L11a48

L11a47

L11a49

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Link Presentations

[edit Notes on L11a48's Link Presentations]

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

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^2-4 v+1\right) \left(v^2-v+1\right)}{\sqrt{u} v^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^{13/2}+4 q^{11/2}-10 q^{9/2}+15 q^{7/2}-20 q^{5/2}+24 q^{3/2}-23 \sqrt{q}+\frac{19}{\sqrt{q}}-\frac{15}{q^{3/2}}+\frac{8}{q^{5/2}}-\frac{4}{q^{7/2}}+\frac{1}{q^{9/2}} }[/math] (db)
Signature	1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^7 a^{-1} +2 a z^5-3 z^5 a^{-1} +2 z^5 a^{-3} -a^3 z^3+4 a z^3-4 z^3 a^{-1} +4 z^3 a^{-3} -z^3 a^{-5} -a^3 z+a z-2 z a^{-1} +3 z a^{-3} -z a^{-5} +a^3 z^{-1} -a z^{-1} - a^{-1} z^{-1} +2 a^{-3} z^{-1} - a^{-5} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^5 a^{-7} -z^3 a^{-7} +4 z^6 a^{-6} -4 z^4 a^{-6} +9 z^7 a^{-5} -15 z^5 a^{-5} +10 z^3 a^{-5} -5 z a^{-5} + a^{-5} z^{-1} +11 z^8 a^{-4} +a^4 z^6-19 z^6 a^{-4} -2 a^4 z^4+14 z^4 a^{-4} +a^4 z^2-4 z^2 a^{-4} + a^{-4} +7 z^9 a^{-3} +4 a^3 z^7-10 a^3 z^5-19 z^5 a^{-3} +8 a^3 z^3+23 z^3 a^{-3} -a^3 z-13 z a^{-3} -a^3 z^{-1} +2 a^{-3} z^{-1} +2 z^{10} a^{-2} +6 a^2 z^8+16 z^8 a^{-2} -11 a^2 z^6-38 z^6 a^{-2} +3 a^2 z^4+26 z^4 a^{-2} +a^2 z^2-6 z^2 a^{-2} +a^2+3 a^{-2} +5 a z^9+12 z^9 a^{-1} -a z^7-14 z^7 a^{-1} -16 a z^5-9 z^5 a^{-1} +14 a z^3+18 z^3 a^{-1} -3 a z-10 z a^{-1} -a z^{-1} + a^{-1} z^{-1} +2 z^{10}+11 z^8-27 z^6+13 z^4-2 z^2+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 6 χ 14                       1 1 12                     3   -3 10                   7 1   6 8                 8 3     -5 6               12 7       5 4             12 8         -4 2           11 12           -1 0         10 14             4 -2       5 9               -4 -4     3 10                 7 -6   1 5                   -4 -8   3                     3 -10 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=-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}^{10}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{14}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{12} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{12} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=6 }[/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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