L11a34

L11a33

L11a35

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Link Presentations

[edit Notes on L11a34's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X18,11,19,12 X16,7,17,8 X8,17,9,18 X22,15,5,16 X12,21,13,22 X20,13,21,14 X14,19,15,20 X2536 X4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, 4, -5, 11, -2, 3, -7, 8, -9, 6, -4, 5, -3, 9, -8, 7, -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{4 u v^4-8 u v^3+8 u v^2-5 u v+2 u+2 v^5-5 v^4+8 v^3-8 v^2+4 v}{\sqrt{u} v^{5/2}} }[/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{2}{q^{25/2}}+\frac{6}{q^{23/2}}-\frac{10}{q^{21/2}}+\frac{14}{q^{19/2}}-\frac{18}{q^{17/2}}+\frac{17}{q^{15/2}}-\frac{16}{q^{13/2}}+\frac{12}{q^{11/2}} }[/math] (db)
Signature	-5 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z a^{13}-2 a^{13} z^{-1} +3 z^3 a^{11}+7 z a^{11}+4 a^{11} z^{-1} -2 z^5 a^9-4 z^3 a^9-2 z a^9-a^9 z^{-1} -3 z^5 a^7-8 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^6-4 a^{16} z^4+5 a^{16} z^2-2 a^{16}+2 a^{15} z^7-5 a^{15} z^5+2 a^{15} z^3+a^{15} z+3 a^{14} z^8-6 a^{14} z^6+2 a^{14} z^4+a^{14} z^2-a^{14}+3 a^{13} z^9-5 a^{13} z^7+6 a^{13} z^5-11 a^{13} z^3+9 a^{13} z-2 a^{13} z^{-1} +a^{12} z^{10}+7 a^{12} z^8-23 a^{12} z^6+32 a^{12} z^4-24 a^{12} z^2+6 a^{12}+7 a^{11} z^9-13 a^{11} z^7+17 a^{11} z^5-22 a^{11} z^3+16 a^{11} z-4 a^{11} z^{-1} +a^{10} z^{10}+10 a^{10} z^8-25 a^{10} z^6+28 a^{10} z^4-18 a^{10} z^2+5 a^{10}+4 a^9 z^9-6 a^9 z^5+3 a^9 z^3+a^9 z-a^9 z^{-1} +6 a^8 z^8-6 a^8 z^6-2 a^8 z^4+3 a^8 z^2-a^8+6 a^7 z^7-11 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                 7 3     -4 -12               9 5       4 -14             9 8         -1 -16           9 8           1 -18         5 9             4 -20       5 9               -4 -22     1 5                 4 -24   1 5                   -4 -26   1                     1 -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}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-9 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/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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