L11a345

L11a344

L11a346

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Link Presentations

[edit Notes on L11a345's Link Presentations]

Planar diagram presentation	X12,1,13,2 X8493 X18,14,19,13 X22,20,11,19 X20,15,21,16 X14,21,15,22 X6,17,7,18 X16,7,17,8 X10,6,1,5 X4,10,5,9 X2,11,3,12
Gauss code	{1, -11, 2, -10, 9, -7, 8, -2, 10, -9}, {11, -1, 3, -6, 5, -8, 7, -3, 4, -5, 6, -4}

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^3 v^3-4 u^3 v^2+6 u^3 v-2 u^3-2 u^2 v^3+6 u^2 v^2-6 u^2 v+2 u^2+2 u v^3-6 u v^2+6 u v-2 u-2 v^3+6 v^2-4 v+1}{u^{3/2} v^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{11/2}-4 q^{9/2}+9 q^{7/2}-14 q^{5/2}+17 q^{3/2}-19 \sqrt{q}+\frac{18}{\sqrt{q}}-\frac{15}{q^{3/2}}+\frac{10}{q^{5/2}}-\frac{6}{q^{7/2}}+\frac{2}{q^{9/2}}-\frac{1}{q^{11/2}} }[/math] (db)
Signature	1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^7 a^{-1} +3 a z^5-4 z^5 a^{-1} +z^5 a^{-3} -3 a^3 z^3+10 a z^3-8 z^3 a^{-1} +2 z^3 a^{-3} +a^5 z-8 a^3 z+12 a z-9 z a^{-1} +2 z a^{-3} +2 a^5 z^{-1} -5 a^3 z^{-1} +6 a z^{-1} -4 a^{-1} z^{-1} + a^{-3} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^4 a^{-6} +a^5 z^7-5 a^5 z^5+4 z^5 a^{-5} +9 a^5 z^3-z^3 a^{-5} -7 a^5 z+2 a^5 z^{-1} +2 a^4 z^8-7 a^4 z^6+9 z^6 a^{-4} +7 a^4 z^4-8 z^4 a^{-4} -a^4 z^2+3 z^2 a^{-4} -a^4- a^{-4} +2 a^3 z^9-a^3 z^7+13 z^7 a^{-3} -16 a^3 z^5-19 z^5 a^{-3} +31 a^3 z^3+11 z^3 a^{-3} -21 a^3 z-4 z a^{-3} +5 a^3 z^{-1} + a^{-3} z^{-1} +a^2 z^{10}+6 a^2 z^8+11 z^8 a^{-2} -24 a^2 z^6-12 z^6 a^{-2} +21 a^2 z^4-5 z^4 a^{-2} -3 a^2 z^2+9 z^2 a^{-2} -a^2-3 a^{-2} +7 a z^9+5 z^9 a^{-1} -6 a z^7+9 z^7 a^{-1} -30 a z^5-42 z^5 a^{-1} +50 a z^3+40 z^3 a^{-1} -29 a z-19 z a^{-1} +6 a z^{-1} +4 a^{-1} z^{-1} +z^{10}+15 z^8-38 z^6+18 z^4+4 z^2-3 }[/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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 12                       1 -1 10                     3   3 8                   6 1   -5 6                 8 3     5 4               9 6       -3 2             10 8         2 0           9 10           1 -2         6 9             -3 -4       4 9               5 -6     2 6                 -4 -8   1 5                   4 -10   1                     -1 -12 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=-6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=5 }[/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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