L11n310

L11n309

L11n311

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Link Presentations

[edit Notes on L11n310's Link Presentations]

Planar diagram presentation	X6172 X3,13,4,12 X13,19,14,18 X17,11,18,22 X7,17,8,16 X21,8,22,9 X9,20,10,21 X15,5,16,10 X19,15,20,14 X2536 X11,1,12,4
Gauss code	{1, -10, -2, 11}, {10, -1, -5, 6, -7, 8}, {-11, 2, -3, 9, -8, 5, -4, 3, -9, 7, -6, 4}

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(2) t(3)^4+t(3)^4+t(1) t(2)^2 t(3)^3+t(1) t(3)^3-3 t(1) t(2) t(3)^3+3 t(2) t(3)^3-2 t(3)^3-2 t(1) t(2)^2 t(3)^2+t(2)^2 t(3)^2-t(1) t(3)^2+4 t(1) t(2) t(3)^2-4 t(2) t(3)^2+2 t(3)^2+2 t(1) t(2)^2 t(3)-t(2)^2 t(3)-3 t(1) t(2) t(3)+3 t(2) t(3)-t(3)-t(1) t(2)^2+t(1) t(2)}{\sqrt{t(1)} t(2) t(3)^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^8+3 q^7-7 q^6+10 q^5-12 q^4+14 q^3-11 q^2+10 q-5+3 q^{-1} }[/math] (db)
Signature	2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^4 a^{-6} -2 z^2 a^{-6} - a^{-6} z^{-2} -3 a^{-6} +z^6 a^{-4} +4 z^4 a^{-4} +10 z^2 a^{-4} +4 a^{-4} z^{-2} +11 a^{-4} -4 z^4 a^{-2} -12 z^2 a^{-2} -5 a^{-2} z^{-2} -13 a^{-2} +3 z^2+2 z^{-2} +5 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^5 a^{-9} -2 z^3 a^{-9} +z a^{-9} +3 z^6 a^{-8} -5 z^4 a^{-8} +z^2 a^{-8} +5 z^7 a^{-7} -9 z^5 a^{-7} +5 z^3 a^{-7} -4 z a^{-7} + a^{-7} z^{-1} +5 z^8 a^{-6} -9 z^6 a^{-6} +8 z^4 a^{-6} -8 z^2 a^{-6} - a^{-6} z^{-2} +5 a^{-6} +2 z^9 a^{-5} +5 z^7 a^{-5} -22 z^5 a^{-5} +32 z^3 a^{-5} -21 z a^{-5} +5 a^{-5} z^{-1} +10 z^8 a^{-4} -29 z^6 a^{-4} +44 z^4 a^{-4} -36 z^2 a^{-4} -4 a^{-4} z^{-2} +18 a^{-4} +2 z^9 a^{-3} +3 z^7 a^{-3} -18 z^5 a^{-3} +35 z^3 a^{-3} -29 z a^{-3} +9 a^{-3} z^{-1} +5 z^8 a^{-2} -17 z^6 a^{-2} +37 z^4 a^{-2} -41 z^2 a^{-2} -5 a^{-2} z^{-2} +21 a^{-2} +3 z^7 a^{-1} -6 z^5 a^{-1} +10 z^3 a^{-1} -13 z a^{-1} +5 a^{-1} z^{-1} +6 z^4-14 z^2-2 z^{-2} +9 }[/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   \ -2 -1 0 1 2 3 4 5 6 7 χ 17                   1 -1 15                 2   2 13               5 1   -4 11             5 2     3 9           7 5       -2 7         7 5         2 5       5 8           3 3     5 6             -1 1   2 7               5 -1 1 3                 -2 -3 3                   3

Integral Khovanov Homology (db, data source)

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

L11n309 L11n311