L11a329

L11a328

L11a330

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Link Presentations

[edit Notes on L11a329's Link Presentations]

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

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{\left(v^2-v+1\right) (u v+1) (u v-u+1) (u v-v+1)}{u^{3/2} v^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{9/2}-3 q^{7/2}+7 q^{5/2}-11 q^{3/2}+14 \sqrt{q}-\frac{18}{\sqrt{q}}+\frac{17}{q^{3/2}}-\frac{15}{q^{5/2}}+\frac{11}{q^{7/2}}-\frac{7}{q^{9/2}}+\frac{3}{q^{11/2}}-\frac{1}{q^{13/2}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^3 z^7+5 a^3 z^5+9 a^3 z^3+7 a^3 z+2 a^3 z^{-1} -a z^9-7 a z^7+z^7 a^{-1} -19 a z^5+5 z^5 a^{-1} -25 a z^3+9 z^3 a^{-1} -16 a z+7 z a^{-1} -3 a z^{-1} + a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -2 a^2 z^{10}-2 z^{10}-5 a^3 z^9-10 a z^9-5 z^9 a^{-1} -6 a^4 z^8-4 a^2 z^8-5 z^8 a^{-2} -3 z^8-5 a^5 z^7+9 a^3 z^7+31 a z^7+14 z^7 a^{-1} -3 z^7 a^{-3} -3 a^6 z^6+11 a^4 z^6+19 a^2 z^6+14 z^6 a^{-2} -z^6 a^{-4} +20 z^6-a^7 z^5+8 a^5 z^5-10 a^3 z^5-45 a z^5-18 z^5 a^{-1} +8 z^5 a^{-3} +5 a^6 z^4-9 a^4 z^4-28 a^2 z^4-12 z^4 a^{-2} +3 z^4 a^{-4} -29 z^4+2 a^7 z^3-3 a^5 z^3+7 a^3 z^3+34 a z^3+18 z^3 a^{-1} -4 z^3 a^{-3} -a^6 z^2+4 a^4 z^2+14 a^2 z^2+7 z^2 a^{-2} -2 z^2 a^{-4} +18 z^2-a^7 z+2 a^5 z-6 a^3 z-17 a z-8 z a^{-1} -3 a^2- a^{-2} -3+2 a^3 z^{-1} +3 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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 10                       1 -1 8                     2   2 6                   5 1   -4 4                 6 2     4 2               8 5       -3 0             10 6         4 -2           8 9           1 -4         7 9             -2 -6       4 8               4 -8     3 7                 -4 -10   1 5                   4 -12   2                     -2 -14 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=0 }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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}^{8}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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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