L11a375

L11a374

L11a376

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Link Presentations

[edit Notes on L11a375's Link Presentations]

Planar diagram presentation	X12,1,13,2 X2,13,3,14 X14,3,15,4 X22,7,11,8 X18,10,19,9 X20,16,21,15 X16,6,17,5 X6,18,7,17 X10,20,1,19 X8,11,9,12 X4,21,5,22
Gauss code	{1, -2, 3, -11, 7, -8, 4, -10, 5, -9}, {10, -1, 2, -3, 6, -7, 8, -5, 9, -6, 11, -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{t(2)^4 t(1)^4-t(2)^3 t(1)^4-2 t(2)^4 t(1)^3+6 t(2)^3 t(1)^3-5 t(2)^2 t(1)^3+t(2) t(1)^3+t(2)^4 t(1)^2-6 t(2)^3 t(1)^2+11 t(2)^2 t(1)^2-6 t(2) t(1)^2+t(1)^2+t(2)^3 t(1)-5 t(2)^2 t(1)+6 t(2) t(1)-2 t(1)-t(2)+1}{t(1)^2 t(2)^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{9/2}-3 q^{7/2}+7 q^{5/2}-12 q^{3/2}+15 \sqrt{q}-\frac{19}{\sqrt{q}}+\frac{18}{q^{3/2}}-\frac{16}{q^{5/2}}+\frac{12}{q^{7/2}}-\frac{7}{q^{9/2}}+\frac{3}{q^{11/2}}-\frac{1}{q^{13/2}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a z^9+a^3 z^7-7 a z^7+z^7 a^{-1} +5 a^3 z^5-19 a z^5+5 z^5 a^{-1} +9 a^3 z^3-24 a z^3+9 z^3 a^{-1} +6 a^3 z-13 a z+6 z a^{-1} +a^3 z^{-1} -a z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^7 z^5-2 a^7 z^3+a^7 z+3 a^6 z^6-5 a^6 z^4+2 a^6 z^2+5 a^5 z^7-7 a^5 z^5+2 a^5 z^3+6 a^4 z^8-9 a^4 z^6+z^6 a^{-4} +7 a^4 z^4-3 z^4 a^{-4} -4 a^4 z^2+2 z^2 a^{-4} +5 a^3 z^9-7 a^3 z^7+3 z^7 a^{-3} +9 a^3 z^5-8 z^5 a^{-3} -10 a^3 z^3+5 z^3 a^{-3} +6 a^3 z-z a^{-3} -a^3 z^{-1} +2 a^2 z^{10}+5 a^2 z^8+5 z^8 a^{-2} -17 a^2 z^6-13 z^6 a^{-2} +21 a^2 z^4+10 z^4 a^{-2} -10 a^2 z^2-4 z^2 a^{-2} +a^2+10 a z^9+5 z^9 a^{-1} -27 a z^7-12 z^7 a^{-1} +38 a z^5+13 z^5 a^{-1} -32 a z^3-13 z^3 a^{-1} +14 a z+6 z a^{-1} -a z^{-1} +2 z^{10}+4 z^8-19 z^6+22 z^4-10 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 10                       1 -1 8                     2   2 6                   5 1   -4 4                 7 2     5 2               8 5       -3 0             11 7         4 -2           8 9           1 -4         8 10             -2 -6       5 9               4 -8     2 7                 -5 -10   1 5                   4 -12   2                     -2 -14 1                       1

Integral Khovanov Homology (db, data source)

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