L11a67

L11a66

L11a68

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Link Presentations

[edit Notes on L11a67's Link Presentations]

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

A Braid Representative	{{{braid_table}}}
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) t(2)^5-2 t(2)^5-5 t(1) t(2)^4+9 t(2)^4+12 t(1) t(2)^3-16 t(2)^3-16 t(1) t(2)^2+12 t(2)^2+9 t(1) t(2)-5 t(2)-2 t(1)+1}{\sqrt{t(1)} t(2)^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{18}{q^{9/2}}-q^{7/2}+\frac{24}{q^{7/2}}+5 q^{5/2}-\frac{29}{q^{5/2}}-13 q^{3/2}+\frac{30}{q^{3/2}}+\frac{1}{q^{15/2}}-\frac{4}{q^{13/2}}+\frac{9}{q^{11/2}}+20 \sqrt{q}-\frac{26}{\sqrt{q}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z a^7+3 z^3 a^5+3 z a^5+2 a^5 z^{-1} -3 z^5 a^3-6 z^3 a^3-7 z a^3-4 a^3 z^{-1} +z^7 a+3 z^5 a+6 z^3 a+5 z a+3 a z^{-1} -z^5 a^{-1} -z^3 a^{-1} -2 z a^{-1} - a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -3 a^4 z^{10}-3 a^2 z^{10}-7 a^5 z^9-19 a^3 z^9-12 a z^9-7 a^6 z^8-18 a^4 z^8-29 a^2 z^8-18 z^8-4 a^7 z^7+2 a^5 z^7+20 a^3 z^7+a z^7-13 z^7 a^{-1} -a^8 z^6+12 a^6 z^6+50 a^4 z^6+70 a^2 z^6-5 z^6 a^{-2} +28 z^6+9 a^7 z^5+22 a^5 z^5+27 a^3 z^5+31 a z^5+16 z^5 a^{-1} -z^5 a^{-3} +2 a^8 z^4-6 a^6 z^4-37 a^4 z^4-45 a^2 z^4+2 z^4 a^{-2} -14 z^4-8 a^7 z^3-31 a^5 z^3-45 a^3 z^3-29 a z^3-7 z^3 a^{-1} -a^8 z^2+a^6 z^2+9 a^4 z^2+11 a^2 z^2+4 z^2+3 a^7 z+16 a^5 z+24 a^3 z+15 a z+4 z a^{-1} -a^6-2 a^4-3 a^2-1-2 a^5 z^{-1} -4 a^3 z^{-1} -3 a z^{-1} - a^{-1} 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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 8                       1 1 6                     4   -4 4                   9 1   8 2                 11 4     -7 0               15 9       6 -2             16 12         -4 -4           13 14           -1 -6         11 16             5 -8       7 13               -6 -10     3 12                 9 -12   1 6                   -5 -14   3                     3 -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}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{16}\oplus{\mathbb Z}_2^{13} }[/math] [math]\displaystyle{ {\mathbb Z}^{13} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{14}\oplus{\mathbb Z}_2^{16} }[/math] [math]\displaystyle{ {\mathbb Z}^{16} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{14} }[/math] [math]\displaystyle{ {\mathbb Z}^{15} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/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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