L11a387

L11a386

L11a388

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Link Presentations

[edit Notes on L11a387's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X14,7,15,8 X8,13,5,14 X18,11,19,12 X20,15,21,16 X22,17,9,18 X16,21,17,22 X12,19,13,20 X2536 X4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -4}, {11, -2, 5, -9, 4, -3, 6, -8, 7, -5, 9, -6, 8, -7}

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 u v w^4-3 u v w^3+3 u v w^2-2 u v w+u v+u w^5-3 u w^4+4 u w^3-4 u w^2+3 u w-u+v w^5-3 v w^4+4 v w^3-4 v w^2+3 v w-v-w^5+2 w^4-3 w^3+3 w^2-2 w}{\sqrt{u} \sqrt{v} w^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{-3} -3 q^{-4} +8 q^{-5} -11 q^{-6} +17 q^{-7} -16 q^{-8} +18 q^{-9} -15 q^{-10} +10 q^{-11} -6 q^{-12} +2 q^{-13} - q^{-14} }[/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^4-22 a^{10} z^2-8 a^{10} z^{-2} -24 a^{10}+3 a^8 z^6+13 a^8 z^4+19 a^8 z^2+3 a^8 z^{-2} +13 a^8+a^6 z^6+3 a^6 z^4+2 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^7-2 a^{15} z^5-3 a^{15} z^3+3 a^{15} z-a^{15} z^{-1} +4 a^{14} z^8-5 a^{14} z^6+7 a^{14} z^4-9 a^{14} z^2-2 a^{14} z^{-2} +7 a^{14}+3 a^{13} z^9+2 a^{13} z^7-14 a^{13} z^5+26 a^{13} z^3-21 a^{13} z+7 a^{13} z^{-1} +a^{12} z^{10}+10 a^{12} z^8-31 a^{12} z^6+45 a^{12} z^4-38 a^{12} z^2-7 a^{12} z^{-2} +22 a^{12}+7 a^{11} z^9-6 a^{11} z^7-23 a^{11} z^5+54 a^{11} z^3-45 a^{11} z+15 a^{11} z^{-1} +a^{10} z^{10}+12 a^{10} z^8-44 a^{10} z^6+64 a^{10} z^4-56 a^{10} z^2-8 a^{10} z^{-2} +28 a^{10}+4 a^9 z^9-2 a^9 z^7-19 a^9 z^5+31 a^9 z^3-24 a^9 z+8 a^9 z^{-1} +6 a^8 z^8-19 a^8 z^6+26 a^8 z^4-25 a^8 z^2-3 a^8 z^{-2} +13 a^8+3 a^7 z^7-7 a^7 z^5+3 a^7 z^3+a^6 z^6-3 a^6 z^4+2 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                   5     5 -11                 6 3     -3 -13               11 5       6 -15             8 9         1 -17           10 8           2 -19         5 8             3 -21       5 10               -5 -23     1 5                 4 -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}^{5}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/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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