 t(1)^2+2 t(2)^2 t(1)^2-2 t(2) t(1)^2+2 t(1)^2+2 t(2)^3 t(1)-2 t(2)^2 t(1)+2 t(2) t(1)-2 t(1)-2 t(2)^3+2 t(2)^2-2 t(2)+1}{t(1)^{3/2} t(2)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^{15/2}+3 q^{13/2}-6 q^{11/2}+9 q^{9/2}-9 q^{7/2}+10 q^{5/2}-9 q^{3/2}+6 \sqrt{q}-\frac{5}{\sqrt{q}}+\frac{1}{q^{3/2}}-\frac{1}{q^{5/2}} }[/math] (db)
Signature	3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^5 a^{-5} -3 z^3 a^{-5} -3 z a^{-5} - a^{-5} z^{-1} +z^7 a^{-3} +5 z^5 a^{-3} +10 z^3 a^{-3} +11 z a^{-3} +5 a^{-3} z^{-1} -2 z^5 a^{-1} +a z^3-9 z^3 a^{-1} +4 a z-14 z a^{-1} +4 a z^{-1} -8 a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^9 a^{-1} -z^9 a^{-3} -5 z^8 a^{-2} -4 z^8 a^{-4} -z^8-a z^7-7 z^7 a^{-3} -8 z^7 a^{-5} +11 z^6 a^{-2} -9 z^6 a^{-6} +2 z^6+6 a z^5+12 z^5 a^{-1} +26 z^5 a^{-3} +14 z^5 a^{-5} -6 z^5 a^{-7} +9 z^4 a^{-2} +22 z^4 a^{-4} +15 z^4 a^{-6} -3 z^4 a^{-8} +5 z^4-13 a z^3-24 z^3 a^{-1} -20 z^3 a^{-3} -4 z^3 a^{-5} +4 z^3 a^{-7} -z^3 a^{-9} -28 z^2 a^{-2} -23 z^2 a^{-4} -9 z^2 a^{-6} -14 z^2+12 a z+20 z a^{-1} +9 z a^{-3} -z a^{-7} +14 a^{-2} +9 a^{-4} +2 a^{-6} +8-4 a z^{-1} -8 a^{-1} z^{-1} -5 a^{-3} z^{-1} - a^{-5} 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   \ -4 -3 -2 -1 0 1 2 3 4 5 6 χ 16                     1 1 14                   2   -2 12                 4 1   3 10               5 2     -3 8             4 4       0 6           6 5         -1 4         3 4           -1 2       4 7             3 0     1 2               -1 -2     4                 4 -4 1 1                   0 -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{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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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