L11a70

L11a69

L11a71

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Link Presentations

[edit Notes on L11a70's Link Presentations]

Planar diagram presentation	X6172 X12,3,13,4 X16,8,17,7 X22,18,5,17 X18,9,19,10 X8,21,9,22 X10,14,11,13 X20,15,21,16 X14,19,15,20 X2536 X4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -6, 5, -7, 11, -2, 7, -9, 8, -3, 4, -5, 9, -8, 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{t(1) t(2)^5-2 t(2)^5-5 t(1) t(2)^4+7 t(2)^4+10 t(1) t(2)^3-10 t(2)^3-10 t(1) t(2)^2+10 t(2)^2+7 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{22}{q^{9/2}}-\frac{23}{q^{7/2}}+\frac{20}{q^{5/2}}+q^{3/2}-\frac{16}{q^{3/2}}-\frac{1}{q^{19/2}}+\frac{3}{q^{17/2}}-\frac{7}{q^{15/2}}+\frac{13}{q^{13/2}}-\frac{19}{q^{11/2}}-5 \sqrt{q}+\frac{10}{\sqrt{q}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z a^9+a^9 z^{-1} -3 z^3 a^7-6 z a^7-3 a^7 z^{-1} +3 z^5 a^5+8 z^3 a^5+8 z a^5+4 a^5 z^{-1} -z^7 a^3-3 z^5 a^3-4 z^3 a^3-4 z a^3-2 a^3 z^{-1} +z^5 a+z^3 a-z a }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{11} z^5-2 a^{11} z^3+a^{11} z+3 a^{10} z^6-5 a^{10} z^4+3 a^{10} z^2-a^{10}+5 a^9 z^7-6 a^9 z^5+3 a^9 z^3-2 a^9 z+a^9 z^{-1} +6 a^8 z^8-4 a^8 z^6-4 a^8 z^4+8 a^8 z^2-3 a^8+5 a^7 z^9+2 a^7 z^7-17 a^7 z^5+24 a^7 z^3-13 a^7 z+3 a^7 z^{-1} +2 a^6 z^{10}+12 a^6 z^8-28 a^6 z^6+16 a^6 z^4+4 a^6 z^2-3 a^6+12 a^5 z^9-14 a^5 z^7-15 a^5 z^5+30 a^5 z^3-18 a^5 z+4 a^5 z^{-1} +2 a^4 z^{10}+15 a^4 z^8-42 a^4 z^6+26 a^4 z^4-2 a^4 z^2-2 a^4+7 a^3 z^9-6 a^3 z^7-15 a^3 z^5+15 a^3 z^3-7 a^3 z+2 a^3 z^{-1} +9 a^2 z^8-20 a^2 z^6+10 a^2 z^4-a^2 z^2+5 a z^7-10 a z^5+4 a z^3+a z+z^6-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                     4   4 0                   6 1   -5 -2                 10 4     6 -4               11 7       -4 -6             12 9         3 -8           10 11           1 -10         9 12             -3 -12       5 11               6 -14     2 8                 -6 -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}^{8}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/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}^{11}\oplus{\mathbb Z}_2^{12} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/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=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\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^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/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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