L11a494

L11a493

L11a495

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Link Presentations

[edit Notes on L11a494's Link Presentations]

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

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-1) (v-1) (w-1)^3 \left(w^2+1\right)}{\sqrt{u} \sqrt{v} w^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^3-4 q^2+9 q-12+19 q^{-1} -20 q^{-2} +21 q^{-3} -17 q^{-4} +13 q^{-5} -8 q^{-6} +3 q^{-7} - q^{-8} }[/math] (db)
Signature	-2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a^6 z^4-3 a^6 z^2-a^6 z^{-2} -3 a^6+2 a^4 z^6+8 a^4 z^4+12 a^4 z^2+4 a^4 z^{-2} +10 a^4-a^2 z^8-5 a^2 z^6-10 a^2 z^4-12 a^2 z^2-5 a^2 z^{-2} -11 a^2+z^6+3 z^4+3 z^2+2 z^{-2} +4 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^9 z^5-2 a^9 z^3+a^9 z+3 a^8 z^6-4 a^8 z^4+a^8 z^2+6 a^7 z^7-10 a^7 z^5+8 a^7 z^3-5 a^7 z+a^7 z^{-1} +7 a^6 z^8-10 a^6 z^6+8 a^6 z^4-9 a^6 z^2-a^6 z^{-2} +5 a^6+5 a^5 z^9+a^5 z^7-20 a^5 z^5+32 a^5 z^3-22 a^5 z+5 a^5 z^{-1} +2 a^4 z^{10}+11 a^4 z^8-38 a^4 z^6+52 a^4 z^4-38 a^4 z^2-4 a^4 z^{-2} +16 a^4+12 a^3 z^9-25 a^3 z^7+8 a^3 z^5+25 a^3 z^3-26 a^3 z+9 a^3 z^{-1} +2 a^2 z^{10}+12 a^2 z^8-50 a^2 z^6+z^6 a^{-2} +65 a^2 z^4-2 z^4 a^{-2} -40 a^2 z^2-5 a^2 z^{-2} +17 a^2+7 a z^9-16 a z^7+4 z^7 a^{-1} +8 a z^5-9 z^5 a^{-1} +5 a z^3+2 z^3 a^{-1} -10 a z+5 a z^{-1} +8 z^8-24 z^6+23 z^4-12 z^2-2 z^{-2} +7 }[/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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 7                       1 1 5                     3   -3 3                   6 1   5 1                 6 3     -3 -1               13 6       7 -3             11 10         -1 -5           10 9           1 -7         7 11             4 -9       6 10               -4 -11     2 7                 5 -13   1 6                   -5 -15   2                     2 -17 1                       -1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-3 }[/math] [math]\displaystyle{ i=-1 }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/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}^{10}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{13} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=4 }[/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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