L11a517

L11a516

L11a518

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Link Presentations

[edit Notes on L11a517's Link Presentations]

Planar diagram presentation	X8192 X14,4,15,3 X12,14,7,13 X2738 X22,10,13,9 X6,22,1,21 X20,16,21,15 X16,5,17,6 X18,11,19,12 X10,17,11,18 X4,19,5,20
Gauss code	{1, -4, 2, -11, 8, -6}, {4, -1, 5, -10, 9, -3}, {3, -2, 7, -8, 10, -9, 11, -7, 6, -5}

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(1)-1) (t(3)-1) \left(t(1) t(3)^2 t(2)^2-t(3)^2 t(2)^2+2 t(3) t(2)^2-t(2)^2-2 t(1) t(3)^2 t(2)+t(3)^2 t(2)-t(1) t(2)+3 t(1) t(3) t(2)-3 t(3) t(2)+2 t(2)+t(1) t(3)^2+t(1)-2 t(1) t(3)-1\right)}{t(1) t(2) t(3)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{-6} -q^5-5 q^{-5} +4 q^4+12 q^{-4} -10 q^3-18 q^{-3} +18 q^2+26 q^{-2} -24 q-28 q^{-1} +29 }[/math] (db)
Signature	0 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^8-2 a^2 z^6-z^6 a^{-2} +5 z^6+a^4 z^4-6 a^2 z^4-3 z^4 a^{-2} +11 z^4+a^4 z^2-7 a^2 z^2-4 z^2 a^{-2} +10 z^2+a^4-4 a^2- a^{-2} +4+a^4 z^{-2} -2 a^2 z^{-2} + z^{-2} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ 4 a^2 z^{10}+4 z^{10}+11 a^3 z^9+22 a z^9+11 z^9 a^{-1} +11 a^4 z^8+17 a^2 z^8+13 z^8 a^{-2} +19 z^8+5 a^5 z^7-18 a^3 z^7-42 a z^7-10 z^7 a^{-1} +9 z^7 a^{-3} +a^6 z^6-24 a^4 z^6-58 a^2 z^6-21 z^6 a^{-2} +4 z^6 a^{-4} -58 z^6-8 a^5 z^5+3 a^3 z^5+20 a z^5-4 z^5 a^{-1} -12 z^5 a^{-3} +z^5 a^{-5} -a^6 z^4+15 a^4 z^4+50 a^2 z^4+18 z^4 a^{-2} -4 z^4 a^{-4} +56 z^4+2 a^5 z^3+2 a^3 z^3+8 z^3 a^{-1} +7 z^3 a^{-3} -z^3 a^{-5} -4 a^4 z^2-21 a^2 z^2-9 z^2 a^{-2} +z^2 a^{-4} -27 z^2-2 a^3 z-4 a z-3 z a^{-1} -z a^{-3} +3 a^4+7 a^2+2 a^{-2} +7+2 a^3 z^{-1} +2 a z^{-1} -a^4 z^{-2} -2 a^2 z^{-2} - 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 4 5 χ 11                       1 -1 9                     3   3 7                   7 1   -6 5                 11 3     8 3               13 7       -6 1             16 11         5 -1           14 15           1 -3         12 14             -2 -5       8 16               8 -7     4 10                 -6 -9   1 8                   7 -11   4                     -4 -13 1                       1

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=1 }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{16}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{14}\oplus{\mathbb Z}_2^{14} }[/math] [math]\displaystyle{ {\mathbb Z}^{14} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{15}\oplus{\mathbb Z}_2^{14} }[/math] [math]\displaystyle{ {\mathbb Z}^{16} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{13} }[/math] [math]\displaystyle{ {\mathbb Z}^{13} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=5 }[/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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