L11n305

L11n304

L11n306

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Link Presentations

[edit Notes on L11n305's Link Presentations]

Planar diagram presentation	X6172 X12,3,13,4 X13,21,14,20 X19,11,20,22 X10,15,5,16 X8,17,9,18 X16,7,17,8 X18,9,19,10 X21,15,22,14 X2536 X4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, 7, -6, 8, -5}, {11, -2, -3, 9, 5, -7, 6, -8, -4, 3, -9, 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(2)^2 t(3)^4+t(1) t(2) t(3)^4-t(2) t(3)^4-t(2)^2 t(3)^3+t(1) t(3)^3-2 t(1) t(2) t(3)^3+2 t(2) t(3)^3-t(3)^3+t(2)^2 t(3)^2-t(1) t(3)^2+2 t(1) t(2) t(3)^2-2 t(2) t(3)^2+t(1) t(2)^2 t(3)-t(2)^2 t(3)+t(1) t(3)-2 t(1) t(2) t(3)+2 t(2) t(3)-t(1)+t(1) t(2)-t(2)}{\sqrt{t(1)} t(2) t(3)^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -2+4 q^{-1} -6 q^{-2} +9 q^{-3} -8 q^{-4} +9 q^{-5} -6 q^{-6} +5 q^{-7} -2 q^{-8} + q^{-9} }[/math] (db)
Signature	-2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^2 a^8+2 a^8 z^{-2} +3 a^8-3 z^4 a^6-12 z^2 a^6-5 a^6 z^{-2} -15 a^6+2 z^6 a^4+11 z^4 a^4+22 z^2 a^4+4 a^4 z^{-2} +17 a^4-2 z^4 a^2-6 z^2 a^2-a^2 z^{-2} -5 a^2 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^6 a^{10}-4 z^4 a^{10}+5 z^2 a^{10}-2 a^{10}+2 z^7 a^9-6 z^5 a^9+3 z^3 a^9+z a^9+2 z^8 a^8-4 z^6 a^8-4 z^2 a^8-2 a^8 z^{-2} +6 a^8+z^9 a^7+2 z^7 a^7-14 z^5 a^7+19 z^3 a^7-16 z a^7+5 a^7 z^{-1} +6 z^8 a^6-22 z^6 a^6+36 z^4 a^6-36 z^2 a^6-5 a^6 z^{-2} +20 a^6+z^9 a^5+4 z^7 a^5-23 z^5 a^5+43 z^3 a^5-33 z a^5+9 a^5 z^{-1} +4 z^8 a^4-16 z^6 a^4+34 z^4 a^4-32 z^2 a^4-4 a^4 z^{-2} +17 a^4+4 z^7 a^3-15 z^5 a^3+30 z^3 a^3-21 z a^3+5 a^3 z^{-1} +z^6 a^2+2 z^4 a^2-5 z^2 a^2-a^2 z^{-2} +4 a^2+3 z^3 a-5 z a+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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 χ 1                   2 -2 -1                 2   2 -3               5 3   -2 -5             4 1     3 -7           4 5       1 -9         5 4         1 -11       2 5           3 -13     3 4             -1 -15   1 4               3 -17   1                 -1 -19 1                   1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-3 }[/math] [math]\displaystyle{ i=-1 }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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}^{4}\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^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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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