L11a340

L11a339

L11a341

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Link Presentations

[edit Notes on L11a340's Link Presentations]

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

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