L11a330

L11a329

L11a331

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Link Presentations

[edit Notes on L11a330's Link Presentations]

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

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-u^2 v^3+u^2 v^2-u v^4+3 u v^3-3 u v^2+3 u v-u+v^2-v+1\right)}{u^{3/2} v^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ 19 q^{9/2}-22 q^{7/2}+21 q^{5/2}-\frac{1}{q^{5/2}}-19 q^{3/2}+\frac{4}{q^{3/2}}+q^{17/2}-4 q^{15/2}+9 q^{13/2}-14 q^{11/2}+13 \sqrt{q}-\frac{9}{\sqrt{q}} }[/math] (db)
Signature	3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^7 a^{-5} +4 z^5 a^{-5} +5 z^3 a^{-5} +3 z a^{-5} + a^{-5} z^{-1} -z^9 a^{-3} -6 z^7 a^{-3} -13 z^5 a^{-3} -13 z^3 a^{-3} -7 z a^{-3} -3 a^{-3} z^{-1} +z^7 a^{-1} +4 z^5 a^{-1} +5 z^3 a^{-1} +4 z a^{-1} +2 a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^4 a^{-10} +4 z^5 a^{-9} -z^3 a^{-9} +9 z^6 a^{-8} -8 z^4 a^{-8} +3 z^2 a^{-8} +13 z^7 a^{-7} -17 z^5 a^{-7} +8 z^3 a^{-7} -z a^{-7} +13 z^8 a^{-6} -18 z^6 a^{-6} +6 z^4 a^{-6} -3 z^2 a^{-6} + a^{-6} +9 z^9 a^{-5} -8 z^7 a^{-5} -9 z^5 a^{-5} +3 z^3 a^{-5} +2 z a^{-5} - a^{-5} z^{-1} +3 z^{10} a^{-4} +12 z^8 a^{-4} -45 z^6 a^{-4} +36 z^4 a^{-4} -13 z^2 a^{-4} +3 a^{-4} +15 z^9 a^{-3} -40 z^7 a^{-3} +28 z^5 a^{-3} -10 z^3 a^{-3} +7 z a^{-3} -3 a^{-3} z^{-1} +3 z^{10} a^{-2} +3 z^8 a^{-2} -31 z^6 a^{-2} +33 z^4 a^{-2} -10 z^2 a^{-2} +3 a^{-2} +6 z^9 a^{-1} +a z^7-18 z^7 a^{-1} -3 a z^5+13 z^5 a^{-1} +3 a z^3-z^3 a^{-1} -a z+3 z a^{-1} -2 a^{-1} z^{-1} +4 z^8-13 z^6+12 z^4-3 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   \ -4 -3 -2 -1 0 1 2 3 4 5 6 7 χ 18                       1 -1 16                     3   3 14                   6 1   -5 12                 8 3     5 10               11 6       -5 8             11 8         3 6           10 11           1 4         9 11             -2 2       6 12               6 0     3 7                 -4 -2   1 6                   5 -4   3                     -3 -6 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=-4 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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}^{12}\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^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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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