L10a53

L10a52

L10a54

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Link Presentations

[edit Notes on L10a53's Link Presentations]

Planar diagram presentation	X8192 X18,9,19,10 X6718 X20,14,7,13 X12,5,13,6 X10,4,11,3 X4,15,5,16 X16,12,17,11 X14,20,15,19 X2,18,3,17
Gauss code	{1, -10, 6, -7, 5, -3}, {3, -1, 2, -6, 8, -5, 4, -9, 7, -8, 10, -2, 9, -4}

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{\left(u v^2-2 u v+u+2 v-1\right) \left(u v^2-2 u v-v^2+2 v-1\right)}{u v^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{9/2}-\frac{4}{q^{9/2}}-4 q^{7/2}+\frac{8}{q^{7/2}}+8 q^{5/2}-\frac{13}{q^{5/2}}-12 q^{3/2}+\frac{15}{q^{3/2}}+\frac{1}{q^{11/2}}+15 \sqrt{q}-\frac{17}{\sqrt{q}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a^3 z^5-2 a^3 z^3+z^3 a^{-3} -a^3 z+z a^{-3} +a^3 z^{-1} +a z^7+4 a z^5-2 z^5 a^{-1} +6 a z^3-5 z^3 a^{-1} +2 a z-3 z a^{-1} -a z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^6 z^4+4 a^5 z^5-2 a^5 z^3+8 a^4 z^6+z^6 a^{-4} -7 a^4 z^4-2 z^4 a^{-4} +a^4 z^2+z^2 a^{-4} +11 a^3 z^7+4 z^7 a^{-3} -16 a^3 z^5-10 z^5 a^{-3} +9 a^3 z^3+7 z^3 a^{-3} -a^3 z-2 z a^{-3} -a^3 z^{-1} +9 a^2 z^8+6 z^8 a^{-2} -11 a^2 z^6-14 z^6 a^{-2} +2 a^2 z^4+7 z^4 a^{-2} +a^2+3 a z^9+3 z^9 a^{-1} +12 a z^7+5 z^7 a^{-1} -39 a z^5-29 z^5 a^{-1} +29 a z^3+25 z^3 a^{-1} -5 a z-6 z a^{-1} -a z^{-1} +15 z^8-34 z^6+19 z^4-2 z^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   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 10                     1 -1 8                   3   3 6                 5 1   -4 4               7 3     4 2             8 5       -3 0           9 7         2 -2         7 9           2 -4       6 8             -2 -6     3 8               5 -8   1 5                 -4 -10   3                   3 -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=-2 }[/math] [math]\displaystyle{ i=0 }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/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=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/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.

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