L11a199

L11a198

L11a200

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Link Presentations

[edit Notes on L11a199's Link Presentations]

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

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^4-6 u^2 v^3+4 u^2 v^2-u^2 v-u v^4+5 u v^3-9 u v^2+5 u v-u-v^3+4 v^2-6 v+2}{u v^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{12}{q^{9/2}}+\frac{8}{q^{7/2}}-\frac{5}{q^{5/2}}+\frac{2}{q^{3/2}}+\frac{1}{q^{23/2}}-\frac{4}{q^{21/2}}+\frac{7}{q^{19/2}}-\frac{11}{q^{17/2}}+\frac{14}{q^{15/2}}-\frac{15}{q^{13/2}}+\frac{14}{q^{11/2}}-\frac{1}{\sqrt{q}} }[/math] (db)
Signature	-5 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^5 a^9-2 z^3 a^9+z a^9+a^9 z^{-1} +z^7 a^7+3 z^5 a^7-4 z a^7-2 a^7 z^{-1} +z^7 a^5+4 z^5 a^5+5 z^3 a^5+4 z a^5+2 a^5 z^{-1} -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}-4 z^5 a^{13}+3 z^3 a^{13}-7 z^6 a^{12}+7 z^4 a^{12}-z^2 a^{12}-8 z^7 a^{11}+8 z^5 a^{11}+z^3 a^{11}-2 z a^{11}-7 z^8 a^{10}+8 z^6 a^{10}-2 z^2 a^{10}-4 z^9 a^9+z^7 a^9+9 z^5 a^9-7 z^3 a^9+3 z a^9-a^9 z^{-1} -z^{10} a^8-8 z^8 a^8+26 z^6 a^8-22 z^4 a^8+6 z^2 a^8-6 z^9 a^7+14 z^7 a^7-2 z^5 a^7-12 z^3 a^7+9 z a^7-2 a^7 z^{-1} -z^{10} a^6-3 z^8 a^6+19 z^6 a^6-23 z^4 a^6+10 z^2 a^6-a^6-2 z^9 a^5+4 z^7 a^5+6 z^5 a^5-15 z^3 a^5+9 z a^5-2 a^5 z^{-1} -2 z^8 a^4+8 z^6 a^4-9 z^4 a^4+3 z^2 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                 5 2     -3 -8               7 3       4 -10             7 5         -2 -12           8 7           1 -14         6 7             1 -16       5 8               -3 -18     3 7                 4 -20   1 4                   -3 -22   3                     3 -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}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/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=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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