L11a208

L11a207

L11a209

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Link Presentations

[edit Notes on L11a208's Link Presentations]

Planar diagram presentation	X8192 X18,9,19,10 X6718 X20,13,21,14 X10,4,11,3 X16,6,17,5 X4,12,5,11 X22,15,7,16 X12,19,13,20 X14,21,15,22 X2,18,3,17
Gauss code	{1, -11, 5, -7, 6, -3}, {3, -1, 2, -5, 7, -9, 4, -10, 8, -6, 11, -2, 9, -4, 10, -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{\left(v^4-v^3+v^2-v+1\right) (u v-u+1) (u v-v+1)}{u v^3} }[/math] (db)
Jones polynomial	[math]\displaystyle{ \frac{13}{q^{9/2}}-\frac{14}{q^{7/2}}-q^{5/2}+\frac{13}{q^{5/2}}+3 q^{3/2}-\frac{13}{q^{3/2}}+\frac{1}{q^{17/2}}-\frac{3}{q^{15/2}}+\frac{6}{q^{13/2}}-\frac{9}{q^{11/2}}-6 \sqrt{q}+\frac{8}{\sqrt{q}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a^5 z^7-5 a^5 z^5-8 a^5 z^3-5 a^5 z-2 a^5 z^{-1} +a^3 z^9+7 a^3 z^7+18 a^3 z^5+21 a^3 z^3+12 a^3 z+5 a^3 z^{-1} -a z^7-5 a z^5-8 a z^3-6 a z-3 a z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{10} z^4-a^{10} z^2+3 a^9 z^5-3 a^9 z^3+5 a^8 z^6-6 a^8 z^4+3 a^8 z^2-a^8+6 a^7 z^7-8 a^7 z^5+4 a^7 z^3+6 a^6 z^8-10 a^6 z^6+5 a^6 z^4+2 a^6 z^2+5 a^5 z^9-12 a^5 z^7+15 a^5 z^5-13 a^5 z^3+6 a^5 z-2 a^5 z^{-1} +2 a^4 z^{10}+a^4 z^8-15 a^4 z^6+19 a^4 z^4-12 a^4 z^2+5 a^4+9 a^3 z^9-35 a^3 z^7+50 a^3 z^5-38 a^3 z^3+15 a^3 z-5 a^3 z^{-1} +2 a^2 z^{10}-2 a^2 z^8-12 a^2 z^6+19 a^2 z^4-12 a^2 z^2+5 a^2+4 a z^9-16 a z^7+z^7 a^{-1} +20 a z^5-4 z^5 a^{-1} -14 a z^3+4 z^3 a^{-1} +8 a z-3 a z^{-1} -z a^{-1} +3 z^8-12 z^6+12 z^4-2 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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 6                       1 1 4                     2   -2 2                   4 1   3 0                 5 3     -2 -2               8 3       5 -4             6 6         0 -6           8 7           1 -8         5 6             1 -10       4 8               -4 -12     2 5                 3 -14   1 4                   -3 -16   2                     2 -18 1                       -1

Integral Khovanov Homology (db, data source)

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