L11a473

L11a472

L11a474

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Link Presentations

[edit Notes on L11a473's Link Presentations]

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

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-1) (w-1) \left(v^2 w^3-v^2 w^2+v^2 w-v w^3+3 v w^2-3 v w+v-w^2+w-1\right)}{\sqrt{u} v w^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^5+3 q^4-7 q^3+12 q^2-15 q+19-17 q^{-1} +16 q^{-2} -11 q^{-3} +7 q^{-4} -3 q^{-5} + q^{-6} }[/math] (db)
Signature	0 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^8-2 a^2 z^6-z^6 a^{-2} +6 z^6+a^4 z^4-9 a^2 z^4-4 z^4 a^{-2} +15 z^4+3 a^4 z^2-15 a^2 z^2-6 z^2 a^{-2} +18 z^2+3 a^4-10 a^2-3 a^{-2} +10+a^4 z^{-2} -2 a^2 z^{-2} + z^{-2} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^2 z^{10}+z^{10}+3 a^3 z^9+8 a z^9+5 z^9 a^{-1} +4 a^4 z^8+10 a^2 z^8+8 z^8 a^{-2} +14 z^8+3 a^5 z^7+a^3 z^7-13 a z^7-5 z^7 a^{-1} +6 z^7 a^{-3} +a^6 z^6-7 a^4 z^6-32 a^2 z^6-22 z^6 a^{-2} +3 z^6 a^{-4} -49 z^6-8 a^5 z^5-15 a^3 z^5-7 z^5 a^{-1} -13 z^5 a^{-3} +z^5 a^{-5} -3 a^6 z^4+40 a^2 z^4+30 z^4 a^{-2} -5 z^4 a^{-4} +72 z^4+6 a^5 z^3+13 a^3 z^3+16 a z^3+21 z^3 a^{-1} +10 z^3 a^{-3} -2 z^3 a^{-5} +3 a^6 z^2-33 a^2 z^2-18 z^2 a^{-2} -48 z^2-a^5 z-7 a^3 z-13 a z-10 z a^{-1} -3 z a^{-3} -a^6+3 a^4+13 a^2+5 a^{-2} +15+2 a^3 z^{-1} +2 a z^{-1} -a^4 z^{-2} -2 a^2 z^{-2} - 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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 11                       1 -1 9                     2   2 7                   5 1   -4 5                 7 2     5 3               8 5       -3 1             11 7         4 -1           8 10           2 -3         8 9             -1 -5       5 10               5 -7     2 6                 -4 -9   1 5                   4 -11   2                     -2 -13 1                       1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-1 }[/math] [math]\displaystyle{ i=1 }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=5 }[/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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