L11a332

L11a331

L11a333

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Link Presentations

[edit Notes on L11a332's Link Presentations]

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

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 t(2)^3 t(1)^3-3 t(2)^2 t(1)^3+t(2) t(1)^3-3 t(2)^3 t(1)^2+8 t(2)^2 t(1)^2-5 t(2) t(1)^2+t(1)^2+t(2)^3 t(1)-5 t(2)^2 t(1)+8 t(2) t(1)-3 t(1)+t(2)^2-3 t(2)+3}{t(1)^{3/2} t(2)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ \frac{2}{q^{9/2}}-\frac{1}{q^{7/2}}+\frac{1}{q^{29/2}}-\frac{4}{q^{27/2}}+\frac{7}{q^{25/2}}-\frac{11}{q^{23/2}}+\frac{14}{q^{21/2}}-\frac{15}{q^{19/2}}+\frac{15}{q^{17/2}}-\frac{12}{q^{15/2}}+\frac{8}{q^{13/2}}-\frac{6}{q^{11/2}} }[/math] (db)
Signature	-7 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^{13} \left(-z^3\right)-2 a^{13} z+3 a^{11} z^5+11 a^{11} z^3+9 a^{11} z-2 a^9 z^7-10 a^9 z^5-15 a^9 z^3-6 a^9 z+a^9 z^{-1} -a^7 z^7-5 a^7 z^5-8 a^7 z^3-5 a^7 z-a^7 z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{18} z^4+4 a^{17} z^5-3 a^{17} z^3+7 a^{16} z^6-7 a^{16} z^4+a^{16} z^2+8 a^{15} z^7-8 a^{15} z^5-a^{15} z^3+2 a^{15} z+7 a^{14} z^8-8 a^{14} z^6+a^{14} z^4+4 a^{13} z^9-12 a^{13} z^5+7 a^{13} z^3+a^{13} z+a^{12} z^{10}+10 a^{12} z^8-37 a^{12} z^6+42 a^{12} z^4-18 a^{12} z^2+7 a^{11} z^9-20 a^{11} z^7+18 a^{11} z^5-12 a^{11} z^3+7 a^{11} z+a^{10} z^{10}+5 a^{10} z^8-29 a^{10} z^6+38 a^{10} z^4-16 a^{10} z^2+3 a^9 z^9-11 a^9 z^7+13 a^9 z^5-9 a^9 z^3+3 a^9 z+a^9 z^{-1} +2 a^8 z^8-7 a^8 z^6+5 a^8 z^4+a^8 z^2-a^8+a^7 z^7-5 a^7 z^5+8 a^7 z^3-5 a^7 z+a^7 z^{-1} }[/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 -6                       1 1 -8                     2 1 -1 -10                   4     4 -12                 4 2     -2 -14               8 4       4 -16             7 4         -3 -18           8 8           0 -20         6 7             1 -22       5 8               -3 -24     3 7                 4 -26   1 4                   -3 -28   3                     3 -30 1                       -1

Integral Khovanov Homology (db, data source)

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