L10a49

L10a48

L10a50

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Link Presentations

[edit Notes on L10a49's Link Presentations]

Planar diagram presentation	X6172 X16,7,17,8 X4,17,1,18 X10,5,11,6 X14,3,15,4 X18,11,19,12 X20,13,5,14 X12,19,13,20 X2,9,3,10 X8,15,9,16
Gauss code	{1, -9, 5, -3}, {4, -1, 2, -10, 9, -4, 6, -8, 7, -5, 10, -2, 3, -6, 8, -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{t(2)^5+4 t(1) t(2)^4-4 t(2)^4-6 t(1) t(2)^3+6 t(2)^3+6 t(1) t(2)^2-6 t(2)^2-4 t(1) t(2)+4 t(2)+t(1)}{\sqrt{t(1)} t(2)^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{7}{q^{9/2}}+\frac{3}{q^{7/2}}-\frac{1}{q^{5/2}}-\frac{1}{q^{25/2}}+\frac{4}{q^{23/2}}-\frac{7}{q^{21/2}}+\frac{11}{q^{19/2}}-\frac{13}{q^{17/2}}+\frac{14}{q^{15/2}}-\frac{14}{q^{13/2}}+\frac{9}{q^{11/2}} }[/math] (db)
Signature	-5 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^3 a^{11}-2 a^{11} z^{-1} -z^5 a^9+z^3 a^9+8 z a^9+5 a^9 z^{-1} -3 z^5 a^7-10 z^3 a^7-10 z a^7-3 a^7 z^{-1} -z^5 a^5-2 z^3 a^5 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^5 a^{15}+z^3 a^{15}-4 z^6 a^{14}+7 z^4 a^{14}-2 z^2 a^{14}-a^{14}-6 z^7 a^{13}+10 z^5 a^{13}-3 z^3 a^{13}+z a^{13}-5 z^8 a^{12}+5 z^6 a^{12}+z^4 a^{12}+z^2 a^{12}-2 z^9 a^{11}-6 z^7 a^{11}+16 z^5 a^{11}-11 z^3 a^{11}+5 z a^{11}-2 a^{11} z^{-1} -10 z^8 a^{10}+20 z^6 a^{10}-13 z^4 a^{10}-4 z^2 a^{10}+5 a^{10}-2 z^9 a^9-6 z^7 a^9+22 z^5 a^9-27 z^3 a^9+14 z a^9-5 a^9 z^{-1} -5 z^8 a^8+8 z^6 a^8-2 z^4 a^8-7 z^2 a^8+5 a^8-6 z^7 a^7+16 z^5 a^7-18 z^3 a^7+10 z a^7-3 a^7 z^{-1} -3 z^6 a^6+5 z^4 a^6-z^5 a^5+2 z^3 a^5 }[/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   \ -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -4                     1 1 -6                   3 1 -2 -8                 4     4 -10               5 3     -2 -12             9 4       5 -14           6 6         0 -16         7 8           -1 -18       4 6             2 -20     3 7               -4 -22   1 4                 3 -24   3                   -3 -26 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=-10 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-9 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/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}^{6}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\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^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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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