L11n335

L11n334

L11n336

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Link Presentations

[edit Notes on L11n335's Link Presentations]

Planar diagram presentation	X6172 X11,16,12,17 X8493 X2,18,3,17 X5,14,6,15 X18,7,19,8 X15,12,16,5 X20,14,21,13 X22,9,13,10 X10,21,11,22 X4,19,1,20
Gauss code	{1, -4, 3, -11}, {-5, -1, 6, -3, 9, -10, -2, 7}, {8, 5, -7, 2, 4, -6, 11, -8, 10, -9}

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-1) (v-1) (w-1) (v w-2 v-2 w+1)}{\sqrt{u} v w} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^2+5 q-9+13 q^{-1} -16 q^{-2} +17 q^{-3} -14 q^{-4} +12 q^{-5} -6 q^{-6} +3 q^{-7} }[/math] (db)
Signature	-2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^8 z^{-2} +2 z^2 a^6-2 a^6 z^{-2} -a^6-3 z^4 a^4-4 z^2 a^4+a^4 z^{-2} +z^6 a^2+2 z^4 a^2+2 z^2 a^2-z^4+1 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ 2 a^5 z^9+2 a^3 z^9+5 a^6 z^8+12 a^4 z^8+7 a^2 z^8+3 a^7 z^7+9 a^5 z^7+15 a^3 z^7+9 a z^7-9 a^6 z^6-18 a^4 z^6-4 a^2 z^6+5 z^6-3 a^7 z^5-23 a^5 z^5-36 a^3 z^5-15 a z^5+z^5 a^{-1} +6 a^8 z^4+16 a^6 z^4+7 a^4 z^4-9 a^2 z^4-6 z^4+7 a^7 z^3+21 a^5 z^3+19 a^3 z^3+5 a z^3-11 a^8 z^2-16 a^6 z^2-3 a^4 z^2+2 a^2 z^2-9 a^7 z-9 a^5 z+6 a^8+9 a^6+3 a^4+1+2 a^7 z^{-1} +2 a^5 z^{-1} -a^8 z^{-2} -2 a^6 z^{-2} -a^4 z^{-2} }[/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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 χ 5                   1 -1 3                 4   4 1               5 1   -4 -1             8 4     4 -3           10 7       -3 -5         7 6         1 -7       7 10           3 -9     5 7             -2 -11   1 7               6 -13 2 5                 -3 -15 3                   3

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-3 }[/math] [math]\displaystyle{ i=-1 }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/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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