L9n4

L9n3

L9n5

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9n4 at Knotilus!
	L9n4 is $9^2_{43}$ in the Rolfsen table of links.

Link Presentations

[edit Notes on L9n4's Link Presentations]

Planar diagram presentation	X6172 X14,7,15,8 X4,15,1,16 X5,12,6,13 X3849 X9,16,10,17 X11,18,12,5 X17,10,18,11 X13,2,14,3
Gauss code	{1, 9, -5, -3}, {-4, -1, 2, 5, -6, 8, -7, 4, -9, -2, 3, 6, -8, 7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$\frac{-u v^5-1}{\sqrt{u} v^{5/2}}$ (db)
Jones polynomial	$-\frac{1}{q^{7/2}}+\frac{1}{q^{21/2}}-\frac{1}{q^{15/2}}-\frac{1}{q^{11/2}}$ (db)
Signature	-7 (db)
HOMFLY-PT polynomial	$a^{11} (-z)-2 a^{11} z^{-1} +a^9 z^5+6 a^9 z^3+10 a^9 z+5 a^9 z^{-1} -a^7 z^7-7 a^7 z^5-15 a^7 z^3-11 a^7 z-3 a^7 z^{-1}$ (db)
Kauffman polynomial	$-a^{14}-z^3 a^{11}+4 z a^{11}-2 a^{11} z^{-1} -z^6 a^{10}+6 z^4 a^{10}-10 z^2 a^{10}+5 a^{10}-z^7 a^9+7 z^5 a^9-16 z^3 a^9+15 z a^9-5 a^9 z^{-1} -z^6 a^8+6 z^4 a^8-10 z^2 a^8+5 a^8-z^7 a^7+7 z^5 a^7-15 z^3 a^7+11 z a^7-3 a^7 z^{-1}$ (db)

Khovanov Homology

The coefficients of the monomials $t^rq^j$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$).

\   r    \    j   \ -7 -6 -5 -4 -3 -2 -1 0 χ -6               1 1 -8               1 1 -10           1     1 -12       1         1 -14       2 1       1 -16   1             1 -18   1 1           0 -20 1               -1 -22 1               -1

Integral Khovanov Homology (db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-8$ $i=-6$ $i=-4$ $r=-7$ ${\mathbb Z}$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}_2$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}_2$ ${\mathbb Z}^{2}$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=-1$ $r=0$ ${\mathbb Z}$ ${\mathbb Z}$

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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