L11a498

L11a497

L11a499

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Link Presentations

[edit Notes on L11a498's Link Presentations]

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

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(3)-1) \left(t(2) t(3)^4+2 t(1) t(2) t(3)^3-3 t(2) t(3)^3+2 t(3)^3+4 t(1) t(3)^2-4 t(1) t(2) t(3)^2+4 t(2) t(3)^2-4 t(3)^2-3 t(1) t(3)+2 t(1) t(2) t(3)+2 t(3)+t(1)\right)}{\sqrt{t(1)} \sqrt{t(2)} t(3)^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^4-4 q^3+10 q^2-14 q+19-20 q^{-1} +21 q^{-2} -16 q^{-3} +12 q^{-4} -7 q^{-5} +3 q^{-6} - q^{-7} }[/math] (db)
Signature	0 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^6 \left(-z^2\right)-a^6 z^{-2} -2 a^6+3 a^4 z^4+9 a^4 z^2+4 a^4 z^{-2} +10 a^4-2 a^2 z^6-8 a^2 z^4+z^4 a^{-2} -15 a^2 z^2-5 a^2 z^{-2} +z^2 a^{-2} -13 a^2+ a^{-2} -z^6-z^4+2 z^2+2 z^{-2} +4 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^4 z^{10}+a^2 z^{10}+3 a^5 z^9+9 a^3 z^9+6 a z^9+3 a^6 z^8+12 a^4 z^8+22 a^2 z^8+13 z^8+a^7 z^7-3 a^5 z^7-8 a^3 z^7+10 a z^7+14 z^7 a^{-1} -11 a^6 z^6-49 a^4 z^6-66 a^2 z^6+10 z^6 a^{-2} -18 z^6-4 a^7 z^5-17 a^5 z^5-39 a^3 z^5-48 a z^5-18 z^5 a^{-1} +4 z^5 a^{-3} +14 a^6 z^4+60 a^4 z^4+66 a^2 z^4-10 z^4 a^{-2} +z^4 a^{-4} +9 z^4+6 a^7 z^3+34 a^5 z^3+66 a^3 z^3+44 a z^3+6 z^3 a^{-1} -8 a^6 z^2-35 a^4 z^2-43 a^2 z^2+6 z^2 a^{-2} -10 z^2-4 a^7 z-21 a^5 z-39 a^3 z-22 a z+3 a^6+16 a^4+21 a^2-2 a^{-2} +7+a^7 z^{-1} +5 a^5 z^{-1} +9 a^3 z^{-1} +5 a z^{-1} -a^6 z^{-2} -4 a^4 z^{-2} -5 a^2 z^{-2} -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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 9                       1 1 7                     4 1 -3 5                   6     6 3                 8 4     -4 1               11 6       5 -1             11 10         -1 -3           10 9           1 -5         6 11             5 -7       6 10               -4 -9     2 7                 5 -11   1 5                   -4 -13   2                     2 -15 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=-7 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/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}^{6}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=4 }[/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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