L11a412

L11a411

L11a413

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Link Presentations

[edit Notes on L11a412's Link Presentations]

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

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{u v^2 w^3-3 u v^2 w^2+3 u v^2 w-u v^2+u v w^4-5 u v w^3+9 u v w^2-6 u v w+2 u v-u w^4+4 u w^3-5 u w^2+3 u w-u+v^2 w^4-3 v^2 w^3+5 v^2 w^2-4 v^2 w+v^2-2 v w^4+6 v w^3-9 v w^2+5 v w-v+w^4-3 w^3+3 w^2-w}{\sqrt{u} v w^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^2+5 q-12+20 q^{-1} -25 q^{-2} +31 q^{-3} -28 q^{-4} +25 q^{-5} -18 q^{-6} +10 q^{-7} -4 q^{-8} + q^{-9} }[/math] (db)
Signature	-2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^8 z^2+a^8-3 a^6 z^4-5 a^6 z^2+a^6 z^{-2} -3 a^6+2 a^4 z^6+5 a^4 z^4+6 a^4 z^2-2 a^4 z^{-2} +a^4+a^2 z^6-a^2 z^2+a^2 z^{-2} +a^2-z^4 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{10} z^6-2 a^{10} z^4+a^{10} z^2+4 a^9 z^7-8 a^9 z^5+6 a^9 z^3-2 a^9 z+8 a^8 z^8-16 a^8 z^6+13 a^8 z^4-7 a^8 z^2+2 a^8+8 a^7 z^9-6 a^7 z^7-14 a^7 z^5+18 a^7 z^3-7 a^7 z+3 a^6 z^{10}+21 a^6 z^8-64 a^6 z^6+59 a^6 z^4-24 a^6 z^2+a^6 z^{-2} +4 a^6+20 a^5 z^9-26 a^5 z^7-13 a^5 z^5+25 a^5 z^3-7 a^5 z-2 a^5 z^{-1} +3 a^4 z^{10}+30 a^4 z^8-79 a^4 z^6+63 a^4 z^4-21 a^4 z^2+2 a^4 z^{-2} +3 a^4+12 a^3 z^9-4 a^3 z^7-22 a^3 z^5+17 a^3 z^3-3 a^3 z-2 a^3 z^{-1} +17 a^2 z^8-27 a^2 z^6+16 a^2 z^4-5 a^2 z^2+a^2 z^{-2} +12 a z^7-14 a z^5+z^5 a^{-1} +4 a z^3-a z+5 z^6-3 z^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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 χ 5                       1 -1 3                     4   4 1                   8 1   -7 -1                 12 4     8 -3               14 9       -5 -5             17 11         6 -7           13 16           3 -9         12 15             -3 -11       7 14               7 -13     3 11                 -8 -15   1 7                   6 -17   3                     -3 -19 1                       1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-3 }[/math] [math]\displaystyle{ i=-1 }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{14}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{15}\oplus{\mathbb Z}_2^{13} }[/math] [math]\displaystyle{ {\mathbb Z}^{13} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{16}\oplus{\mathbb Z}_2^{15} }[/math] [math]\displaystyle{ {\mathbb Z}^{17} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{14} }[/math] [math]\displaystyle{ {\mathbb Z}^{14} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/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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