L11a42

L11a41

L11a43

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Link Presentations

[edit Notes on L11a42's Link Presentations]

Planar diagram presentation	X6172 X10,4,11,3 X18,8,19,7 X20,13,21,14 X22,15,5,16 X14,21,15,22 X16,19,17,20 X12,10,13,9 X8,18,9,17 X2536 X4,12,1,11
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -9, 8, -2, 11, -8, 4, -6, 5, -7, 9, -3, 7, -4, 6, -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{(u-1) (v-1) \left(v^4-4 v^3+4 v^2-4 v+1\right)}{\sqrt{u} v^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{9/2}-\frac{7}{q^{9/2}}-3 q^{7/2}+\frac{12}{q^{7/2}}+6 q^{5/2}-\frac{16}{q^{5/2}}-12 q^{3/2}+\frac{18}{q^{3/2}}-\frac{1}{q^{13/2}}+\frac{3}{q^{11/2}}+15 \sqrt{q}-\frac{18}{\sqrt{q}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a z^7-2 a^3 z^5+4 a z^5-2 z^5 a^{-1} +a^5 z^3-6 a^3 z^3+8 a z^3-6 z^3 a^{-1} +z^3 a^{-3} +2 a^5 z-7 a^3 z+9 a z-6 z a^{-1} +2 z a^{-3} +a^5 z^{-1} -3 a^3 z^{-1} +4 a z^{-1} -2 a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -a^2 z^{10}-z^{10}-3 a^3 z^9-6 a z^9-3 z^9 a^{-1} -5 a^4 z^8-9 a^2 z^8-4 z^8 a^{-2} -8 z^8-5 a^5 z^7-7 a^3 z^7-z^7 a^{-1} -3 z^7 a^{-3} -3 a^6 z^6+2 a^4 z^6+14 a^2 z^6+8 z^6 a^{-2} -z^6 a^{-4} +18 z^6-a^7 z^5+7 a^5 z^5+23 a^3 z^5+24 a z^5+18 z^5 a^{-1} +9 z^5 a^{-3} +5 a^6 z^4+7 a^4 z^4-2 z^4 a^{-2} +3 z^4 a^{-4} -7 z^4+2 a^7 z^3-4 a^5 z^3-24 a^3 z^3-33 a z^3-24 z^3 a^{-1} -9 z^3 a^{-3} -3 a^6 z^2-8 a^4 z^2-8 a^2 z^2-2 z^2 a^{-2} -2 z^2 a^{-4} -3 z^2-a^7 z+2 a^5 z+13 a^3 z+19 a z+13 z a^{-1} +4 z a^{-3} +a^6+3 a^4+3 a^2+2-a^5 z^{-1} -3 a^3 z^{-1} -4 a z^{-1} -2 a^{-1} z^{-1} }[/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 10                       1 -1 8                     2   2 6                   4 1   -3 4                 8 2     6 2               7 4       -3 0             11 8         3 -2           9 9           0 -4         7 9             -2 -6       5 9               4 -8     2 7                 -5 -10   1 5                   4 -12   2                     -2 -14 1                       1

Integral Khovanov Homology (db, data source)

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

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