L11a392

L11a391

L11a393

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Link Presentations

[edit Notes on L11a392's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X16,7,17,8 X8,15,5,16 X18,11,19,12 X22,17,9,18 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, 3, -4}, {11, -2, 5, -7, 8, -9, 4, -3, 6, -5, 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{2 t(1) t(3)^3+2 t(2) t(3)^3-2 t(3)^3-7 t(1) t(3)^2+4 t(1) t(2) t(3)^2-7 t(2) t(3)^2+5 t(3)^2+7 t(1) t(3)-5 t(1) t(2) t(3)+7 t(2) t(3)-4 t(3)-2 t(1)+2 t(1) t(2)-2 t(2)}{\sqrt{t(1)} \sqrt{t(2)} t(3)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{-2} -4 q^{-3} +10 q^{-4} -13 q^{-5} +18 q^{-6} -18 q^{-7} +19 q^{-8} -14 q^{-9} +10 q^{-10} -6 q^{-11} +2 q^{-12} - q^{-13} }[/math] (db)
Signature	-4 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a^{14} z^{-2} +3 a^{12} z^{-2} +4 a^{12}-6 z^2 a^{10}-2 a^{10} z^{-2} -7 a^{10}+3 z^4 a^8+2 z^2 a^8-a^8 z^{-2} -a^8+4 z^4 a^6+7 z^2 a^6+a^6 z^{-2} +4 a^6+z^4 a^4 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^7 a^{15}-5 z^5 a^{15}+9 z^3 a^{15}-7 z a^{15}+2 a^{15} z^{-1} +2 z^8 a^{14}-7 z^6 a^{14}+7 z^4 a^{14}-2 z^2 a^{14}-a^{14} z^{-2} +a^{14}+2 z^9 a^{13}-z^7 a^{13}-17 z^5 a^{13}+35 z^3 a^{13}-27 z a^{13}+8 a^{13} z^{-1} +z^{10} a^{12}+6 z^8 a^{12}-25 z^6 a^{12}+24 z^4 a^{12}-7 z^2 a^{12}-3 a^{12} z^{-2} +5 a^{12}+7 z^9 a^{11}-7 z^7 a^{11}-27 z^5 a^{11}+50 z^3 a^{11}-34 z a^{11}+10 a^{11} z^{-1} +z^{10} a^{10}+15 z^8 a^{10}-40 z^6 a^{10}+27 z^4 a^{10}-8 z^2 a^{10}-2 a^{10} z^{-2} +4 a^{10}+5 z^9 a^9+8 z^7 a^9-35 z^5 a^9+26 z^3 a^9-10 z a^9+2 a^9 z^{-1} +11 z^8 a^8-12 z^6 a^8-2 z^4 a^8+4 z^2 a^8+a^8 z^{-2} -3 a^8+13 z^7 a^7-16 z^5 a^7+2 z^3 a^7+4 z a^7-2 a^7 z^{-1} +10 z^6 a^6-11 z^4 a^6+7 z^2 a^6+a^6 z^{-2} -4 a^6+4 z^5 a^5+z^4 a^4 }[/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 -3                       1 1 -5                     4 1 -3 -7                   6     6 -9                 7 4     -3 -11               11 6       5 -13             10 10         0 -15           9 8           1 -17         5 10             5 -19       5 9               -4 -21     1 5                 4 -23   1 5                   -4 -25   1                     1 -27 1                       -1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-5 }[/math] [math]\displaystyle{ i=-3 }[/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}^{10}\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^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/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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