L11a430

L11a429

L11a431

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Link Presentations

[edit Notes on L11a430's Link Presentations]

Planar diagram presentation	X6172 X14,6,15,5 X8493 X2,16,3,15 X16,7,17,8 X18,10,19,9 X4,17,1,18 X22,19,13,20 X10,14,11,13 X12,21,5,22 X20,11,21,12
Gauss code	{1, -4, 3, -7}, {2, -1, 5, -3, 6, -9, 11, -10}, {9, -2, 4, -5, 7, -6, 8, -11, 10, -8}

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{(u-1) (v-1) (w-1) (v w-v+1) (v w-w+1)}{\sqrt{u} v^{3/2} w^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^5+4 q^4-8 q^3+14 q^2-19 q+24-22 q^{-1} +21 q^{-2} -15 q^{-3} +10 q^{-4} -5 q^{-5} + q^{-6} }[/math] (db)
Signature	0 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^4 z^4+a^4 z^2-a^4-2 a^2 z^6-z^6 a^{-2} -6 a^2 z^4-3 z^4 a^{-2} -3 a^2 z^2-2 z^2 a^{-2} +a^2 z^{-2} + a^{-2} z^{-2} +3 a^2+ a^{-2} +z^8+5 z^6+9 z^4+4 z^2-2 z^{-2} -3 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ 2 a^2 z^{10}+2 z^{10}+7 a^3 z^9+13 a z^9+6 z^9 a^{-1} +9 a^4 z^8+16 a^2 z^8+8 z^8 a^{-2} +15 z^8+5 a^5 z^7-7 a^3 z^7-19 a z^7+7 z^7 a^{-3} +a^6 z^6-21 a^4 z^6-48 a^2 z^6-9 z^6 a^{-2} +4 z^6 a^{-4} -39 z^6-10 a^5 z^5-11 a^3 z^5-2 a z^5-12 z^5 a^{-1} -10 z^5 a^{-3} +z^5 a^{-5} -a^6 z^4+12 a^4 z^4+36 a^2 z^4+2 z^4 a^{-2} -6 z^4 a^{-4} +31 z^4+3 a^5 z^3+7 a^3 z^3+8 a z^3+9 z^3 a^{-1} +4 z^3 a^{-3} -z^3 a^{-5} -a^4 z^2-3 a^2 z^2+z^2 a^{-2} +2 z^2 a^{-4} -3 z^2+a^5 z+3 a^3 z+3 a z+z a^{-1} -2 a^4-6 a^2-2 a^{-2} -5-2 a z^{-1} -2 a^{-1} z^{-1} +a^2 z^{-2} + a^{-2} z^{-2} +2 z^{-2} }[/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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 11                       1 -1 9                     3   3 7                   5 1   -4 5                 9 3     6 3               10 5       -5 1             14 9         5 -1           12 14           2 -3         9 10             -1 -5       6 12               6 -7     4 9                 -5 -9   1 6                   5 -11   4                     -4 -13 1                       1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-1 }[/math] [math]\displaystyle{ i=1 }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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}^{9}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{12} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{14}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{14} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=5 }[/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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