L11a106

L11a105

L11a107

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Link Presentations

[edit Notes on L11a106's Link Presentations]

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

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{u v^5-4 u v^4+8 u v^3-8 u v^2+6 u v-2 u-2 v^5+6 v^4-8 v^3+8 v^2-4 v+1}{\sqrt{u} v^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{3/2}-4 \sqrt{q}+\frac{7}{\sqrt{q}}-\frac{12}{q^{3/2}}+\frac{16}{q^{5/2}}-\frac{19}{q^{7/2}}+\frac{18}{q^{9/2}}-\frac{16}{q^{11/2}}+\frac{12}{q^{13/2}}-\frac{7}{q^{15/2}}+\frac{3}{q^{17/2}}-\frac{1}{q^{19/2}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^9 z+a^9 z^{-1} -3 a^7 z^3-6 a^7 z-3 a^7 z^{-1} +3 a^5 z^5+9 a^5 z^3+9 a^5 z+4 a^5 z^{-1} -a^3 z^7-4 a^3 z^5-7 a^3 z^3-6 a^3 z-2 a^3 z^{-1} +a z^5+2 a z^3 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^5 a^{11}+2 z^3 a^{11}-z a^{11}-3 z^6 a^{10}+5 z^4 a^{10}-3 z^2 a^{10}+a^{10}-5 z^7 a^9+7 z^5 a^9-4 z^3 a^9+2 z a^9-a^9 z^{-1} -5 z^8 a^8+2 z^6 a^8+7 z^4 a^8-8 z^2 a^8+3 a^8-3 z^9 a^7-7 z^7 a^7+22 z^5 a^7-22 z^3 a^7+12 z a^7-3 a^7 z^{-1} -z^{10} a^6-10 z^8 a^6+18 z^6 a^6-6 z^4 a^6-5 z^2 a^6+3 a^6-7 z^9 a^5+4 z^7 a^5+17 z^5 a^5-26 z^3 a^5+17 z a^5-4 a^5 z^{-1} -z^{10} a^4-11 z^8 a^4+30 z^6 a^4-22 z^4 a^4+2 z^2 a^4+2 a^4-4 z^9 a^3+2 z^7 a^3+14 z^5 a^3-17 z^3 a^3+8 z a^3-2 a^3 z^{-1} -6 z^8 a^2+16 z^6 a^2-12 z^4 a^2+2 z^2 a^2-4 z^7 a+11 z^5 a-7 z^3 a-z^6+2 z^4 }[/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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 χ 4                       1 -1 2                     3   3 0                   4 1   -3 -2                 8 3     5 -4               9 5       -4 -6             10 7         3 -8           8 9           1 -10         8 10             -2 -12       5 9               4 -14     2 7                 -5 -16   1 5                   4 -18   2                     -2 -20 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=-8 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=3 }[/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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