L11n116

L11n115

L11n117

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Link Presentations

[edit Notes on L11n116's Link Presentations]

Planar diagram presentation	X6172 X7,17,8,16 X20,17,21,18 X18,13,19,14 X14,19,15,20 X4,21,1,22 X10,5,11,6 X12,3,13,4 X22,11,5,12 X2,9,3,10 X15,9,16,8
Gauss code	{1, -10, 8, -6}, {7, -1, -2, 11, 10, -7, 9, -8, 4, -5, -11, 2, 3, -4, 5, -3, 6, -9}

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(2)^5+2 t(1) t(2)^4-4 t(2)^4-7 t(1) t(2)^3+7 t(2)^3+7 t(1) t(2)^2-7 t(2)^2-4 t(1) t(2)+2 t(2)+t(1)}{\sqrt{t(1)} t(2)^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{3}{q^{3/2}}+\frac{6}{q^{5/2}}-\frac{11}{q^{7/2}}+\frac{13}{q^{9/2}}-\frac{14}{q^{11/2}}+\frac{14}{q^{13/2}}-\frac{11}{q^{15/2}}+\frac{7}{q^{17/2}}-\frac{4}{q^{19/2}}+\frac{1}{q^{21/2}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^9 \left(-z^3\right)+a^9 z^{-1} +a^7 z^5-3 a^7 z-3 a^7 z^{-1} +2 a^5 z^5+5 a^5 z^3+6 a^5 z+4 a^5 z^{-1} -3 a^3 z^3-5 a^3 z-2 a^3 z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{12} z^6-2 a^{12} z^4+a^{12} z^2+4 a^{11} z^7-11 a^{11} z^5+8 a^{11} z^3+5 a^{10} z^8-11 a^{10} z^6+4 a^{10} z^4+a^{10} z^2-a^{10}+2 a^9 z^9+7 a^9 z^7-28 a^9 z^5+19 a^9 z^3-4 a^9 z+a^9 z^{-1} +11 a^8 z^8-22 a^8 z^6+7 a^8 z^4+2 a^8 z^2-3 a^8+2 a^7 z^9+10 a^7 z^7-31 a^7 z^5+27 a^7 z^3-12 a^7 z+3 a^7 z^{-1} +6 a^6 z^8-7 a^6 z^6+a^6 z^4+5 a^6 z^2-3 a^6+7 a^5 z^7-14 a^5 z^5+22 a^5 z^3-15 a^5 z+4 a^5 z^{-1} +3 a^4 z^6+3 a^4 z^2-2 a^4+6 a^3 z^3-7 a^3 z+2 a^3 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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -2                   3 3 -4                 4 1 -3 -6               7 2   5 -8             6 4     -2 -10           8 7       1 -12         7 7         0 -14       4 7           -3 -16     3 7             4 -18   1 4               -3 -20   3                 3 -22 1                   -1

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=-9 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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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