L11n262

L11n261

L11n263

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Link Presentations

[edit Notes on L11n262's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X14,7,15,8 X8,13,5,14 X11,18,12,19 X19,22,20,9 X15,20,16,21 X21,16,22,17 X17,12,18,13 X2536 X4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -4}, {11, -2, -5, 9, 4, -3, -7, 8, -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{u v w^3-u v w+u v+u w^3-3 u w^2+3 u w-u+v w^3-3 v w^2+3 v w-v-w^3+w^2-1}{\sqrt{u} \sqrt{v} w^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ - q^{-13} +2 q^{-12} -5 q^{-11} +6 q^{-10} -6 q^{-9} +7 q^{-8} -4 q^{-7} +5 q^{-6} - q^{-5} + q^{-3} }[/math] (db)
Signature	-4 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a^{14} z^{-2} +3 a^{12} z^{-2} +4 a^{12}-5 z^2 a^{10}-2 a^{10} z^{-2} -7 a^{10}+z^4 a^8-a^8 z^{-2} -a^8+z^6 a^6+6 z^4 a^6+8 z^2 a^6+a^6 z^{-2} +4 a^6 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{15} z^7-5 a^{15} z^5+9 a^{15} z^3-7 a^{15} z+2 a^{15} z^{-1} +2 a^{14} z^8-8 a^{14} z^6+8 a^{14} z^4-2 a^{14} z^2-a^{14} z^{-2} +a^{14}+a^{13} z^9+2 a^{13} z^7-24 a^{13} z^5+39 a^{13} z^3-27 a^{13} z+8 a^{13} z^{-1} +6 a^{12} z^8-24 a^{12} z^6+24 a^{12} z^4-10 a^{12} z^2-3 a^{12} z^{-2} +5 a^{12}+a^{11} z^9+4 a^{11} z^7-33 a^{11} z^5+52 a^{11} z^3-34 a^{11} z+10 a^{11} z^{-1} +4 a^{10} z^8-18 a^{10} z^6+24 a^{10} z^4-13 a^{10} z^2-2 a^{10} z^{-2} +4 a^{10}+3 a^9 z^7-15 a^9 z^5+22 a^9 z^3-10 a^9 z+2 a^9 z^{-1} -a^8 z^6+2 a^8 z^4+3 a^8 z^2+a^8 z^{-2} -3 a^8-a^7 z^5+4 a^7 z-2 a^7 z^{-1} +a^6 z^6-6 a^6 z^4+8 a^6 z^2+a^6 z^{-2} -4 a^6 }[/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 -5                       1 1 -7                       1 1 -9                 2 1     -1 -11               4         4 -13             4 6 1       1 -15           4 1           3 -17         2 4 1           1 -19       4 4               0 -21     1 2                 1 -23   1 4                   -3 -25   1                     1 -27 1                       -1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-7 }[/math] [math]\displaystyle{ i=-5 }[/math] [math]\displaystyle{ i=-3 }[/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}^{2}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/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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