L11n354

L11n353

L11n355

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Link Presentations

[edit Notes on L11n354's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X11,18,12,19 X7,16,8,17 X15,8,16,9 X13,21,14,20 X19,22,20,15 X21,13,22,12 X17,14,18,5 X2536 X4,9,1,10
Gauss code	{1, -10, 2, -11}, {-5, 4, -9, 3, -7, 6, -8, 7}, {10, -1, -4, 5, 11, -2, -3, 8, -6, 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{(t(3)-1) \left(t(2) t(3)^3-t(3)^3+t(1) t(2)^2 t(3)^2-t(1) t(2) t(3)^2-t(2) t(3)^2-t(1) t(2) t(3)-t(2) t(3)+t(3)-t(1) t(2)^2+t(1) t(2)\right)}{\sqrt{t(1)} t(2) t(3)^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ - q^{-10} +2 q^{-9} -2 q^{-8} +3 q^{-7} - q^{-6} +2 q^{-5} + q^{-2} - q^{-1} +1 }[/math] (db)
Signature	-2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a^{10}+2 a^8 z^2+a^8 z^{-2} +2 a^8+a^6 z^2-2 a^6 z^{-2} -2 a^6-a^4 z^6-5 a^4 z^4-5 a^4 z^2+a^4 z^{-2} -a^4+a^2 z^4+4 a^2 z^2+2 a^2 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{11} z^7-5 a^{11} z^5+6 a^{11} z^3-2 a^{11} z+2 a^{10} z^8-11 a^{10} z^6+17 a^{10} z^4-11 a^{10} z^2+3 a^{10}+a^9 z^9-3 a^9 z^7-6 a^9 z^5+16 a^9 z^3-7 a^9 z+4 a^8 z^8-25 a^8 z^6+46 a^8 z^4-35 a^8 z^2-a^8 z^{-2} +12 a^8+a^7 z^9-4 a^7 z^7-4 a^7 z^5+19 a^7 z^3-14 a^7 z+2 a^7 z^{-1} +2 a^6 z^8-14 a^6 z^6+28 a^6 z^4-26 a^6 z^2-2 a^6 z^{-2} +12 a^6+a^5 z^7-8 a^5 z^5+14 a^5 z^3-10 a^5 z+2 a^5 z^{-1} +a^4 z^6-6 a^4 z^4+4 a^4 z^2-a^4 z^{-2} +2 a^4+a^3 z^7-5 a^3 z^5+5 a^3 z^3-a^3 z+a^2 z^6-5 a^2 z^4+6 a^2 z^2-2 a^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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 1                       1 1 -1                         0 -3                 1 2 1   0 -5               1 1 1     1 -7             2 3 1       0 -9           2 2 2         2 -11         2 4 1           1 -13       2 1 1             2 -15     1 3 1               1 -17   1 1                   0 -19   1                     1 -21 1                       -1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-5 }[/math] [math]\displaystyle{ i=-3 }[/math] [math]\displaystyle{ i=-1 }[/math] [math]\displaystyle{ r=-9 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=2 }[/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.

L11n353 L11n355