L11a216

L11a215

L11a217

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Link Presentations

[edit Notes on L11a216's Link Presentations]

Planar diagram presentation	X8192 X12,3,13,4 X14,17,15,18 X16,5,17,6 X4,15,5,16 X18,13,19,14 X22,19,7,20 X20,9,21,10 X10,21,11,22 X2738 X6,11,1,12
Gauss code	{1, -10, 2, -5, 4, -11}, {10, -1, 8, -9, 11, -2, 6, -3, 5, -4, 3, -6, 7, -8, 9, -7}

A Braid Representative	{{{braid_table}}}
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{4 u^2 v^3-6 u^2 v^2+4 u^2 v-u^2+2 u v^4-8 u v^3+13 u v^2-8 u v+2 u-v^4+4 v^3-6 v^2+4 v}{u v^2} }[/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{13}{q^{21/2}}+\frac{18}{q^{19/2}}-\frac{20}{q^{17/2}}+\frac{20}{q^{15/2}}-\frac{18}{q^{13/2}}+\frac{12}{q^{11/2}} }[/math] (db)
Signature	-5 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^{13} (-z)+3 a^{11} z^3+3 a^{11} z-a^{11} z^{-1} -2 a^9 z^5-2 a^9 z^3+3 a^9 z+3 a^9 z^{-1} -3 a^7 z^5-8 a^7 z^3-7 a^7 z-2 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+7 a^{15} z^3-a^{15} z+6 a^{14} z^8-13 a^{14} z^6+7 a^{14} z^4-a^{14} z^2+4 a^{13} z^9+2 a^{13} z^7-22 a^{13} z^5+18 a^{13} z^3-3 a^{13} z+a^{12} z^{10}+14 a^{12} z^8-36 a^{12} z^6+27 a^{12} z^4-8 a^{12} z^2-a^{12}+8 a^{11} z^9-3 a^{11} z^7-19 a^{11} z^5+16 a^{11} z^3-3 a^{11} z+a^{11} z^{-1} +a^{10} z^{10}+14 a^{10} z^8-29 a^{10} z^6+17 a^{10} z^4+a^{10} z^2-3 a^{10}+4 a^9 z^9+5 a^9 z^7-19 a^9 z^5+19 a^9 z^3-10 a^9 z+3 a^9 z^{-1} +6 a^8 z^8-4 a^8 z^6-5 a^8 z^4+8 a^8 z^2-3 a^8+6 a^7 z^7-11 a^7 z^5+12 a^7 z^3-8 a^7 z+2 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               11 5       6 -14             9 7         -2 -16           11 11           0 -18         8 10             2 -20       5 10               -5 -22     3 8                 5 -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}^{8}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/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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