L11a204

L11a203

L11a205

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Link Presentations

[edit Notes on L11a204's Link Presentations]

Planar diagram presentation	X8192 X2,9,3,10 X10,3,11,4 X16,5,17,6 X18,13,19,14 X14,17,15,18 X22,20,7,19 X20,12,21,11 X12,22,13,21 X6718 X4,15,5,16
Gauss code	{1, -2, 3, -11, 4, -10}, {10, -1, 2, -3, 8, -9, 5, -6, 11, -4, 6, -5, 7, -8, 9, -7}

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