L11a203

L11a202

L11a204

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Link Presentations

[edit Notes on L11a203's Link Presentations]

Planar diagram presentation	X8192 X2,9,3,10 X10,3,11,4 X6718 X18,11,19,12 X16,6,17,5 X4,18,5,17 X22,15,7,16 X12,21,13,22 X20,13,21,14 X14,19,15,20
Gauss code	{1, -2, 3, -7, 6, -4}, {4, -1, 2, -3, 5, -9, 10, -11, 8, -6, 7, -5, 11, -10, 9, -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{2 u^2 v^4-3 u^2 v^3+3 u^2 v^2-2 u^2 v-2 u v^4+3 u v^3-3 u v^2+3 u v-2 u-2 v^3+3 v^2-3 v+2}{u v^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{1}{\sqrt{q}}+\frac{2}{q^{3/2}}-\frac{5}{q^{5/2}}+\frac{6}{q^{7/2}}-\frac{9}{q^{9/2}}+\frac{10}{q^{11/2}}-\frac{10}{q^{13/2}}+\frac{9}{q^{15/2}}-\frac{7}{q^{17/2}}+\frac{4}{q^{19/2}}-\frac{2}{q^{21/2}}+\frac{1}{q^{23/2}} }[/math] (db)
Signature	-5 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^5 a^9-4 z^3 a^9-4 z a^9-a^9 z^{-1} +z^7 a^7+5 z^5 a^7+9 z^3 a^7+8 z a^7+2 a^7 z^{-1} +z^7 a^5+4 z^5 a^5+3 z^3 a^5-z a^5-z^5 a^3-4 z^3 a^3-4 z a^3-a^3 z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^4 a^{14}+2 z^2 a^{14}-2 z^5 a^{13}+3 z^3 a^{13}-3 z^6 a^{12}+5 z^4 a^{12}-3 z^2 a^{12}-4 z^7 a^{11}+10 z^5 a^{11}-12 z^3 a^{11}+2 z a^{11}-4 z^8 a^{10}+12 z^6 a^{10}-18 z^4 a^{10}+9 z^2 a^{10}-2 a^{10}-3 z^9 a^9+9 z^7 a^9-13 z^5 a^9+9 z^3 a^9-4 z a^9+a^9 z^{-1} -z^{10} a^8-2 z^8 a^8+17 z^6 a^8-31 z^4 a^8+25 z^2 a^8-5 a^8-5 z^9 a^7+20 z^7 a^7-31 z^5 a^7+29 z^3 a^7-12 z a^7+2 a^7 z^{-1} -z^{10} a^6+10 z^6 a^6-14 z^4 a^6+10 z^2 a^6-3 a^6-2 z^9 a^5+6 z^7 a^5-z^5 a^5-3 z^3 a^5-z a^5-2 z^8 a^4+8 z^6 a^4-7 z^4 a^4-z^2 a^4+a^4-z^7 a^3+5 z^5 a^3-8 z^3 a^3+5 z a^3-a^3 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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 0                       1 1 -2                     1   -1 -4                   4 1   3 -6                 3 2     -1 -8               6 3       3 -10             5 4         -1 -12           5 5           0 -14         4 5             1 -16       3 5               -2 -18     1 4                 3 -20   1 3                   -2 -22   1                     1 -24 1                       -1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-6 }[/math] [math]\displaystyle{ i=-4 }[/math] [math]\displaystyle{ r=-9 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=2 }[/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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