L10a30

L10a29

L10a31

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Link Presentations

[edit Notes on L10a30's Link Presentations]

Planar diagram presentation	X6172 X12,3,13,4 X14,8,15,7 X18,10,19,9 X20,16,5,15 X16,20,17,19 X8,18,9,17 X10,14,11,13 X2536 X4,11,1,12
Gauss code	{1, -9, 2, -10}, {9, -1, 3, -7, 4, -8, 10, -2, 8, -3, 5, -6, 7, -4, 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{2 t(1) t(2)^3-4 t(2)^3-6 t(1) t(2)^2+7 t(2)^2+7 t(1) t(2)-6 t(2)-4 t(1)+2}{\sqrt{t(1)} t(2)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^{13/2}+4 q^{11/2}-7 q^{9/2}+10 q^{7/2}-12 q^{5/2}+13 q^{3/2}-12 \sqrt{q}+\frac{8}{\sqrt{q}}-\frac{6}{q^{3/2}}+\frac{2}{q^{5/2}}-\frac{1}{q^{7/2}} }[/math] (db)
Signature	1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^3 a^{-5} +z^5 a^{-3} +z^3 a^{-3} +a^3 z+z a^{-3} +a^3 z^{-1} + a^{-3} z^{-1} +z^5 a^{-1} -2 a z^3-2 a z-2 z a^{-1} -2 a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^5 a^{-7} -z^3 a^{-7} +4 z^6 a^{-6} -7 z^4 a^{-6} +2 z^2 a^{-6} +6 z^7 a^{-5} -11 z^5 a^{-5} +4 z^3 a^{-5} +4 z^8 a^{-4} -z^6 a^{-4} -9 z^4 a^{-4} +7 z^2 a^{-4} -2 a^{-4} +z^9 a^{-3} +9 z^7 a^{-3} +a^3 z^5-18 z^5 a^{-3} -3 a^3 z^3+8 z^3 a^{-3} +3 a^3 z-z a^{-3} -a^3 z^{-1} + a^{-3} z^{-1} +6 z^8 a^{-2} +2 a^2 z^6-3 z^6 a^{-2} -3 a^2 z^4-12 z^4 a^{-2} +16 z^2 a^{-2} +a^2-5 a^{-2} +z^9 a^{-1} +3 a z^7+6 z^7 a^{-1} -4 a z^5-11 z^5 a^{-1} +2 a z^3+8 z^3 a^{-1} -a z-5 z a^{-1} +2 a^{-1} z^{-1} +2 z^8+4 z^6-13 z^4+11 z^2-3 }[/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   \ -4 -3 -2 -1 0 1 2 3 4 5 6 χ 14                     1 1 12                   3   -3 10                 4 1   3 8               6 3     -3 6             6 4       2 4           7 6         -1 2         5 6           -1 0       4 8             4 -2     2 4               -2 -4     4                 4 -6 1 2                   -1 -8 1                     1

Integral Khovanov Homology (db, data source)

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

L10a29 L10a31