L11a539

L11a538

L11a540

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Link Presentations

[edit Notes on L11a539's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X20,14,21,13 X16,12,17,11 X12,20,13,19 X8,16,5,15 X14,8,15,7 X22,17,19,18 X18,21,9,22 X2536 X4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, 7, -6}, {5, -3, 9, -8}, {11, -2, 4, -5, 3, -7, 6, -4, 8, -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{(w-1) (x-1) \left(u v x^2-u v x+u v-u x^2+2 u x-2 u-2 v x^2+2 v x-v+x^2-x+1\right)}{\sqrt{u} \sqrt{v} \sqrt{w} x^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -4 q^{9/2}+\frac{2}{q^{9/2}}+10 q^{7/2}-\frac{7}{q^{7/2}}-16 q^{5/2}+\frac{10}{q^{5/2}}+18 q^{3/2}-\frac{18}{q^{3/2}}+q^{11/2}-\frac{1}{q^{11/2}}-22 \sqrt{q}+\frac{19}{\sqrt{q}} }[/math] (db)
Signature	1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^7 a^{-1} +3 a z^5-4 z^5 a^{-1} +z^5 a^{-3} -3 a^3 z^3+10 a z^3-9 z^3 a^{-1} +2 z^3 a^{-3} +a^5 z-8 a^3 z+15 a z-11 z a^{-1} +3 z a^{-3} +2 a^5 z^{-1} -8 a^3 z^{-1} +11 a z^{-1} -6 a^{-1} z^{-1} + a^{-3} z^{-1} +a^5 z^{-3} -3 a^3 z^{-3} +3 a z^{-3} - a^{-1} z^{-3} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^4 a^{-6} +a^5 z^7-5 a^5 z^5+4 z^5 a^{-5} +10 a^5 z^3-a^5 z^{-3} -10 a^5 z+5 a^5 z^{-1} +2 a^4 z^8-6 a^4 z^6+10 z^6 a^{-4} +3 a^4 z^4-8 z^4 a^{-4} +7 a^4 z^2+4 z^2 a^{-4} +3 a^4 z^{-2} -9 a^4- a^{-4} +2 a^3 z^9+a^3 z^7+16 z^7 a^{-3} -23 a^3 z^5-25 z^5 a^{-3} +43 a^3 z^3+17 z^3 a^{-3} -3 a^3 z^{-3} -34 a^3 z-9 z a^{-3} +14 a^3 z^{-1} +2 a^{-3} z^{-1} +a^2 z^{10}+8 a^2 z^8+14 z^8 a^{-2} -27 a^2 z^6-17 z^6 a^{-2} +14 a^2 z^4-6 z^4 a^{-2} +19 a^2 z^2+15 z^2 a^{-2} +6 a^2 z^{-2} -21 a^2-6 a^{-2} +8 a z^9+6 z^9 a^{-1} -3 a z^7+13 z^7 a^{-1} -48 a z^5-59 z^5 a^{-1} +77 a z^3+61 z^3 a^{-1} -3 a z^{-3} - a^{-1} z^{-3} -50 a z-35 z a^{-1} +18 a z^{-1} +11 a^{-1} z^{-1} +z^{10}+20 z^8-48 z^6+14 z^4+23 z^2+3 z^{-2} -18 }[/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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 12                       1 -1 10                     3   3 8                   7 1   -6 6                 9 3     6 4               9 7       -2 2             13 9         4 0           12 15           3 -2         6 7             -1 -4       4 12               8 -6     3 6                 -3 -8   1 6                   5 -10   1                     -1 -12 1                       1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=0 }[/math] [math]\displaystyle{ i=2 }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{12} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{15}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{13} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/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=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=5 }[/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.

L11a538 L11a540