L11a109

L11a108

L11a110

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Link Presentations

[edit Notes on L11a109's Link Presentations]

Planar diagram presentation	X6172 X14,3,15,4 X22,15,5,16 X16,7,17,8 X8,21,9,22 X20,11,21,12 X18,9,19,10 X10,19,11,20 X12,17,13,18 X2536 X4,13,1,14
Gauss code	{1, -10, 2, -11}, {10, -1, 4, -5, 7, -8, 6, -9, 11, -2, 3, -4, 9, -7, 8, -6, 5, -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^4-7 u v^3+8 u v^2-6 u v+2 u+2 v^5-6 v^4+8 v^3-7 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{3}{q^{25/2}}+\frac{7}{q^{23/2}}-\frac{11}{q^{21/2}}+\frac{15}{q^{19/2}}-\frac{17}{q^{17/2}}+\frac{17}{q^{15/2}}-\frac{16}{q^{13/2}}+\frac{10}{q^{11/2}} }[/math] (db)
Signature	-5 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^{13} (-z)-a^{13} z^{-1} +3 a^{11} z^3+5 a^{11} z+a^{11} z^{-1} -2 a^9 z^5-3 a^9 z^3+2 a^9 z+2 a^9 z^{-1} -3 a^7 z^5-9 a^7 z^3-8 a^7 z-2 a^7 z^{-1} -a^5 z^5-2 a^5 z^3 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{16} z^6-3 a^{16} z^4+3 a^{16} z^2-a^{16}+3 a^{15} z^7-8 a^{15} z^5+6 a^{15} z^3-a^{15} z+4 a^{14} z^8-7 a^{14} z^6-a^{14} z^4+3 a^{14} z^2+3 a^{13} z^9-11 a^{13} z^5+6 a^{13} z^3-a^{13} z^{-1} +a^{12} z^{10}+8 a^{12} z^8-22 a^{12} z^6+20 a^{12} z^4-13 a^{12} z^2+3 a^{12}+7 a^{11} z^9-11 a^{11} z^7+6 a^{11} z^5-4 a^{11} z^3+3 a^{11} z-a^{11} z^{-1} +a^{10} z^{10}+10 a^{10} z^8-28 a^{10} z^6+30 a^{10} z^4-10 a^{10} z^2+4 a^9 z^9-2 a^9 z^7-7 a^9 z^5+14 a^9 z^3-6 a^9 z+2 a^9 z^{-1} +6 a^8 z^8-11 a^8 z^6+7 a^8 z^4+3 a^8 z^2-3 a^8+6 a^7 z^7-15 a^7 z^5+16 a^7 z^3-8 a^7 z+2 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               10 4       6 -14             8 7         -1 -16           9 9           0 -18         6 8             2 -20       5 9               -4 -22     2 6                 4 -24   1 5                   -4 -26   2                     2 -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}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-9 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/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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