L11a391

L11a390

L11a392

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Link Presentations

[edit Notes on L11a391'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 X20,13,21,14 X12,19,13,20 X14,21,15,22 X2536 X4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -4}, {11, -2, 5, -8, 7, -9, 4, -3, 6, -5, 8, -7, 9, -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{t(1) t(3)^5+t(2) t(3)^5-t(3)^5-3 t(1) t(3)^4+2 t(1) t(2) t(3)^4-3 t(2) t(3)^4+2 t(3)^4+3 t(1) t(3)^3-2 t(1) t(2) t(3)^3+3 t(2) t(3)^3-2 t(3)^3-3 t(1) t(3)^2+2 t(1) t(2) t(3)^2-3 t(2) t(3)^2+2 t(3)^2+3 t(1) t(3)-2 t(1) t(2) t(3)+3 t(2) t(3)-2 t(3)-t(1)+t(1) t(2)-t(2)}{\sqrt{t(1)} \sqrt{t(2)} t(3)^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ - q^{-14} +2 q^{-13} -6 q^{-12} +9 q^{-11} -12 q^{-10} +15 q^{-9} -14 q^{-8} +14 q^{-7} -8 q^{-6} +7 q^{-5} -3 q^{-4} + q^{-3} }[/math] (db)
Signature	-6 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -2 a^{14} z^{-2} -a^{14}+4 a^{12} z^2+7 a^{12} z^{-2} +12 a^{12}-6 a^{10} z^4-23 a^{10} z^2-8 a^{10} z^{-2} -24 a^{10}+3 a^8 z^6+14 a^8 z^4+21 a^8 z^2+3 a^8 z^{-2} +13 a^8+a^6 z^6+3 a^6 z^4+a^6 z^2 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{17} z^5-3 a^{17} z^3+3 a^{17} z-a^{17} z^{-1} +2 a^{16} z^6-3 a^{16} z^4+a^{16}+3 a^{15} z^7-3 a^{15} z^5-2 a^{15} z^3+3 a^{15} z-a^{15} z^{-1} +3 a^{14} z^8-2 a^{14} z^6+2 a^{14} z^4-9 a^{14} z^2-2 a^{14} z^{-2} +7 a^{14}+2 a^{13} z^9+3 a^{13} z^7-13 a^{13} z^5+22 a^{13} z^3-21 a^{13} z+7 a^{13} z^{-1} +a^{12} z^{10}+5 a^{12} z^8-18 a^{12} z^6+33 a^{12} z^4-35 a^{12} z^2-7 a^{12} z^{-2} +22 a^{12}+6 a^{11} z^9-11 a^{11} z^7-8 a^{11} z^5+46 a^{11} z^3-45 a^{11} z+15 a^{11} z^{-1} +a^{10} z^{10}+8 a^{10} z^8-38 a^{10} z^6+62 a^{10} z^4-51 a^{10} z^2-8 a^{10} z^{-2} +28 a^{10}+4 a^9 z^9-8 a^9 z^7-7 a^9 z^5+27 a^9 z^3-24 a^9 z+8 a^9 z^{-1} +6 a^8 z^8-23 a^8 z^6+31 a^8 z^4-24 a^8 z^2-3 a^8 z^{-2} +13 a^8+3 a^7 z^7-8 a^7 z^5+2 a^7 z^3+a^6 z^6-3 a^6 z^4+a^6 z^2 }[/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 -5                       1 1 -7                     3 1 -2 -9                   4     4 -11                 4 3     -1 -13               10 4       6 -15             7 7         0 -17           8 7           1 -19         4 7             3 -21       5 8               -3 -23     1 4                 3 -25   1 5                   -4 -27   1                     1 -29 1                       -1

Integral Khovanov Homology (db, data source)

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