L11a525

L11a524

L11a526

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Link Presentations

[edit Notes on L11a525's Link Presentations]

Planar diagram presentation	X8192 X14,3,15,4 X12,15,7,16 X22,19,13,20 X16,9,17,10 X10,22,11,21 X20,12,21,11 X18,5,19,6 X2738 X4,13,5,14 X6,17,1,18
Gauss code	{1, -9, 2, -10, 8, -11}, {9, -1, 5, -6, 7, -3}, {10, -2, 3, -5, 11, -8, 4, -7, 6, -4}

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^2 v^2 w+u^2 v^2-2 u^2 v w^2+5 u^2 v w-2 u^2 v+2 u^2 w^2-3 u^2 w+u^2-2 u v^2 w^2+3 u v^2 w-u v^2-u v w^3+6 u v w^2-6 u v w+u v+u w^3-3 u w^2+2 u w-v^2 w^3+3 v^2 w^2-2 v^2 w+2 v w^3-5 v w^2+2 v w-w^3+w^2}{u v w^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q-3+7 q^{-1} -12 q^{-2} +17 q^{-3} -18 q^{-4} +20 q^{-5} -16 q^{-6} +13 q^{-7} -8 q^{-8} +4 q^{-9} - q^{-10} }[/math] (db)
Signature	-2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a^{10}+4 z^2 a^8+a^8 z^{-2} +3 a^8-3 z^4 a^6-4 z^2 a^6-2 a^6 z^{-2} -5 a^6-3 z^4 a^4-z^2 a^4+a^4 z^{-2} +2 a^4-z^4 a^2+2 z^2 a^2+a^2+z^2 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{11} z^7-3 a^{11} z^5+3 a^{11} z^3-a^{11} z+4 a^{10} z^8-14 a^{10} z^6+15 a^{10} z^4-6 a^{10} z^2+2 a^{10}+5 a^9 z^9-13 a^9 z^7+2 a^9 z^5+10 a^9 z^3-3 a^9 z+2 a^8 z^{10}+8 a^8 z^8-46 a^8 z^6+59 a^8 z^4-34 a^8 z^2-a^8 z^{-2} +10 a^8+12 a^7 z^9-30 a^7 z^7+12 a^7 z^5+8 a^7 z^3-8 a^7 z+2 a^7 z^{-1} +2 a^6 z^{10}+14 a^6 z^8-56 a^6 z^6+66 a^6 z^4-41 a^6 z^2-2 a^6 z^{-2} +12 a^6+7 a^5 z^9-7 a^5 z^7-9 a^5 z^5+12 a^5 z^3-6 a^5 z+2 a^5 z^{-1} +10 a^4 z^8-18 a^4 z^6+15 a^4 z^4-8 a^4 z^2-a^4 z^{-2} +4 a^4+9 a^3 z^7-13 a^3 z^5+9 a^3 z^3+6 a^2 z^6-6 a^2 z^4+4 a^2 z^2-a^2+3 a z^5-2 a z^3+z^4-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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 3                       1 1 1                     2   -2 -1                   5 1   4 -3                 8 3     -5 -5               9 4       5 -7             10 9         -1 -9           10 8           2 -11         7 11             4 -13       6 9               -3 -15     3 8                 5 -17   1 5                   -4 -19   3                     3 -21 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=-9 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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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