L11a472

L11a471

L11a473

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Link Presentations

[edit Notes on L11a472's Link Presentations]

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

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) (w-1)^2 \left(v^2 w^2-v^2 w-v w^2+v w-v-w+1\right)}{\sqrt{u} v w^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^7-4 q^6+8 q^5-12 q^4- q^{-4} +17 q^3+3 q^{-3} -17 q^2-6 q^{-2} +18 q+11 q^{-1} -14 }[/math] (db)
Signature	2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^8 a^{-2} -5 z^6 a^{-2} +z^6 a^{-4} +2 z^6-a^2 z^4-9 z^4 a^{-2} +3 z^4 a^{-4} +8 z^4-3 a^2 z^2-8 z^2 a^{-2} +2 z^2 a^{-4} +9 z^2-a^2-4 a^{-2} + a^{-4} +4-2 a^{-2} z^{-2} + a^{-4} z^{-2} + z^{-2} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^4 a^{-8} +4 z^5 a^{-7} -2 z^3 a^{-7} +8 z^6 a^{-6} -8 z^4 a^{-6} +2 z^2 a^{-6} +10 z^7 a^{-5} -11 z^5 a^{-5} +z^3 a^{-5} +10 z^8 a^{-4} -15 z^6 a^{-4} +6 z^4 a^{-4} -3 z^2 a^{-4} - a^{-4} z^{-2} +3 a^{-4} +7 z^9 a^{-3} +a^3 z^7-10 z^7 a^{-3} -4 a^3 z^5+z^5 a^{-3} +4 a^3 z^3+z^3 a^{-3} -a^3 z-2 z a^{-3} +2 a^{-3} z^{-1} +2 z^{10} a^{-2} +3 a^2 z^8+10 z^8 a^{-2} -12 a^2 z^6-43 z^6 a^{-2} +15 a^2 z^4+48 z^4 a^{-2} -9 a^2 z^2-23 z^2 a^{-2} -2 a^{-2} z^{-2} +2 a^2+7 a^{-2} +4 a z^9+11 z^9 a^{-1} -13 a z^7-34 z^7 a^{-1} +9 a z^5+29 z^5 a^{-1} +2 a z^3-4 z^3 a^{-1} -3 a z-4 z a^{-1} +2 a^{-1} z^{-1} +2 z^{10}+3 z^8-32 z^6+48 z^4-27 z^2- z^{-2} +7 }[/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   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 6 χ 15                       1 1 13                     3   -3 11                   5 1   4 9                 7 3     -4 7               10 5       5 5             9 9         0 3           9 8           1 1         7 11             4 -1       4 7               -3 -3     2 7                 5 -5   1 4                   -3 -7   2                     2 -9 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=3 }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=6 }[/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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