L11a140

L11a139

L11a141

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Link Presentations

[edit Notes on L11a140's Link Presentations]

Planar diagram presentation	X8192 X10,4,11,3 X22,10,7,9 X2738 X16,11,17,12 X12,5,13,6 X4,18,5,17 X14,19,15,20 X20,13,21,14 X18,22,19,21 X6,15,1,16
Gauss code	{1, -4, 2, -7, 6, -11}, {4, -1, 3, -2, 5, -6, 9, -8, 11, -5, 7, -10, 8, -9, 10, -3}

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{\left(t(1) t(2)^2-t(2)^2-3 t(1) t(2)+2 t(2)+t(1)-1\right) \left(t(1) t(2)^2-t(2)^2-2 t(1) t(2)+3 t(2)+t(1)-1\right)}{t(1) t(2)^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^{7/2}+5 q^{5/2}-11 q^{3/2}+17 \sqrt{q}-\frac{24}{\sqrt{q}}+\frac{26}{q^{3/2}}-\frac{26}{q^{5/2}}+\frac{22}{q^{7/2}}-\frac{16}{q^{9/2}}+\frac{9}{q^{11/2}}-\frac{4}{q^{13/2}}+\frac{1}{q^{15/2}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z a^7+3 z^3 a^5+3 z a^5-3 z^5 a^3-6 z^3 a^3-3 z a^3+a^3 z^{-1} +z^7 a+3 z^5 a+4 z^3 a-a z^{-1} -z^5 a^{-1} -z^3 a^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^8 z^6-2 a^8 z^4+a^8 z^2+4 a^7 z^7-9 a^7 z^5+7 a^7 z^3-2 a^7 z+7 a^6 z^8-14 a^6 z^6+9 a^6 z^4-2 a^6 z^2+6 a^5 z^9-a^5 z^7-20 a^5 z^5+21 a^5 z^3-6 a^5 z+2 a^4 z^{10}+18 a^4 z^8-47 a^4 z^6+34 a^4 z^4-8 a^4 z^2+14 a^3 z^9-9 a^3 z^7-29 a^3 z^5+z^5 a^{-3} +29 a^3 z^3-5 a^3 z-a^3 z^{-1} +2 a^2 z^{10}+24 a^2 z^8-53 a^2 z^6+5 z^6 a^{-2} +31 a^2 z^4-4 z^4 a^{-2} -6 a^2 z^2+a^2+8 a z^9+7 a z^7+11 z^7 a^{-1} -34 a z^5-15 z^5 a^{-1} +21 a z^3+6 z^3 a^{-1} -a z-a z^{-1} +13 z^8-16 z^6+4 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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 8                       1 1 6                     4   -4 4                   7 1   6 2                 10 4     -6 0               14 7       7 -2             13 11         -2 -4           13 13           0 -6         10 14             4 -8       6 12               -6 -10     3 10                 7 -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}^{10}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{14}\oplus{\mathbb Z}_2^{12} }[/math] [math]\displaystyle{ {\mathbb Z}^{13} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{13} }[/math] [math]\displaystyle{ {\mathbb Z}^{13} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{13} }[/math] [math]\displaystyle{ {\mathbb Z}^{14} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/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=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.

L11a139 L11a141