L11a413

L11a412

L11a414

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Link Presentations

[edit Notes on L11a413's Link Presentations]

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

A Braid Representative	{{{braid_table}}}
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^2 w^3-2 u v^2 w^2+u v^2 w+u v w^4-4 u v w^3+6 u v w^2-4 u v w+u v+u w^3-3 u w^2+3 u w-u+v^2 w^4-3 v^2 w^3+3 v^2 w^2-v^2 w-v w^4+4 v w^3-6 v w^2+4 v w-v-w^3+2 w^2-2 w}{\sqrt{u} v w^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ - q^{-14} +3 q^{-13} -7 q^{-12} +12 q^{-11} -16 q^{-10} +19 q^{-9} -18 q^{-8} +17 q^{-7} -11 q^{-6} +8 q^{-5} -3 q^{-4} + q^{-3} }[/math] (db)
Signature	-6 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a^{14} z^{-2} -a^{14}+4 a^{12} z^2+4 a^{12} z^{-2} +9 a^{12}-6 a^{10} z^4-20 a^{10} z^2-5 a^{10} z^{-2} -19 a^{10}+3 a^8 z^6+13 a^8 z^4+19 a^8 z^2+2 a^8 z^{-2} +11 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^5-2 a^{17} z^3+a^{17} z+3 a^{16} z^6-5 a^{16} z^4+3 a^{16} z^2-a^{16}+5 a^{15} z^7-7 a^{15} z^5+4 a^{15} z^3-2 a^{15} z+a^{15} z^{-1} +5 a^{14} z^8-2 a^{14} z^6-6 a^{14} z^4+5 a^{14} z^2-a^{14} z^{-2} +3 a^{13} z^9+8 a^{13} z^7-27 a^{13} z^5+31 a^{13} z^3-19 a^{13} z+5 a^{13} z^{-1} +a^{12} z^{10}+11 a^{12} z^8-24 a^{12} z^6+21 a^{12} z^4-16 a^{12} z^2-4 a^{12} z^{-2} +13 a^{12}+7 a^{11} z^9-2 a^{11} z^7-29 a^{11} z^5+50 a^{11} z^3-35 a^{11} z+9 a^{11} z^{-1} +a^{10} z^{10}+12 a^{10} z^8-39 a^{10} z^6+51 a^{10} z^4-43 a^{10} z^2-5 a^{10} z^{-2} +22 a^{10}+4 a^9 z^9-2 a^9 z^7-17 a^9 z^5+28 a^9 z^3-19 a^9 z+5 a^9 z^{-1} +6 a^8 z^8-19 a^8 z^6+26 a^8 z^4-23 a^8 z^2-2 a^8 z^{-2} +11 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 7         -1 -17           11 10           1 -19         7 10             3 -21       5 9               -4 -23     2 7                 5 -25   1 5                   -4 -27   2                     2 -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}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-9 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\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^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{10} }[/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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