L11n104

L11n103

L11n105

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Link Presentations

[edit Notes on L11n104's Link Presentations]

Planar diagram presentation	X6172 X12,3,13,4 X7,16,8,17 X22,18,5,17 X18,14,19,13 X21,14,22,15 X9,20,10,21 X15,8,16,9 X19,10,20,11 X2536 X4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, -3, 8, -7, 9, 11, -2, 5, 6, -8, 3, 4, -5, -9, 7, -6, -4}

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