L11a66

L11a65

L11a67

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Link Presentations

[edit Notes on L11a66's Link Presentations]

Planar diagram presentation	X6172 X12,4,13,3 X14,8,15,7 X22,12,5,11 X20,17,21,18 X10,16,11,15 X8,20,9,19 X18,10,19,9 X16,21,17,22 X2536 X4,14,1,13
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -7, 8, -6, 4, -2, 11, -3, 6, -9, 5, -8, 7, -5, 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{(u-1) (v-1) \left(v^4-4 v^3+3 v^2-4 v+1\right)}{\sqrt{u} v^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ 14 q^{9/2}-17 q^{7/2}+16 q^{5/2}-\frac{1}{q^{5/2}}-14 q^{3/2}+\frac{3}{q^{3/2}}+q^{17/2}-3 q^{15/2}+7 q^{13/2}-11 q^{11/2}+10 \sqrt{q}-\frac{7}{\sqrt{q}} }[/math] (db)
Signature	3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^7 a^{-3} -2 z^5 a^{-1} +4 z^5 a^{-3} -2 z^5 a^{-5} +a z^3-6 z^3 a^{-1} +7 z^3 a^{-3} -6 z^3 a^{-5} +z^3 a^{-7} +2 a z-5 z a^{-1} +7 z a^{-3} -6 z a^{-5} +2 z a^{-7} +a z^{-1} -2 a^{-1} z^{-1} +3 a^{-3} z^{-1} -3 a^{-5} z^{-1} + a^{-7} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^{10} a^{-2} -z^{10} a^{-4} -3 z^9 a^{-1} -7 z^9 a^{-3} -4 z^9 a^{-5} -8 z^8 a^{-2} -12 z^8 a^{-4} -7 z^8 a^{-6} -3 z^8-a z^7+5 z^7 a^{-1} +10 z^7 a^{-3} -4 z^7 a^{-5} -8 z^7 a^{-7} +35 z^6 a^{-2} +37 z^6 a^{-4} +7 z^6 a^{-6} -6 z^6 a^{-8} +11 z^6+4 a z^5+11 z^5 a^{-1} +18 z^5 a^{-3} +26 z^5 a^{-5} +12 z^5 a^{-7} -3 z^5 a^{-9} -36 z^4 a^{-2} -30 z^4 a^{-4} +2 z^4 a^{-6} +7 z^4 a^{-8} -z^4 a^{-10} -12 z^4-6 a z^3-23 z^3 a^{-1} -31 z^3 a^{-3} -27 z^3 a^{-5} -11 z^3 a^{-7} +2 z^3 a^{-9} +12 z^2 a^{-2} +9 z^2 a^{-4} -5 z^2 a^{-6} -5 z^2 a^{-8} +z^2 a^{-10} +4 z^2+4 a z+11 z a^{-1} +16 z a^{-3} +14 z a^{-5} +5 z a^{-7} -2 a^{-2} +2 a^{-6} + a^{-8} -a z^{-1} -2 a^{-1} z^{-1} -3 a^{-3} z^{-1} -3 a^{-5} z^{-1} - a^{-7} 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   \ -4 -3 -2 -1 0 1 2 3 4 5 6 7 χ 18                       1 -1 16                     2   2 14                   5 1   -4 12                 6 2     4 10               8 5       -3 8             9 6         3 6           7 8           1 4         7 9             -2 2       5 9               4 0     2 5                 -3 -2   1 5                   4 -4   2                     -2 -6 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=4 }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=7 }[/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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