L10a13

L10a12

L10a14

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Link Presentations

[edit Notes on L10a13's Link Presentations]

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

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)^5+2 t(1) t(2)^4-4 t(2)^4-5 t(1) t(2)^3+7 t(2)^3+7 t(1) t(2)^2-5 t(2)^2-4 t(1) t(2)+2 t(2)+t(1)}{\sqrt{t(1)} t(2)^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{5/2}-3 q^{3/2}+6 \sqrt{q}-\frac{10}{\sqrt{q}}+\frac{12}{q^{3/2}}-\frac{13}{q^{5/2}}+\frac{11}{q^{7/2}}-\frac{10}{q^{9/2}}+\frac{6}{q^{11/2}}-\frac{3}{q^{13/2}}+\frac{1}{q^{15/2}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^7 (-z)-a^7 z^{-1} +3 a^5 z^3+6 a^5 z+4 a^5 z^{-1} -2 a^3 z^5-6 a^3 z^3-8 a^3 z-4 a^3 z^{-1} -a z^5-a z^3+z^3 a^{-1} +a z^{-1} +z a^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -a^5 z^9-a^3 z^9-3 a^6 z^8-7 a^4 z^8-4 a^2 z^8-3 a^7 z^7-9 a^5 z^7-12 a^3 z^7-6 a z^7-a^8 z^6+3 a^6 z^6+7 a^4 z^6-2 a^2 z^6-5 z^6+9 a^7 z^5+32 a^5 z^5+33 a^3 z^5+7 a z^5-3 z^5 a^{-1} +3 a^8 z^4+10 a^6 z^4+16 a^4 z^4+15 a^2 z^4-z^4 a^{-2} +5 z^4-9 a^7 z^3-32 a^5 z^3-31 a^3 z^3-5 a z^3+3 z^3 a^{-1} -3 a^8 z^2-13 a^6 z^2-21 a^4 z^2-15 a^2 z^2+z^2 a^{-2} -3 z^2+4 a^7 z+17 a^5 z+16 a^3 z+2 a z-z a^{-1} +a^8+4 a^6+7 a^4+4 a^2+1-a^7 z^{-1} -4 a^5 z^{-1} -4 a^3 z^{-1} -a 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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 χ 6                     1 -1 4                   2   2 2                 4 1   -3 0               6 2     4 -2             7 5       -2 -4           6 5         1 -6         5 7           2 -8       5 6             -1 -10     2 6               4 -12   1 4                 -3 -14   2                   2 -16 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=0 }[/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}^{5} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/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=2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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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