L11n398

L11n397

L11n399

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Link Presentations

[edit Notes on L11n398's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X15,22,16,19 X7,20,8,21 X19,8,20,9 X13,18,14,5 X11,14,12,15 X17,12,18,13 X21,16,22,17 X2536 X4,9,1,10
Gauss code	{1, -10, 2, -11}, {-5, 4, -9, 3}, {10, -1, -4, 5, 11, -2, -7, 8, -6, 7, -3, 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{(w-1) \left(u v w^3+u v w^2-u v w+u v-u w^2+u w+v w^3-v w^2+w^4-w^3+w^2+w\right)}{\sqrt{u} \sqrt{v} w^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{-3} - q^{-5} +4 q^{-6} -4 q^{-7} +6 q^{-8} -5 q^{-9} +6 q^{-10} -4 q^{-11} +2 q^{-12} - q^{-13} }[/math] (db)
Signature	-4 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a^{14} z^{-2} +4 a^{12} z^{-2} +4 a^{12}-5 z^2 a^{10}-5 a^{10} z^{-2} -9 a^{10}+z^4 a^8+z^2 a^8+2 a^8 z^{-2} +2 a^8+z^6 a^6+6 z^4 a^6+8 z^2 a^6+3 a^6 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^7 a^{15}-5 z^5 a^{15}+8 z^3 a^{15}-5 z a^{15}+a^{15} z^{-1} +2 z^8 a^{14}-9 z^6 a^{14}+12 z^4 a^{14}-8 z^2 a^{14}-a^{14} z^{-2} +4 a^{14}+z^9 a^{13}+z^7 a^{13}-21 z^5 a^{13}+36 z^3 a^{13}-23 z a^{13}+5 a^{13} z^{-1} +6 z^8 a^{12}-28 z^6 a^{12}+41 z^4 a^{12}-34 z^2 a^{12}-4 a^{12} z^{-2} +18 a^{12}+z^9 a^{11}+3 z^7 a^{11}-32 z^5 a^{11}+57 z^3 a^{11}-39 z a^{11}+9 a^{11} z^{-1} +4 z^8 a^{10}-21 z^6 a^{10}+38 z^4 a^{10}-37 z^2 a^{10}-5 a^{10} z^{-2} +21 a^{10}+3 z^7 a^9-17 z^5 a^9+30 z^3 a^9-20 z a^9+5 a^9 z^{-1} -z^6 a^8+3 z^4 a^8-3 z^2 a^8-2 a^8 z^{-2} +5 a^8-z^5 a^7+z^3 a^7+z a^7+z^6 a^6-6 z^4 a^6+8 z^2 a^6-3 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               3         3 -13             4 5 1       0 -15           3 1           2 -17         2 4 1           1 -19       4 3               1 -21     1 3                 2 -23   1 3                   -2 -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}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\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^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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}^{5}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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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