L11n66

L11n65

L11n67

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Link Presentations

[edit Notes on L11n66's Link Presentations]

Planar diagram presentation	X6172 X3,10,4,11 X7,14,8,15 X15,22,16,5 X9,17,10,16 X21,9,22,8 X17,21,18,20 X13,18,14,19 X19,12,20,13 X2536 X11,4,12,1
Gauss code	{1, -10, -2, 11}, {10, -1, -3, 6, -5, 2, -11, 9, -8, 3, -4, 5, -7, 8, -9, 7, -6, 4}

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{u v^5-5 u v^4+8 u v^3-4 u v^2+u v+v^4-4 v^3+8 v^2-5 v+1}{\sqrt{u} v^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ \frac{12}{q^{9/2}}-\frac{13}{q^{7/2}}+\frac{12}{q^{5/2}}+q^{3/2}-\frac{11}{q^{3/2}}-\frac{2}{q^{15/2}}+\frac{5}{q^{13/2}}-\frac{9}{q^{11/2}}-4 \sqrt{q}+\frac{7}{\sqrt{q}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^9 z^{-1} -a^7 z^3-4 a^7 z-3 a^7 z^{-1} +2 a^5 z^5+7 a^5 z^3+8 a^5 z+4 a^5 z^{-1} -a^3 z^7-4 a^3 z^5-6 a^3 z^3-6 a^3 z-2 a^3 z^{-1} +a z^5+2 a z^3 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ 3 a^9 z^3-4 a^9 z+a^9 z^{-1} +a^8 z^6+4 a^8 z^4-4 a^8 z^2+a^8+5 a^7 z^7-13 a^7 z^5+25 a^7 z^3-16 a^7 z+3 a^7 z^{-1} +6 a^6 z^8-16 a^6 z^6+23 a^6 z^4-12 a^6 z^2+3 a^6+2 a^5 z^9+8 a^5 z^7-37 a^5 z^5+49 a^5 z^3-25 a^5 z+4 a^5 z^{-1} +11 a^4 z^8-29 a^4 z^6+23 a^4 z^4-9 a^4 z^2+2 a^4+2 a^3 z^9+7 a^3 z^7-35 a^3 z^5+35 a^3 z^3-15 a^3 z+2 a^3 z^{-1} +5 a^2 z^8-11 a^2 z^6+2 a^2 z^4+a^2+4 a z^7-11 a z^5+8 a z^3-2 a z+z^6-2 z^4+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 χ 4                   1 -1 2                 3   3 0               4 1   -3 -2             7 3     4 -4           6 5       -1 -6         7 6         1 -8       5 6           1 -10     4 7             -3 -12   2 6               4 -14   3                 -3 -16 2                   2

Integral Khovanov Homology (db, data source)

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