L11a404

L11a403

L11a405

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Link Presentations

[edit Notes on L11a404's Link Presentations]

Planar diagram presentation	X6172 X12,3,13,4 X10,13,5,14 X8,17,9,18 X14,7,15,8 X18,9,19,10 X22,20,11,19 X20,16,21,15 X16,22,17,21 X2536 X4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, 5, -4, 6, -3}, {11, -2, 3, -5, 8, -9, 4, -6, 7, -8, 9, -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{u v^2 w^3-2 u v^2 w^2+2 u v^2 w+u v w^4-4 u v w^3+7 u v w^2-5 u v w+u v+2 u w^3-4 u w^2+3 u w-u+v^2 w^4-3 v^2 w^3+4 v^2 w^2-2 v^2 w-v w^4+5 v w^3-7 v w^2+4 v w-v-2 w^3+2 w^2-w}{\sqrt{u} v w^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{-9} -3 q^{-8} +8 q^{-7} -12 q^{-6} +19 q^{-5} -20 q^{-4} +22 q^{-3} -q^2-19 q^{-2} +4 q+14 q^{-1} -9 }[/math] (db)
Signature	-2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^8 z^2+2 a^8 z^{-2} +2 a^8-3 a^6 z^4-9 a^6 z^2-5 a^6 z^{-2} -12 a^6+2 a^4 z^6+8 a^4 z^4+16 a^4 z^2+4 a^4 z^{-2} +14 a^4+a^2 z^6+a^2 z^4-2 a^2 z^2-a^2 z^{-2} -4 a^2-z^4-z^2 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^6 a^{10}-3 z^4 a^{10}+3 z^2 a^{10}-a^{10}+3 z^7 a^9-7 z^5 a^9+4 z^3 a^9+5 z^8 a^8-11 z^6 a^8+10 z^4 a^8-11 z^2 a^8-2 a^8 z^{-2} +8 a^8+4 z^9 a^7-18 z^5 a^7+25 z^3 a^7-18 z a^7+5 a^7 z^{-1} +z^{10} a^6+16 z^8 a^6-50 z^6 a^6+62 z^4 a^6-45 z^2 a^6-5 a^6 z^{-2} +20 a^6+9 z^9 a^5-3 z^7 a^5-34 z^5 a^5+53 z^3 a^5-33 z a^5+9 a^5 z^{-1} +z^{10} a^4+20 z^8 a^4-55 z^6 a^4+62 z^4 a^4-38 z^2 a^4-4 a^4 z^{-2} +15 a^4+5 z^9 a^3+8 z^7 a^3-37 z^5 a^3+41 z^3 a^3-19 z a^3+5 a^3 z^{-1} +9 z^8 a^2-13 z^6 a^2+8 z^4 a^2-6 z^2 a^2-a^2 z^{-2} +3 a^2+8 z^7 a-13 z^5 a+8 z^3 a-4 z a+a z^{-1} +4 z^6-5 z^4+z^2+z^5 a^{-1} -z^3 a^{-1} }[/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                     3   3 1                   6 1   -5 -1                 8 3     5 -3               12 7       -5 -5             10 7         3 -7           10 12           2 -9         9 10             -1 -11       4 11               7 -13     4 8                 -4 -15   1 6                   5 -17   2                     -2 -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}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{12} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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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