L11a124

L11a123

L11a125

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Link Presentations

[edit Notes on L11a124's Link Presentations]

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