L11a138

L11a137

L11a139

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Link Presentations

[edit Notes on L11a138's Link Presentations]

Planar diagram presentation	X6172 X2,11,3,12 X14,3,15,4 X12,5,13,6 X22,13,5,14 X4,21,1,22 X20,16,21,15 X16,8,17,7 X8,20,9,19 X18,10,19,9 X10,18,11,17
Gauss code	{1, -2, 3, -6}, {4, -1, 8, -9, 10, -11, 2, -4, 5, -3, 7, -8, 11, -10, 9, -7, 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{4 t(1) t(2)^3-4 t(2)^3-12 t(1) t(2)^2+13 t(2)^2+13 t(1) t(2)-12 t(2)-4 t(1)+4}{\sqrt{t(1)} t(2)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{9/2}-4 q^{7/2}+8 q^{5/2}-13 q^{3/2}+18 \sqrt{q}-\frac{21}{\sqrt{q}}+\frac{21}{q^{3/2}}-\frac{19}{q^{5/2}}+\frac{13}{q^{7/2}}-\frac{9}{q^{9/2}}+\frac{4}{q^{11/2}}-\frac{1}{q^{13/2}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^5 z^3+a^5 z^{-1} -a^3 z^5+z^3 a^{-3} -a^3 z-a^3 z^{-1} -2 a z^5-z^5 a^{-1} -2 a z^3-a z }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^7 z^5-a^7 z^3+4 a^6 z^6-5 a^6 z^4+8 a^5 z^7-14 a^5 z^5+8 a^5 z^3-3 a^5 z+a^5 z^{-1} +9 a^4 z^8-15 a^4 z^6+z^6 a^{-4} +10 a^4 z^4-2 z^4 a^{-4} -3 a^4 z^2-a^4+6 a^3 z^9-2 a^3 z^7+4 z^7 a^{-3} -10 a^3 z^5-10 z^5 a^{-3} +9 a^3 z^3+5 z^3 a^{-3} -2 a^3 z+a^3 z^{-1} +2 a^2 z^{10}+12 a^2 z^8+7 z^8 a^{-2} -35 a^2 z^6-19 z^6 a^{-2} +34 a^2 z^4+15 z^4 a^{-2} -11 a^2 z^2-4 z^2 a^{-2} +12 a z^9+6 z^9 a^{-1} -25 a z^7-11 z^7 a^{-1} +18 a z^5+3 z^5 a^{-1} -4 a z^3+z^3 a^{-1} +a z+2 z^{10}+10 z^8-36 z^6+36 z^4-12 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 4 5 χ 10                       1 -1 8                     3   3 6                   5 1   -4 4                 8 3     5 2               10 5       -5 0             11 8         3 -2           11 11           0 -4         8 10             -2 -6       5 11               6 -8     4 8                 -4 -10   1 6                   5 -12   3                     -3 -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}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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}^{8}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\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^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/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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