L11a249

L11a248

L11a250

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Link Presentations

[edit Notes on L11a249's Link Presentations]

Planar diagram presentation	X10,1,11,2 X16,8,17,7 X18,12,19,11 X2,19,3,20 X12,4,13,3 X20,13,21,14 X14,5,15,6 X6,9,7,10 X22,16,9,15 X8,18,1,17 X4,22,5,21
Gauss code	{1, -4, 5, -11, 7, -8, 2, -10}, {8, -1, 3, -5, 6, -7, 9, -2, 10, -3, 4, -6, 11, -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) (v-1) \left(u^2 v^4-2 u^2 v^3+u^2 v^2-u v^4+3 u v^3-6 u v^2+3 u v-u+v^2-2 v+1\right)}{u^{3/2} v^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^{13/2}+5 q^{11/2}-11 q^{9/2}+18 q^{7/2}-25 q^{5/2}+28 q^{3/2}-29 \sqrt{q}+\frac{24}{\sqrt{q}}-\frac{18}{q^{3/2}}+\frac{11}{q^{5/2}}-\frac{5}{q^{7/2}}+\frac{1}{q^{9/2}} }[/math] (db)
Signature	1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^9 a^{-1} -a z^7+5 z^7 a^{-1} -z^7 a^{-3} -3 a z^5+7 z^5 a^{-1} -3 z^5 a^{-3} -a z^3-z^3 a^{-3} +2 a z-4 z a^{-1} +2 z a^{-3} +a z^{-1} - a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^5 a^{-7} +5 z^6 a^{-6} -4 z^4 a^{-6} +11 z^7 a^{-5} -14 z^5 a^{-5} +4 z^3 a^{-5} +14 z^8 a^{-4} +a^4 z^6-18 z^6 a^{-4} -a^4 z^4+4 z^4 a^{-4} +z^2 a^{-4} +11 z^9 a^{-3} +5 a^3 z^7-6 z^7 a^{-3} -9 a^3 z^5-14 z^5 a^{-3} +3 a^3 z^3+13 z^3 a^{-3} -4 z a^{-3} +4 z^{10} a^{-2} +10 a^2 z^8+18 z^8 a^{-2} -21 a^2 z^6-48 z^6 a^{-2} +11 a^2 z^4+30 z^4 a^{-2} -a^2 z^2-3 z^2 a^{-2} +10 a z^9+21 z^9 a^{-1} -15 a z^7-37 z^7 a^{-1} -2 a z^5+8 z^5 a^{-1} +9 a z^3+15 z^3 a^{-1} -4 a z-8 z a^{-1} +a z^{-1} + a^{-1} z^{-1} +4 z^{10}+14 z^8-47 z^6+34 z^4-5 z^2-1 }[/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 14                       1 1 12                     4   -4 10                   7 1   6 8                 11 4     -7 6               14 7       7 4             14 11         -3 2           15 14           1 0         11 16             5 -2       7 13               -6 -4     4 11                 7 -6   1 7                   -6 -8   4                     4 -10 1                       -1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=0 }[/math] [math]\displaystyle{ i=2 }[/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}^{7} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{16}\oplus{\mathbb Z}_2^{13} }[/math] [math]\displaystyle{ {\mathbb Z}^{15} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{14}\oplus{\mathbb Z}_2^{14} }[/math] [math]\displaystyle{ {\mathbb Z}^{14} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{14} }[/math] [math]\displaystyle{ {\mathbb Z}^{14} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/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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