L11a543

L11a542

L11a544

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Link Presentations

[edit Notes on L11a543's Link Presentations]

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

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{2 u v^2 w x-u v^2 w-u v^2 x+u v^2+2 u v w^2 x-u v w^2-5 u v w x+4 u v w+2 u v x-3 u v-u w^2 x+u w^2+2 u w x-3 u w-u x+2 u+2 v^2 w^2 x+v^2 \left(-w^2\right)-3 v^2 w x+2 v^2 w+v^2 x-v^2-3 v w^2 x+2 v w^2+4 v w x-5 v w-v x+2 v+w^2 x-w^2-w x+2 w}{\sqrt{u} v w \sqrt{x}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{8}{q^{9/2}}+\frac{3}{q^{7/2}}-\frac{1}{q^{5/2}}+\frac{1}{q^{27/2}}-\frac{4}{q^{25/2}}+\frac{8}{q^{23/2}}-\frac{14}{q^{21/2}}+\frac{17}{q^{19/2}}-\frac{22}{q^{17/2}}+\frac{19}{q^{15/2}}-\frac{19}{q^{13/2}}+\frac{12}{q^{11/2}} }[/math] (db)
Signature	-5 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^{13} z^{-3} +a^{13} (-z)+3 a^{11} z^3-3 a^{11} z^{-3} +2 a^{11} z-5 a^{11} z^{-1} -2 a^9 z^5-a^9 z^3+3 a^9 z^{-3} +8 a^9 z+10 a^9 z^{-1} -3 a^7 z^5-8 a^7 z^3-a^7 z^{-3} -8 a^7 z-5 a^7 z^{-1} -a^5 z^5-2 a^5 z^3-a^5 z }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{16} z^6-2 a^{16} z^4+a^{16} z^2+4 a^{15} z^7-10 a^{15} z^5+8 a^{15} z^3-3 a^{15} z+6 a^{14} z^8-12 a^{14} z^6+4 a^{14} z^4+a^{14} z^2+4 a^{13} z^9+4 a^{13} z^7-31 a^{13} z^5+33 a^{13} z^3-a^{13} z^{-3} -15 a^{13} z+5 a^{13} z^{-1} +a^{12} z^{10}+15 a^{12} z^8-38 a^{12} z^6+23 a^{12} z^4+4 a^{12} z^2+3 a^{12} z^{-2} -10 a^{12}+8 a^{11} z^9+a^{11} z^7-38 a^{11} z^5+49 a^{11} z^3-3 a^{11} z^{-3} -29 a^{11} z+12 a^{11} z^{-1} +a^{10} z^{10}+15 a^{10} z^8-31 a^{10} z^6+12 a^{10} z^4+18 a^{10} z^2+6 a^{10} z^{-2} -19 a^{10}+4 a^9 z^9+7 a^9 z^7-29 a^9 z^5+39 a^9 z^3-3 a^9 z^{-3} -29 a^9 z+12 a^9 z^{-1} +6 a^8 z^8-3 a^8 z^6-9 a^8 z^4+15 a^8 z^2+3 a^8 z^{-2} -10 a^8+6 a^7 z^7-11 a^7 z^5+13 a^7 z^3-a^7 z^{-3} -11 a^7 z+5 a^7 z^{-1} +3 a^6 z^6-4 a^6 z^4+a^6 z^2+a^5 z^5-2 a^5 z^3+a^5 z }[/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 -4                       1 1 -6                     3 1 -2 -8                   5     5 -10                 7 3     -4 -12               12 5       7 -14             9 9         0 -16           13 10           3 -18         8 13             5 -20       6 9               -3 -22     3 9                 6 -24   1 5                   -4 -26   3                     3 -28 1                       -1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-6 }[/math] [math]\displaystyle{ i=-4 }[/math] [math]\displaystyle{ r=-11 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-10 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-9 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{13} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/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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