L11a306

L11a305

L11a307

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Link Presentations

[edit Notes on L11a306's Link Presentations]

Planar diagram presentation	X10,1,11,2 X2,11,3,12 X12,3,13,4 X4,9,5,10 X16,5,17,6 X22,17,9,18 X18,13,19,14 X14,21,15,22 X20,7,21,8 X6,15,7,16 X8,19,1,20
Gauss code	{1, -2, 3, -4, 5, -10, 9, -11}, {4, -1, 2, -3, 7, -8, 10, -5, 6, -7, 11, -9, 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{-t(1) t(2)^5-3 t(1)^2 t(2)^4+4 t(1) t(2)^4-t(2)^4-3 t(1)^3 t(2)^3+7 t(1)^2 t(2)^3-8 t(1) t(2)^3+3 t(2)^3+3 t(1)^3 t(2)^2-8 t(1)^2 t(2)^2+7 t(1) t(2)^2-3 t(2)^2-t(1)^3 t(2)+4 t(1)^2 t(2)-3 t(1) t(2)-t(1)^2}{t(1)^{3/2} 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^{27/2}}-\frac{4}{q^{25/2}}+\frac{8}{q^{23/2}}-\frac{13}{q^{21/2}}+\frac{17}{q^{19/2}}-\frac{19}{q^{17/2}}+\frac{19}{q^{15/2}}-\frac{17}{q^{13/2}}+\frac{11}{q^{11/2}} }[/math] (db)
Signature	-5 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^{13} (-z)-a^{13} z^{-1} +4 a^{11} z^3+8 a^{11} z+3 a^{11} z^{-1} -3 a^9 z^5-8 a^9 z^3-6 a^9 z-2 a^9 z^{-1} -3 a^7 z^5-8 a^7 z^3-5 a^7 z-a^5 z^5-2 a^5 z^3 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{16} z^6-2 a^{16} z^4+a^{16} z^2+4 a^{15} z^7-10 a^{15} z^5+6 a^{15} z^3+6 a^{14} z^8-13 a^{14} z^6+5 a^{14} z^4-a^{14} z^2+a^{14}+5 a^{13} z^9-6 a^{13} z^7-5 a^{13} z^5+2 a^{13} z^3+a^{13} z-a^{13} z^{-1} +2 a^{12} z^{10}+7 a^{12} z^8-27 a^{12} z^6+30 a^{12} z^4-19 a^{12} z^2+3 a^{12}+11 a^{11} z^9-28 a^{11} z^7+34 a^{11} z^5-27 a^{11} z^3+14 a^{11} z-3 a^{11} z^{-1} +2 a^{10} z^{10}+8 a^{10} z^8-31 a^{10} z^6+42 a^{10} z^4-20 a^{10} z^2+3 a^{10}+6 a^9 z^9-12 a^9 z^7+14 a^9 z^5-8 a^9 z^3+8 a^9 z-2 a^9 z^{-1} +7 a^8 z^8-15 a^8 z^6+14 a^8 z^4-3 a^8 z^2+6 a^7 z^7-14 a^7 z^5+13 a^7 z^3-5 a^7 z+3 a^6 z^6-5 a^6 z^4+a^5 z^5-2 a^5 z^3 }[/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   \ -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -4                       1 1 -6                     3 1 -2 -8                   4     4 -10                 7 3     -4 -12               10 4       6 -14             9 7         -2 -16           10 10           0 -18         7 9             2 -20       6 10               -4 -22     3 8                 5 -24   1 5                   -4 -26   3                     3 -28 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=-11 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-10 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-9 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/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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