L10a54

L10a53

L10a55

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Link Presentations

[edit Notes on L10a54's Link Presentations]

Planar diagram presentation	X8192 X2,9,3,10 X10,3,11,4 X18,11,19,12 X12,6,13,5 X4,18,5,17 X14,7,15,8 X16,14,17,13 X20,15,7,16 X6,19,1,20
Gauss code	{1, -2, 3, -6, 5, -10}, {7, -1, 2, -3, 4, -5, 8, -7, 9, -8, 6, -4, 10, -9}

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-t(1) t(2)^4-4 t(1)^2 t(2)^3+5 t(1) t(2)^3-t(2)^3+4 t(1)^2 t(2)^2-7 t(1) t(2)^2+4 t(2)^2-t(1)^2 t(2)+5 t(1) t(2)-4 t(2)-t(1)+1}{t(1) t(2)^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{3/2}-4 \sqrt{q}+\frac{6}{\sqrt{q}}-\frac{10}{q^{3/2}}+\frac{12}{q^{5/2}}-\frac{13}{q^{7/2}}+\frac{12}{q^{9/2}}-\frac{10}{q^{11/2}}+\frac{6}{q^{13/2}}-\frac{3}{q^{15/2}}+\frac{1}{q^{17/2}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^7 \left(-z^3\right)-2 a^7 z+2 a^5 z^5+6 a^5 z^3+4 a^5 z-a^3 z^7-4 a^3 z^5-5 a^3 z^3-2 a^3 z+a^3 z^{-1} +a z^5+2 a z^3-a z-a z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^4 a^{10}+z^2 a^{10}-3 z^5 a^9+3 z^3 a^9-z a^9-5 z^6 a^8+5 z^4 a^8-2 z^2 a^8-6 z^7 a^7+7 z^5 a^7-4 z^3 a^7+z a^7-5 z^8 a^6+5 z^6 a^6-z^4 a^6-2 z^9 a^5-6 z^7 a^5+21 z^5 a^5-18 z^3 a^5+5 z a^5-10 z^8 a^4+25 z^6 a^4-19 z^4 a^4+6 z^2 a^4-2 z^9 a^3-4 z^7 a^3+23 z^5 a^3-19 z^3 a^3+2 z a^3+a^3 z^{-1} -5 z^8 a^2+14 z^6 a^2-10 z^4 a^2+3 z^2 a^2-a^2-4 z^7 a+12 z^5 a-8 z^3 a-z a+a z^{-1} -z^6+2 z^4 }[/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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 χ 4                     1 -1 2                   3   3 0                 3 1   -2 -2               7 3     4 -4             6 4       -2 -6           7 6         1 -8         6 7           1 -10       4 6             -2 -12     2 6               4 -14   1 4                 -3 -16   2                   2 -18 1                     -1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-4 }[/math] [math]\displaystyle{ i=-2 }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=3 }[/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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