L11a458

L11a457

L11a459

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Link Presentations

[edit Notes on L11a458's Link Presentations]

Planar diagram presentation	X6172 X14,4,15,3 X16,5,17,6 X20,9,21,10 X18,8,19,7 X22,19,13,20 X8,14,9,13 X12,15,5,16 X10,21,11,22 X2,11,3,12 X4,18,1,17
Gauss code	{1, -10, 2, -11}, {3, -1, 5, -7, 4, -9, 10, -8}, {7, -2, 8, -3, 11, -5, 6, -4, 9, -6}

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 v^3 w^3-2 u v^3 w^2+u v^3 w-2 u v^2 w^3+7 u v^2 w^2-6 u v^2 w+u v^2+u v w^3-5 u v w^2+7 u v w-2 u v+u w^2-2 u w+u-v^3 w^3+2 v^3 w^2-v^3 w+2 v^2 w^3-7 v^2 w^2+5 v^2 w-v^2-v w^3+6 v w^2-7 v w+2 v-w^2+2 w-1}{\sqrt{u} v^{3/2} w^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^3-4 q^2+9 q-14+22 q^{-1} -24 q^{-2} +26 q^{-3} -22 q^{-4} +17 q^{-5} -11 q^{-6} +5 q^{-7} - q^{-8} }[/math] (db)
Signature	-2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a^6 z^4-a^6 z^2+2 a^4 z^6+6 a^4 z^4+5 a^4 z^2+a^4 z^{-2} +a^4-a^2 z^8-5 a^2 z^6-10 a^2 z^4-8 a^2 z^2-2 a^2 z^{-2} -3 a^2+z^6+3 z^4+3 z^2+ z^{-2} +2 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ 2 a^4 z^{10}+2 a^2 z^{10}+8 a^5 z^9+14 a^3 z^9+6 a z^9+13 a^6 z^8+22 a^4 z^8+16 a^2 z^8+7 z^8+11 a^7 z^7+3 a^5 z^7-17 a^3 z^7-5 a z^7+4 z^7 a^{-1} +5 a^8 z^6-18 a^6 z^6-53 a^4 z^6-47 a^2 z^6+z^6 a^{-2} -16 z^6+a^9 z^5-15 a^7 z^5-24 a^5 z^5-7 a^3 z^5-8 a z^5-9 z^5 a^{-1} -4 a^8 z^4+6 a^6 z^4+40 a^4 z^4+45 a^2 z^4-2 z^4 a^{-2} +13 z^4+4 a^7 z^3+13 a^5 z^3+12 a^3 z^3+8 a z^3+5 z^3 a^{-1} -a^6 z^2-14 a^4 z^2-20 a^2 z^2+z^2 a^{-2} -6 z^2-3 a^3 z-3 a z+3 a^4+5 a^2+3+2 a^3 z^{-1} +2 a z^{-1} -a^4 z^{-2} -2 a^2 z^{-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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 7                       1 1 5                     3   -3 3                   6 1   5 1                 8 3     -5 -1               14 6       8 -3             13 11         -2 -5           13 11           2 -7         9 13             4 -9       8 13               -5 -11     4 10                 6 -13   1 7                   -6 -15   4                     4 -17 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=-7 }[/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} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{13} }[/math] [math]\displaystyle{ {\mathbb Z}^{13} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{13} }[/math] [math]\displaystyle{ {\mathbb Z}^{13} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{14} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=4 }[/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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