L11a314

L11a313

L11a315

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Link Presentations

[edit Notes on L11a314's Link Presentations]

Planar diagram presentation	X10,1,11,2 X20,11,21,12 X8,21,1,22 X16,10,17,9 X14,8,15,7 X12,4,13,3 X18,6,19,5 X6,18,7,17 X4,14,5,13 X22,16,9,15 X2,20,3,19
Gauss code	{1, -11, 6, -9, 7, -8, 5, -3}, {4, -1, 2, -6, 9, -5, 10, -4, 8, -7, 11, -2, 3, -10}

A Braid Representative	{{{braid_table}}}
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^3 v^4-2 u^3 v^3+2 u^3 v^2-u^3 v+u^2 v^5-5 u^2 v^4+11 u^2 v^3-10 u^2 v^2+5 u^2 v-u^2-u v^5+5 u v^4-10 u v^3+11 u v^2-5 u v+u-v^4+2 v^3-2 v^2+v}{u^{3/2} v^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ 25 q^{9/2}-26 q^{7/2}+21 q^{5/2}-16 q^{3/2}+\frac{1}{q^{3/2}}-q^{19/2}+4 q^{17/2}-10 q^{15/2}+17 q^{13/2}-22 q^{11/2}+9 \sqrt{q}-\frac{4}{\sqrt{q}} }[/math] (db)
Signature	3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^7 a^{-3} +z^7 a^{-5} -z^5 a^{-1} +3 z^5 a^{-3} +3 z^5 a^{-5} -z^5 a^{-7} -2 z^3 a^{-1} +4 z^3 a^{-3} +4 z^3 a^{-5} -2 z^3 a^{-7} -z a^{-1} +3 z a^{-3} +2 z a^{-5} -2 z a^{-7} + a^{-3} z^{-1} - a^{-5} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -3 z^{10} a^{-4} -3 z^{10} a^{-6} -7 z^9 a^{-3} -16 z^9 a^{-5} -9 z^9 a^{-7} -7 z^8 a^{-2} -10 z^8 a^{-4} -15 z^8 a^{-6} -12 z^8 a^{-8} -4 z^7 a^{-1} +9 z^7 a^{-3} +28 z^7 a^{-5} +6 z^7 a^{-7} -9 z^7 a^{-9} +14 z^6 a^{-2} +30 z^6 a^{-4} +38 z^6 a^{-6} +19 z^6 a^{-8} -4 z^6 a^{-10} -z^6+9 z^5 a^{-1} +2 z^5 a^{-3} -14 z^5 a^{-5} +7 z^5 a^{-7} +13 z^5 a^{-9} -z^5 a^{-11} -7 z^4 a^{-2} -19 z^4 a^{-4} -27 z^4 a^{-6} -13 z^4 a^{-8} +4 z^4 a^{-10} +2 z^4-6 z^3 a^{-1} -4 z^3 a^{-3} +5 z^3 a^{-5} -6 z^3 a^{-7} -8 z^3 a^{-9} +z^3 a^{-11} +3 z^2 a^{-4} +8 z^2 a^{-6} +5 z^2 a^{-8} -z^2 a^{-10} -z^2+z a^{-1} +2 z a^{-3} -z a^{-5} +2 z a^{-9} + a^{-4} - a^{-3} z^{-1} - a^{-5} z^{-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   \ -3 -2 -1 0 1 2 3 4 5 6 7 8 χ 20                       1 1 18                     3   -3 16                   7 1   6 14                 10 3     -7 12               12 7       5 10             14 11         -3 8           12 11           1 6         9 14             5 4       7 12               -5 2     3 10                 7 0   1 6                   -5 -2   3                     3 -4 1                       -1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=2 }[/math] [math]\displaystyle{ i=4 }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{14}\oplus{\mathbb Z}_2^{12} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{14} }[/math] [math]\displaystyle{ {\mathbb Z}^{14} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=7 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=8 }[/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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