L11a258

L11a257

L11a259

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Link Presentations

[edit Notes on L11a258's Link Presentations]

Planar diagram presentation	X10,1,11,2 X14,5,15,6 X12,3,13,4 X4,13,5,14 X22,20,9,19 X18,7,19,8 X6,17,7,18 X16,22,17,21 X20,16,21,15 X2,9,3,10 X8,11,1,12
Gauss code	{1, -10, 3, -4, 2, -7, 6, -11}, {10, -1, 11, -3, 4, -2, 9, -8, 7, -6, 5, -9, 8, -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{u^3 v^4-2 u^3 v^3+2 u^3 v^2+u^2 v^5-3 u^2 v^4+6 u^2 v^3-5 u^2 v^2+2 u^2 v+2 u v^4-5 u v^3+6 u v^2-3 u v+u+2 v^3-2 v^2+v}{u^{3/2} v^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{3/2}-3 \sqrt{q}+\frac{6}{\sqrt{q}}-\frac{10}{q^{3/2}}+\frac{12}{q^{5/2}}-\frac{14}{q^{7/2}}+\frac{14}{q^{9/2}}-\frac{12}{q^{11/2}}+\frac{8}{q^{13/2}}-\frac{5}{q^{15/2}}+\frac{2}{q^{17/2}}-\frac{1}{q^{19/2}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^5 a^7+4 z^3 a^7+5 z a^7+2 a^7 z^{-1} -z^7 a^5-5 z^5 a^5-10 z^3 a^5-9 z a^5-3 a^5 z^{-1} -z^7 a^3-4 z^5 a^3-5 z^3 a^3-2 z a^3+a^3 z^{-1} +z^5 a+3 z^3 a+2 z a }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^5 a^{11}+3 z^3 a^{11}-2 z a^{11}-2 z^6 a^{10}+4 z^4 a^{10}-z^2 a^{10}-3 z^7 a^9+6 z^5 a^9-5 z^3 a^9+3 z a^9-3 z^8 a^8+4 z^6 a^8-2 z^4 a^8-3 z^9 a^7+7 z^7 a^7-14 z^5 a^7+13 z^3 a^7-8 z a^7+2 a^7 z^{-1} -z^{10} a^6-4 z^8 a^6+15 z^6 a^6-23 z^4 a^6+12 z^2 a^6-3 a^6-6 z^9 a^5+17 z^7 a^5-26 z^5 a^5+24 z^3 a^5-14 z a^5+3 a^5 z^{-1} -z^{10} a^4-5 z^8 a^4+20 z^6 a^4-25 z^4 a^4+15 z^2 a^4-3 a^4-3 z^9 a^3+4 z^7 a^3+4 z^5 a^3-4 z^3 a^3+z a^3+a^3 z^{-1} -4 z^8 a^2+10 z^6 a^2-5 z^4 a^2+2 z^2 a^2-a^2-3 z^7 a+9 z^5 a-7 z^3 a+2 z a-z^6+3 z^4-2 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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 χ 4                       1 -1 2                     2   2 0                   4 1   -3 -2                 6 2     4 -4               7 5       -2 -6             7 5         2 -8           7 7           0 -10         5 7             -2 -12       3 7               4 -14     2 5                 -3 -16     3                   3 -18 1 2                     -1 -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{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\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^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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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