L9a49

L9a48

L9a50

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a49 at Knotilus!
	L9a49 is $9^3_{6}$ in the Rolfsen table of links.

Link Presentations

[edit Notes on L9a49's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$\frac{u v^2 w-u v^2+u v w^2-3 u v w+2 u v-u w^2+2 u w-2 v^2 w+v^2-2 v w^2+3 v w-v+w^2-w}{\sqrt{u} v w}$ (db)
Jones polynomial	$-q^5+3 q^4+ q^{-4} -4 q^3-2 q^{-3} +7 q^2+5 q^{-2} -7 q-6 q^{-1} +8$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$-z^2 a^{-4} +a^4+z^4 a^{-2} -2 a^2 z^2+a^2 z^{-2} +z^2 a^{-2} + a^{-2} z^{-2} +2 a^{-2} +z^4-z^2-2 z^{-2} -3$ (db)
Kauffman polynomial	$z^8 a^{-2} +z^8+2 a z^7+5 z^7 a^{-1} +3 z^7 a^{-3} +3 a^2 z^6+4 z^6 a^{-2} +3 z^6 a^{-4} +4 z^6+2 a^3 z^5+2 a z^5-7 z^5 a^{-1} -6 z^5 a^{-3} +z^5 a^{-5} +a^4 z^4-4 a^2 z^4-13 z^4 a^{-2} -8 z^4 a^{-4} -10 z^4-2 a^3 z^3-7 a z^3-z^3 a^{-1} +2 z^3 a^{-3} -2 z^3 a^{-5} -2 a^4 z^2+4 a^2 z^2+11 z^2 a^{-2} +5 z^2 a^{-4} +12 z^2+6 a z+6 z a^{-1} +a^4-3 a^2-5 a^{-2} -8-2 a z^{-1} -2 a^{-1} z^{-1} +a^2 z^{-2} + a^{-2} z^{-2} +2 z^{-2}$ (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   \ -4 -3 -2 -1 0 1 2 3 4 5 χ 11                   1 -1 9                 2   2 7               2 1   -1 5             5 2     3 3           4 4       0 1         4 3         1 -1       3 5           2 -3     2 3             -1 -5     3               3 -7 1 2                 -1 -9 1                   1

Integral Khovanov Homology (db, data source)

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

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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