L11a226

L11a225

L11a227

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Link Presentations

[edit Notes on L11a226's Link Presentations]

Planar diagram presentation	X8192 X20,9,21,10 X4758 X16,5,17,6 X6,15,1,16 X22,17,7,18 X18,13,19,14 X14,21,15,22 X2,11,3,12 X12,3,13,4 X10,19,11,20
Gauss code	{1, -9, 10, -3, 4, -5}, {3, -1, 2, -11, 9, -10, 7, -8, 5, -4, 6, -7, 11, -2, 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{5 u^2 v^3-8 u^2 v^2+5 u^2 v-u^2+3 u v^4-11 u v^3+17 u v^2-11 u v+3 u-v^4+5 v^3-8 v^2+5 v}{u v^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{11}{q^{9/2}}+\frac{4}{q^{7/2}}-\frac{1}{q^{5/2}}+\frac{1}{q^{27/2}}-\frac{4}{q^{25/2}}+\frac{10}{q^{23/2}}-\frac{17}{q^{21/2}}+\frac{23}{q^{19/2}}-\frac{27}{q^{17/2}}+\frac{27}{q^{15/2}}-\frac{24}{q^{13/2}}+\frac{17}{q^{11/2}} }[/math] (db)
Signature	-5 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z a^{13}-a^{13} z^{-1} +4 z^3 a^{11}+6 z a^{11}+2 a^{11} z^{-1} -3 z^5 a^9-5 z^3 a^9-z a^9-4 z^5 a^7-10 z^3 a^7-7 z a^7-a^7 z^{-1} -z^5 a^5-z^3 a^5 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{16} z^6-2 a^{16} z^4+a^{16} z^2+4 a^{15} z^7-8 a^{15} z^5+5 a^{15} z^3-a^{15} z+8 a^{14} z^8-17 a^{14} z^6+13 a^{14} z^4-6 a^{14} z^2+2 a^{14}+8 a^{13} z^9-10 a^{13} z^7-3 a^{13} z^5+4 a^{13} z^3+a^{13} z-a^{13} z^{-1} +3 a^{12} z^{10}+16 a^{12} z^8-51 a^{12} z^6+49 a^{12} z^4-25 a^{12} z^2+5 a^{12}+18 a^{11} z^9-30 a^{11} z^7+13 a^{11} z^5-10 a^{11} z^3+9 a^{11} z-2 a^{11} z^{-1} +3 a^{10} z^{10}+21 a^{10} z^8-56 a^{10} z^6+46 a^{10} z^4-17 a^{10} z^2+3 a^{10}+10 a^9 z^9-6 a^9 z^7-9 a^9 z^5+6 a^9 z^3+13 a^8 z^8-19 a^8 z^6+9 a^8 z^4+a^8 z^2-a^8+10 a^7 z^7-16 a^7 z^5+14 a^7 z^3-7 a^7 z+a^7 z^{-1} +4 a^6 z^6-3 a^6 z^4+a^5 z^5-a^5 z^3 }[/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                     4 1 -3 -8                   7     7 -10                 10 4     -6 -12               14 7       7 -14             13 10         -3 -16           14 14           0 -18         10 14             4 -20       7 13               -6 -22     3 10                 7 -24   1 7                   -6 -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}^{7}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{14}\oplus{\mathbb Z}_2^{13} }[/math] [math]\displaystyle{ {\mathbb Z}^{14} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{14}\oplus{\mathbb Z}_2^{13} }[/math] [math]\displaystyle{ {\mathbb Z}^{13} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{14} }[/math] [math]\displaystyle{ {\mathbb Z}^{14} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/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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