L11a336

L11a335

L11a337

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Link Presentations

[edit Notes on L11a336's Link Presentations]

Planar diagram presentation	X10,1,11,2 X8,9,1,10 X14,3,15,4 X22,11,9,12 X2,15,3,16 X4,19,5,20 X20,5,21,6 X18,21,19,22 X6,13,7,14 X16,7,17,8 X12,17,13,18
Gauss code	{1, -5, 3, -6, 7, -9, 10, -2}, {2, -1, 4, -11, 9, -3, 5, -10, 11, -8, 6, -7, 8, -4}

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^3 v^2-3 u^3 v+u^3+4 u^2 v^3-10 u^2 v^2+11 u^2 v-4 u^2-4 u v^3+11 u v^2-10 u v+4 u+v^3-3 v^2+3 v}{u^{3/2} v^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{1}{q^{5/2}}+\frac{3}{q^{7/2}}-\frac{8}{q^{9/2}}+\frac{13}{q^{11/2}}-\frac{20}{q^{13/2}}+\frac{23}{q^{15/2}}-\frac{23}{q^{17/2}}+\frac{21}{q^{19/2}}-\frac{16}{q^{21/2}}+\frac{10}{q^{23/2}}-\frac{5}{q^{25/2}}+\frac{1}{q^{27/2}} }[/math] (db)
Signature	-5 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z a^{13}+4 z^3 a^{11}+5 z a^{11}-3 z^5 a^9-6 z^3 a^9-2 z a^9+a^9 z^{-1} -3 z^5 a^7-7 z^3 a^7-5 z a^7-a^7 z^{-1} -z^5 a^5-2 z^3 a^5-z a^5 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{16} z^6-a^{16} z^4+5 a^{15} z^7-10 a^{15} z^5+4 a^{15} z^3+a^{15} z+9 a^{14} z^8-20 a^{14} z^6+11 a^{14} z^4-2 a^{14} z^2+7 a^{13} z^9-5 a^{13} z^7-17 a^{13} z^5+14 a^{13} z^3-2 a^{13} z+2 a^{12} z^{10}+17 a^{12} z^8-51 a^{12} z^6+43 a^{12} z^4-15 a^{12} z^2+13 a^{11} z^9-18 a^{11} z^7-3 a^{11} z^5+7 a^{11} z^3+a^{11} z+2 a^{10} z^{10}+15 a^{10} z^8-40 a^{10} z^6+36 a^{10} z^4-11 a^{10} z^2+6 a^9 z^9-2 a^9 z^7-7 a^9 z^5+9 a^9 z^3-3 a^9 z+a^9 z^{-1} +7 a^8 z^8-7 a^8 z^6+a^8 z^4+3 a^8 z^2-a^8+6 a^7 z^7-10 a^7 z^5+10 a^7 z^3-6 a^7 z+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                 8 3     -5 -12               12 5       7 -14             11 8         -3 -16           12 12           0 -18         9 11             2 -20       7 12               -5 -22     4 10                 6 -24   1 6                   -5 -26   4                     4 -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}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-9 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{12} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=-4 }[/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}^{5}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/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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