L10a57

L10a56

L10a58

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Link Presentations

[edit Notes on L10a57's Link Presentations]

Planar diagram presentation	X8192 X10,3,11,4 X14,6,15,5 X16,11,17,12 X20,17,7,18 X18,14,19,13 X12,20,13,19 X4,16,5,15 X2738 X6,9,1,10
Gauss code	{1, -9, 2, -8, 3, -10}, {9, -1, 10, -2, 4, -7, 6, -3, 8, -4, 5, -6, 7, -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) t(2)^4+t(2)^4-t(1)^2 t(2)^3+5 t(1) t(2)^3-3 t(2)^3+3 t(1)^2 t(2)^2-9 t(1) t(2)^2+3 t(2)^2-3 t(1)^2 t(2)+5 t(1) t(2)-t(2)+t(1)^2-t(1)}{t(1) t(2)^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{6}{q^{9/2}}-q^{7/2}+\frac{9}{q^{7/2}}+3 q^{5/2}-\frac{12}{q^{5/2}}-6 q^{3/2}+\frac{12}{q^{3/2}}-\frac{1}{q^{13/2}}+\frac{3}{q^{11/2}}+9 \sqrt{q}-\frac{12}{\sqrt{q}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^3 a^5+z a^5+a^5 z^{-1} -z^5 a^3-2 z^3 a^3-4 z a^3-2 a^3 z^{-1} -z^5 a+2 z a+2 a z^{-1} +2 z^3 a^{-1} +z a^{-1} - a^{-1} z^{-1} -z a^{-3} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^7 z^5-2 a^7 z^3+3 a^6 z^6-6 a^6 z^4+2 a^6 z^2+5 a^5 z^7-12 a^5 z^5+11 a^5 z^3-5 a^5 z+a^5 z^{-1} +4 a^4 z^8-6 a^4 z^6+3 a^4 z^4+a^3 z^9+9 a^3 z^7-27 a^3 z^5+z^5 a^{-3} +31 a^3 z^3-2 z^3 a^{-3} -14 a^3 z+z a^{-3} +2 a^3 z^{-1} +7 a^2 z^8-12 a^2 z^6+3 z^6 a^{-2} +9 a^2 z^4-6 z^4 a^{-2} -3 a^2 z^2+3 z^2 a^{-2} +a^2+a z^9+8 a z^7+4 z^7 a^{-1} -21 a z^5-6 z^5 a^{-1} +22 a z^3+2 z^3 a^{-1} -12 a z-2 z a^{-1} +2 a z^{-1} + a^{-1} z^{-1} +3 z^8-6 z^4+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 χ 8                     1 1 6                   2   -2 4                 4 1   3 2               5 2     -3 0             7 4       3 -2           6 6         0 -4         6 6           0 -6       4 7             3 -8     2 5               -3 -10   1 4                 3 -12   2                   -2 -14 1                     1

Integral Khovanov Homology (db, data source)

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