L11a524

L11a523

L11a525

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Link Presentations

[edit Notes on L11a524's Link Presentations]

Planar diagram presentation	X8192 X22,10,13,9 X18,8,19,7 X20,17,21,18 X6,22,1,21 X4,11,5,12 X16,6,17,5 X10,16,11,15 X12,20,7,19 X2,13,3,14 X14,3,15,4
Gauss code	{1, -10, 11, -6, 7, -5}, {3, -1, 2, -8, 6, -9}, {10, -11, 8, -7, 4, -3, 9, -4, 5, -2}

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{(v-1) \left(u^2 v w^3-3 u^2 v w^2+2 u^2 v w-u^2 w^3+2 u^2 w^2-u^2 w-u v w^3+4 u v w^2-5 u v w+u v+u w^3-5 u w^2+4 u w-u-v w^2+2 v w-v+2 w^2-3 w+1\right)}{u v w^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^6-4 q^5- q^{-5} +10 q^4+5 q^{-4} -16 q^3-11 q^{-3} +24 q^2+18 q^{-2} -26 q-24 q^{-1} +28 }[/math] (db)
Signature	0 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^4 a^{-4} +2 z^2 a^{-4} + a^{-4} z^{-2} + a^{-4} -a^2 z^6-2 z^6 a^{-2} -2 a^2 z^4-6 z^4 a^{-2} -5 z^2 a^{-2} -2 a^{-2} z^{-2} +a^2-2 a^{-2} +z^8+4 z^6+5 z^4+z^2+ z^{-2} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^6 a^{-6} -2 z^4 a^{-6} +z^2 a^{-6} +4 z^7 a^{-5} +a^5 z^5-8 z^5 a^{-5} +4 z^3 a^{-5} +8 z^8 a^{-4} +5 a^4 z^6-18 z^6 a^{-4} -4 a^4 z^4+16 z^4 a^{-4} -10 z^2 a^{-4} - a^{-4} z^{-2} +4 a^{-4} +8 z^9 a^{-3} +11 a^3 z^7-11 z^7 a^{-3} -14 a^3 z^5-z^5 a^{-3} +4 a^3 z^3+6 z^3 a^{-3} -5 z a^{-3} +2 a^{-3} z^{-1} +3 z^{10} a^{-2} +14 a^2 z^8+16 z^8 a^{-2} -19 a^2 z^6-52 z^6 a^{-2} +6 a^2 z^4+50 z^4 a^{-2} -22 z^2 a^{-2} -2 a^{-2} z^{-2} -a^2+6 a^{-2} +10 a z^9+18 z^9 a^{-1} -2 a z^7-28 z^7 a^{-1} -20 a z^5+2 z^5 a^{-1} +13 a z^3+11 z^3 a^{-1} -5 z a^{-1} +2 a^{-1} z^{-1} +3 z^{10}+22 z^8-57 z^6+42 z^4-11 z^2- z^{-2} +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   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 6 χ 13                       1 1 11                     3   -3 9                   7 1   6 7                 10 4     -6 5               14 6       8 3             13 11         -2 1           15 13           2 -1         10 14             4 -3       8 14               -6 -5     4 11                 7 -7   1 7                   -6 -9   4                     4 -11 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=-5 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{14}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{14}\oplus{\mathbb Z}_2^{14} }[/math] [math]\displaystyle{ {\mathbb Z}^{15} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{13} }[/math] [math]\displaystyle{ {\mathbb Z}^{13} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{13} }[/math] [math]\displaystyle{ {\mathbb Z}^{14} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/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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