L11a108

L11a107

L11a109

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Link Presentations

[edit Notes on L11a108's Link Presentations]

Planar diagram presentation	X6172 X14,3,15,4 X22,15,5,16 X16,7,17,8 X18,9,19,10 X8,17,9,18 X10,21,11,22 X20,11,21,12 X12,19,13,20 X2536 X4,13,1,14
Gauss code	{1, -10, 2, -11}, {10, -1, 4, -6, 5, -7, 8, -9, 11, -2, 3, -4, 6, -5, 9, -8, 7, -3}

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{4 u v^4-6 u v^3+6 u v^2-5 u v+2 u+2 v^5-5 v^4+6 v^3-6 v^2+4 v}{\sqrt{u} v^{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{2}{q^{25/2}}+\frac{5}{q^{23/2}}-\frac{8}{q^{21/2}}+\frac{12}{q^{19/2}}-\frac{15}{q^{17/2}}+\frac{14}{q^{15/2}}-\frac{14}{q^{13/2}}+\frac{10}{q^{11/2}} }[/math] (db)
Signature	-5 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^{13} (-z)-2 a^{13} z^{-1} +3 a^{11} z^3+8 a^{11} z+4 a^{11} z^{-1} -2 a^9 z^5-5 a^9 z^3-3 a^9 z-a^9 z^{-1} -3 a^7 z^5-9 a^7 z^3-6 a^7 z-a^7 z^{-1} -a^5 z^5-2 a^5 z^3 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{16} z^6-4 a^{16} z^4+5 a^{16} z^2-2 a^{16}+2 a^{15} z^7-6 a^{15} z^5+4 a^{15} z^3+a^{15} z+2 a^{14} z^8-2 a^{14} z^6-6 a^{14} z^4+7 a^{14} z^2-a^{14}+2 a^{13} z^9-3 a^{13} z^7+5 a^{13} z^5-12 a^{13} z^3+8 a^{13} z-2 a^{13} z^{-1} +a^{12} z^{10}+2 a^{12} z^8-7 a^{12} z^6+13 a^{12} z^4-17 a^{12} z^2+6 a^{12}+6 a^{11} z^9-18 a^{11} z^7+35 a^{11} z^5-36 a^{11} z^3+16 a^{11} z-4 a^{11} z^{-1} +a^{10} z^{10}+6 a^{10} z^8-20 a^{10} z^6+30 a^{10} z^4-19 a^{10} z^2+5 a^{10}+4 a^9 z^9-7 a^9 z^7+8 a^9 z^5-4 a^9 z^3+3 a^9 z-a^9 z^{-1} +6 a^8 z^8-13 a^8 z^6+10 a^8 z^4-a^8+6 a^7 z^7-15 a^7 z^5+14 a^7 z^3-6 a^7 z+a^7 z^{-1} +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                 6 3     -3 -12               8 4       4 -14             7 7         0 -16           8 7           1 -18         4 7             3 -20       4 8               -4 -22     1 4                 3 -24   1 4                   -3 -26   1                     1 -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}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-9 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/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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