L11a466

L11a465

L11a467

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Link Presentations

[edit Notes on L11a466's Link Presentations]

Planar diagram presentation	X6172 X10,4,11,3 X14,8,5,7 X18,21,19,22 X16,9,17,10 X8,15,9,16 X20,13,21,14 X12,19,13,20 X22,17,15,18 X2536 X4,12,1,11
Gauss code	{1, -10, 2, -11}, {6, -5, 9, -4, 8, -7, 4, -9}, {10, -1, 3, -6, 5, -2, 11, -8, 7, -3}

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)-1) (t(3)-1) \left(2 t(2) t(3)^2-t(3)^2-4 t(2) t(3)+4 t(3)+t(2)-2\right)}{\sqrt{t(1)} \sqrt{t(2)} t(3)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{-9} -3 q^{-8} +7 q^{-7} -12 q^{-6} +16 q^{-5} -17 q^{-4} +19 q^{-3} -q^2-15 q^{-2} +3 q+12 q^{-1} -6 }[/math] (db)
Signature	-2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^2 a^8+a^8-2 z^4 a^6-3 z^2 a^6+a^6 z^{-2} -a^6+z^6 a^4+z^4 a^4-2 z^2 a^4-2 a^4 z^{-2} -4 a^4+z^6 a^2+3 z^4 a^2+6 z^2 a^2+a^2 z^{-2} +5 a^2-z^4-2 z^2-1 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^6 a^{10}-3 z^4 a^{10}+3 z^2 a^{10}-a^{10}+3 z^7 a^9-8 z^5 a^9+7 z^3 a^9-2 z a^9+4 z^8 a^8-6 z^6 a^8-3 z^4 a^8+6 z^2 a^8-a^8+3 z^9 a^7+2 z^7 a^7-17 z^5 a^7+15 z^3 a^7-6 z a^7+z^{10} a^6+8 z^8 a^6-15 z^6 a^6+6 z^2 a^6+a^6 z^{-2} -3 a^6+6 z^9 a^5-z^7 a^5-15 z^5 a^5+12 z^3 a^5-z a^5-2 a^5 z^{-1} +z^{10} a^4+8 z^8 a^4-9 z^6 a^4-9 z^4 a^4+16 z^2 a^4+2 a^4 z^{-2} -8 a^4+3 z^9 a^3+4 z^7 a^3-10 z^5 a^3+3 z^3 a^3+5 z a^3-2 a^3 z^{-1} +4 z^8 a^2+2 z^6 a^2-15 z^4 a^2+18 z^2 a^2+a^2 z^{-2} -8 a^2+4 z^7 a-3 z^5 a-3 z^3 a+3 z a+3 z^6-6 z^4+5 z^2-2+z^5 a^{-1} -2 z^3 a^{-1} +z a^{-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 2 3 χ 5                       1 -1 3                     2   2 1                   4 1   -3 -1                 8 2     6 -3               9 6       -3 -5             10 6         4 -7           7 9           2 -9         9 10             -1 -11       5 9               4 -13     2 7                 -5 -15   1 5                   4 -17   2                     -2 -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}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/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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