L11a491

L11a490

L11a492

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Link Presentations

[edit Notes on L11a491's Link Presentations]

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

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)-1) (t(3)-1) \left(t(3)^4+2 t(1) t(3)^3-2 t(3)^3-2 t(1) t(3)^2+2 t(3)^2+2 t(1) t(3)-2 t(3)-t(1)\right)}{\sqrt{t(1)} \sqrt{t(2)} t(3)^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ - q^{-11} +3 q^{-10} -8 q^{-9} +13 q^{-8} -16 q^{-7} +19 q^{-6} -17 q^{-5} +16 q^{-4} -10 q^{-3} +6 q^{-2} -2 q^{-1} +1 }[/math] (db)
Signature	-4 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^{10} \left(-z^2\right)-a^{10} z^{-2} -2 a^{10}+3 a^8 z^4+8 a^8 z^2+4 a^8 z^{-2} +8 a^8-2 a^6 z^6-7 a^6 z^4-10 a^6 z^2-5 a^6 z^{-2} -10 a^6-a^4 z^6-2 a^4 z^4+2 a^4 z^{-2} +2 a^4+a^2 z^4+3 a^2 z^2+2 a^2 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^5 a^{13}-2 z^3 a^{13}+z a^{13}+3 z^6 a^{12}-4 z^4 a^{12}+z^2 a^{12}+6 z^7 a^{11}-10 z^5 a^{11}+8 z^3 a^{11}-5 z a^{11}+a^{11} z^{-1} +7 z^8 a^{10}-11 z^6 a^{10}+10 z^4 a^{10}-11 z^2 a^{10}-a^{10} z^{-2} +6 a^{10}+4 z^9 a^9+5 z^7 a^9-29 z^5 a^9+41 z^3 a^9-25 z a^9+5 a^9 z^{-1} +z^{10} a^8+13 z^8 a^8-39 z^6 a^8+55 z^4 a^8-45 z^2 a^8-4 a^8 z^{-2} +21 a^8+7 z^9 a^7-6 z^7 a^7-18 z^5 a^7+43 z^3 a^7-35 z a^7+9 a^7 z^{-1} +z^{10} a^6+9 z^8 a^6-32 z^6 a^6+49 z^4 a^6-44 z^2 a^6-5 a^6 z^{-2} +22 a^6+3 z^9 a^5-3 z^7 a^5-5 z^5 a^5+14 z^3 a^5-15 z a^5+5 a^5 z^{-1} +3 z^8 a^4-6 z^6 a^4+4 z^4 a^4-6 z^2 a^4-2 a^4 z^{-2} +6 a^4+2 z^7 a^3-5 z^5 a^3+2 z^3 a^3+z a^3+z^6 a^2-4 z^4 a^2+5 z^2 a^2-2 a^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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 1                       1 1 -1                     1   -1 -3                   5 1   4 -5                 7 3     -4 -7               9 3       6 -9             8 7         -1 -11           11 9           2 -13         7 10             3 -15       6 9               -3 -17     2 7                 5 -19   1 6                   -5 -21   2                     2 -23 1                       -1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-5 }[/math] [math]\displaystyle{ i=-3 }[/math] [math]\displaystyle{ r=-9 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\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^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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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