L11a38

L11a37

L11a39

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Link Presentations

[edit Notes on L11a38's Link Presentations]

Planar diagram presentation	X6172 X10,4,11,3 X12,10,13,9 X18,13,19,14 X16,7,17,8 X8,17,9,18 X22,19,5,20 X20,16,21,15 X14,22,15,21 X2536 X4,12,1,11
Gauss code	{1, -10, 2, -11}, {10, -1, 5, -6, 3, -2, 11, -3, 4, -9, 8, -5, 6, -4, 7, -8, 9, -7}

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)-1) (t(2)-1)^5}{\sqrt{t(1)} t(2)^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{9/2}-4 q^{7/2}+8 q^{5/2}-14 q^{3/2}+18 \sqrt{q}-\frac{21}{\sqrt{q}}+\frac{20}{q^{3/2}}-\frac{18}{q^{5/2}}+\frac{13}{q^{7/2}}-\frac{7}{q^{9/2}}+\frac{3}{q^{11/2}}-\frac{1}{q^{13/2}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^5 z^3+2 a^5 z+a^5 z^{-1} -2 a^3 z^5-6 a^3 z^3+z^3 a^{-3} -8 a^3 z+z a^{-3} -3 a^3 z^{-1} +a z^7+4 a z^5-2 z^5 a^{-1} +9 a z^3-5 z^3 a^{-1} +10 a z-5 z a^{-1} +4 a z^{-1} -2 a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^7 z^5-2 a^7 z^3+a^7 z+3 a^6 z^6-5 a^6 z^4+3 a^6 z^2-a^6+5 a^5 z^7-6 a^5 z^5+3 a^5 z^3-2 a^5 z+a^5 z^{-1} +6 a^4 z^8-5 a^4 z^6+z^6 a^{-4} -2 a^4 z^4-2 z^4 a^{-4} +7 a^4 z^2+z^2 a^{-4} -3 a^4+4 a^3 z^9+6 a^3 z^7+4 z^7 a^{-3} -24 a^3 z^5-10 z^5 a^{-3} +28 a^3 z^3+8 z^3 a^{-3} -15 a^3 z-2 z a^{-3} +3 a^3 z^{-1} +a^2 z^{10}+15 a^2 z^8+6 z^8 a^{-2} -32 a^2 z^6-12 z^6 a^{-2} +19 a^2 z^4+5 z^4 a^{-2} +2 a^2 z^2-3 a^2+8 a z^9+4 z^9 a^{-1} +a z^7+4 z^7 a^{-1} -36 a z^5-29 z^5 a^{-1} +43 a z^3+28 z^3 a^{-1} -21 a z-11 z a^{-1} +4 a z^{-1} +2 a^{-1} z^{-1} +z^{10}+15 z^8-37 z^6+23 z^4-3 z^2-2 }[/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 5 χ 10                       1 -1 8                     3   3 6                   5 1   -4 4                 9 3     6 2               9 5       -4 0             12 9         3 -2           10 11           1 -4         8 10             -2 -6       5 10               5 -8     2 8                 -6 -10   1 5                   4 -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}^{5}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/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}^{5}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=5 }[/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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