L11a541

L11a540

L11a542

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Link Presentations

[edit Notes on L11a541's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X20,12,21,11 X22,13,19,14 X18,22,9,21 X12,17,13,18 X8,16,5,15 X14,8,15,7 X16,19,17,20 X2536 X4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, 8, -7}, {9, -3, 5, -4}, {11, -2, 3, -6, 4, -8, 7, -9, 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(3)-1) (t(4)-1)^2 (-t(2) t(1)+t(2) t(4) t(1)-t(4) t(1)+2 t(1)+t(2)-2 t(2) t(4)+t(4)-1)}{\sqrt{t(1)} \sqrt{t(2)} \sqrt{t(3)} t(4)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^{7/2}+5 q^{5/2}-13 q^{3/2}+18 \sqrt{q}-\frac{25}{\sqrt{q}}+\frac{25}{q^{3/2}}-\frac{27}{q^{5/2}}+\frac{20}{q^{7/2}}-\frac{15}{q^{9/2}}+\frac{7}{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-a^7 z^{-1} +3 z^3 a^5+6 z a^5+5 a^5 z^{-1} +a^5 z^{-3} -3 z^5 a^3-8 z^3 a^3-12 z a^3-10 a^3 z^{-1} -3 a^3 z^{-3} +z^7 a+3 z^5 a+6 z^3 a+9 z a+9 a z^{-1} +3 a z^{-3} -z^5 a^{-1} -z^3 a^{-1} -2 z a^{-1} -3 a^{-1} z^{-1} - a^{-1} z^{-3} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -2 a^4 z^{10}-2 a^2 z^{10}-4 a^5 z^9-13 a^3 z^9-9 a z^9-4 a^6 z^8-12 a^4 z^8-24 a^2 z^8-16 z^8-3 a^7 z^7-4 a^5 z^7+5 a^3 z^7-7 a z^7-13 z^7 a^{-1} -a^8 z^6+3 a^6 z^6+24 a^4 z^6+48 a^2 z^6-5 z^6 a^{-2} +23 z^6+8 a^7 z^5+28 a^5 z^5+44 a^3 z^5+43 a z^5+18 z^5 a^{-1} -z^5 a^{-3} +3 a^8 z^4+9 a^6 z^4+2 a^4 z^4-11 a^2 z^4+2 z^4 a^{-2} -5 z^4-9 a^7 z^3-43 a^5 z^3-68 a^3 z^3-43 a z^3-9 z^3 a^{-1} -3 a^8 z^2-14 a^6 z^2-30 a^4 z^2-24 a^2 z^2-5 z^2+6 a^7 z+31 a^5 z+48 a^3 z+30 a z+7 z a^{-1} +a^8+6 a^6+18 a^4+21 a^2+9-2 a^7 z^{-1} -11 a^5 z^{-1} -18 a^3 z^{-1} -14 a z^{-1} -5 a^{-1} z^{-1} -3 a^4 z^{-2} -6 a^2 z^{-2} -3 z^{-2} +a^5 z^{-3} +3 a^3 z^{-3} +3 a z^{-3} + a^{-1} z^{-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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 8                       1 1 6                     4   -4 4                   9 1   8 2                 9 4     -5 0               16 9       7 -2             15 15         0 -4           12 10           2 -6         8 15             7 -8       7 12               -5 -10     2 10                 8 -12   1 5                   -4 -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}^{5}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{15}\oplus{\mathbb Z}_2^{12} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{15} }[/math] [math]\displaystyle{ {\mathbb Z}^{15} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{15}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{16} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/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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