L9a46

L9a45

L9a47

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a46 at Knotilus!
	L9a46 is [math]\displaystyle{ 9^3_{10} }[/math] in the Rolfsen table of links.

Link Presentations

[edit Notes on L9a46's Link Presentations]

Planar diagram presentation	X6172 X12,6,13,5 X8493 X2,14,3,13 X14,7,15,8 X16,12,17,11 X18,9,11,10 X10,17,5,18 X4,15,1,16
Gauss code	{1, -4, 3, -9}, {2, -1, 5, -3, 7, -8}, {6, -2, 4, -5, 9, -6, 8, -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-1) (v-1)^2 (w-1)^2}{\sqrt{u} v w} }[/math] (db)
Jones polynomial	[math]\displaystyle{ - q^{-5} +q^4+4 q^{-4} -4 q^3-6 q^{-3} +7 q^2+10 q^{-2} -9 q-10 q^{-1} +12 }[/math] (db)
Signature	0 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a^4 z^2+a^4 z^{-2} +2 a^2 z^4+z^4 a^{-2} +3 a^2 z^2+z^2 a^{-2} -2 a^2 z^{-2} -a^2-z^6-3 z^4-3 z^2+ z^{-2} +1 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^5 z^5-a^5 z^3+4 a^4 z^6-8 a^4 z^4+z^4 a^{-4} +4 a^4 z^2-a^4 z^{-2} +a^4+5 a^3 z^7-8 a^3 z^5+4 z^5 a^{-3} +2 a^3 z^3-3 z^3 a^{-3} -a^3 z+2 a^3 z^{-1} +2 a^2 z^8+8 a^2 z^6+7 z^6 a^{-2} -23 a^2 z^4-9 z^4 a^{-2} +12 a^2 z^2+4 z^2 a^{-2} -2 a^2 z^{-2} +a^2+11 a z^7+6 z^7 a^{-1} -17 a z^5-4 z^5 a^{-1} +4 a z^3-2 z^3 a^{-1} -a z+2 a z^{-1} +2 z^8+11 z^6-25 z^4+12 z^2- z^{-2} +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   \ -5 -4 -3 -2 -1 0 1 2 3 4 χ 9                   1 1 7                 3   -3 5               4 1   3 3             5 3     -2 1           7 4       3 -1         5 7         2 -3       5 5           0 -5     3 7             4 -7   1 3               -2 -9   3                 3 -11 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=-5 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=4 }[/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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