L11a388

L11a387

L11a389

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Link Presentations

[edit Notes on L11a388's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X14,7,15,8 X8,13,5,14 X18,11,19,12 X22,15,9,16 X20,17,21,18 X16,21,17,22 X12,19,13,20 X2536 X4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -4}, {11, -2, 5, -9, 4, -3, 6, -8, 7, -5, 9, -7, 8, -6}

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{3 u v w^2-4 u v w+2 u v+2 u w^3-6 u w^2+6 u w-2 u+2 v w^3-6 v w^2+6 v w-2 v-2 w^3+4 w^2-3 w}{\sqrt{u} \sqrt{v} w^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{-2} -3 q^{-3} +7 q^{-4} -10 q^{-5} +16 q^{-6} -15 q^{-7} +16 q^{-8} -13 q^{-9} +10 q^{-10} -6 q^{-11} +2 q^{-12} - q^{-13} }[/math] (db)
Signature	-4 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a^{14} z^{-2} +3 a^{12} z^{-2} +4 a^{12}-6 a^{10} z^2-2 a^{10} z^{-2} -7 a^{10}+3 a^8 z^4+3 a^8 z^2-a^8 z^{-2} -a^8+3 a^6 z^4+5 a^6 z^2+a^6 z^{-2} +4 a^6+a^4 z^4+a^4 z^2 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{15} z^7-5 a^{15} z^5+9 a^{15} z^3-7 a^{15} z+2 a^{15} z^{-1} +2 a^{14} z^8-7 a^{14} z^6+7 a^{14} z^4-2 a^{14} z^2-a^{14} z^{-2} +a^{14}+2 a^{13} z^9-a^{13} z^7-18 a^{13} z^5+36 a^{13} z^3-27 a^{13} z+8 a^{13} z^{-1} +a^{12} z^{10}+5 a^{12} z^8-22 a^{12} z^6+19 a^{12} z^4-7 a^{12} z^2-3 a^{12} z^{-2} +5 a^{12}+6 a^{11} z^9-6 a^{11} z^7-27 a^{11} z^5+49 a^{11} z^3-34 a^{11} z+10 a^{11} z^{-1} +a^{10} z^{10}+10 a^{10} z^8-28 a^{10} z^6+17 a^{10} z^4-5 a^{10} z^2-2 a^{10} z^{-2} +4 a^{10}+4 a^9 z^9+3 a^9 z^7-23 a^9 z^5+23 a^9 z^3-10 a^9 z+2 a^9 z^{-1} +7 a^8 z^8-7 a^8 z^6-4 a^8 z^4+9 a^8 z^2+a^8 z^{-2} -3 a^8+7 a^7 z^7-6 a^7 z^5-a^7 z^3+4 a^7 z-2 a^7 z^{-1} +6 a^6 z^6-8 a^6 z^4+8 a^6 z^2+a^6 z^{-2} -4 a^6+3 a^5 z^5-2 a^5 z^3+a^4 z^4-a^4 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   \ -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -3                       1 1 -5                     3 1 -2 -7                   4     4 -9                 6 3     -3 -11               10 4       6 -13             8 9         1 -15           8 7           1 -17         5 8             3 -19       5 8               -3 -21     1 5                 4 -23   1 5                   -4 -25   1                     1 -27 1                       -1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-5 }[/math] [math]\displaystyle{ i=-3 }[/math] [math]\displaystyle{ r=-11 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-10 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-9 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=-5 }[/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}^{9}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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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