L11a155

L11a154

L11a156

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Link Presentations

[edit Notes on L11a155's Link Presentations]

Planar diagram presentation	X8192 X10,3,11,4 X14,6,15,5 X20,11,21,12 X22,18,7,17 X18,22,19,21 X16,13,17,14 X12,19,13,20 X4,16,5,15 X2738 X6,9,1,10
Gauss code	{1, -10, 2, -9, 3, -11}, {10, -1, 11, -2, 4, -8, 7, -3, 9, -7, 5, -6, 8, -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{t(1) t(2)^4-t(2)^4+t(1)^2 t(2)^3-7 t(1) t(2)^3+5 t(2)^3-5 t(1)^2 t(2)^2+13 t(1) t(2)^2-5 t(2)^2+5 t(1)^2 t(2)-7 t(1) t(2)+t(2)-t(1)^2+t(1)}{t(1) t(2)^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^{7/2}+3 q^{5/2}-7 q^{3/2}+11 \sqrt{q}-\frac{15}{\sqrt{q}}+\frac{17}{q^{3/2}}-\frac{17}{q^{5/2}}+\frac{14}{q^{7/2}}-\frac{11}{q^{9/2}}+\frac{6}{q^{11/2}}-\frac{3}{q^{13/2}}+\frac{1}{q^{15/2}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z a^7+2 z^3 a^5+z a^5+a^5 z^{-1} -z^5 a^3-2 z a^3-2 a^3 z^{-1} -z^5 a+z^3 a+2 z a+2 a z^{-1} +2 z^3 a^{-1} - a^{-1} z^{-1} -z a^{-3} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -a^4 z^{10}-a^2 z^{10}-3 a^5 z^9-6 a^3 z^9-3 a z^9-4 a^6 z^8-7 a^4 z^8-8 a^2 z^8-5 z^8-3 a^7 z^7+a^5 z^7+5 a^3 z^7-4 a z^7-5 z^7 a^{-1} -a^8 z^6+9 a^6 z^6+19 a^4 z^6+17 a^2 z^6-3 z^6 a^{-2} +5 z^6+9 a^7 z^5+12 a^5 z^5+10 a^3 z^5+16 a z^5+8 z^5 a^{-1} -z^5 a^{-3} +3 a^8 z^4-4 a^6 z^4-14 a^4 z^4-13 a^2 z^4+5 z^4 a^{-2} -z^4-8 a^7 z^3-15 a^5 z^3-18 a^3 z^3-18 a z^3-5 z^3 a^{-1} +2 z^3 a^{-3} -2 a^8 z^2+3 a^4 z^2+4 a^2 z^2-2 z^2 a^{-2} +z^2+3 a^7 z+7 a^5 z+10 a^3 z+10 a z+3 z a^{-1} -z a^{-3} -a^2-a^5 z^{-1} -2 a^3 z^{-1} -2 a z^{-1} - a^{-1} z^{-1} }[/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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 8                       1 1 6                     2   -2 4                   5 1   4 2                 6 2     -4 0               9 5       4 -2             9 7         -2 -4           8 8           0 -6         7 10             3 -8       4 7               -3 -10     2 7                 5 -12   1 4                   -3 -14   2                     2 -16 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=-7 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/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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