L11a134

L11a133

L11a135

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Link Presentations

[edit Notes on L11a134's Link Presentations]

Planar diagram presentation	X6172 X16,7,17,8 X4,17,1,18 X10,5,11,6 X14,3,15,4 X18,13,19,14 X22,20,5,19 X20,12,21,11 X12,22,13,21 X2,9,3,10 X8,15,9,16
Gauss code	{1, -10, 5, -3}, {4, -1, 2, -11, 10, -4, 8, -9, 6, -5, 11, -2, 3, -6, 7, -8, 9, -7}

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{t(2)^5+4 t(1) t(2)^4-6 t(2)^4-10 t(1) t(2)^3+12 t(2)^3+12 t(1) t(2)^2-10 t(2)^2-6 t(1) t(2)+4 t(2)+t(1)}{\sqrt{t(1)} t(2)^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{5/2}-4 q^{3/2}+9 \sqrt{q}-\frac{14}{\sqrt{q}}+\frac{18}{q^{3/2}}-\frac{22}{q^{5/2}}+\frac{20}{q^{7/2}}-\frac{18}{q^{9/2}}+\frac{13}{q^{11/2}}-\frac{8}{q^{13/2}}+\frac{4}{q^{15/2}}-\frac{1}{q^{17/2}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^3 a^7-a^7 z^{-1} -z^5 a^5+z^3 a^5+5 z a^5+4 a^5 z^{-1} -3 z^5 a^3-8 z^3 a^3-10 z a^3-4 a^3 z^{-1} -z^5 a+z^3 a+3 z a+a z^{-1} +z^3 a^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^9 z^7-3 a^9 z^5+3 a^9 z^3-a^9 z+4 a^8 z^8-14 a^8 z^6+15 a^8 z^4-4 a^8 z^2-a^8+5 a^7 z^9-13 a^7 z^7+4 a^7 z^5+8 a^7 z^3-4 a^7 z+a^7 z^{-1} +2 a^6 z^{10}+8 a^6 z^8-39 a^6 z^6+35 a^6 z^4-2 a^6 z^2-4 a^6+12 a^5 z^9-23 a^5 z^7-8 a^5 z^5+27 a^5 z^3-14 a^5 z+4 a^5 z^{-1} +2 a^4 z^{10}+16 a^4 z^8-47 a^4 z^6+24 a^4 z^4+8 a^4 z^2-7 a^4+7 a^3 z^9+4 a^3 z^7-37 a^3 z^5+34 a^3 z^3-15 a^3 z+4 a^3 z^{-1} +12 a^2 z^8-13 a^2 z^6-5 a^2 z^4+z^4 a^{-2} +9 a^2 z^2-4 a^2+13 a z^7-18 a z^5+4 z^5 a^{-1} +11 a z^3-z^3 a^{-1} -4 a z+a z^{-1} +9 z^6-8 z^4+3 z^2-1 }[/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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 χ 6                       1 -1 4                     3   3 2                   6 1   -5 0                 8 3     5 -2               11 7       -4 -4             11 7         4 -6           9 11           2 -8         9 11             -2 -10       5 10               5 -12     3 8                 -5 -14   1 5                   4 -16   3                     -3 -18 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=-8 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\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^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/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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