L9a29

L9a28

L9a30

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a29 at Knotilus!
	L9a29 is $9^2_{19}$ in the Rolfsen table of links.

Link Presentations

[edit Notes on L9a29's Link Presentations]

Planar diagram presentation	X8192 X2,9,3,10 X10,3,11,4 X6718 X16,11,17,12 X14,6,15,5 X4,16,5,15 X18,13,7,14 X12,17,13,18
Gauss code	{1, -2, 3, -7, 6, -4}, {4, -1, 2, -3, 5, -9, 8, -6, 7, -5, 9, -8}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$\frac{u^2 v^4-u^2 v^3+u^2 v^2-u^2 v-u v^4+u v^3-u v^2+u v-u-v^3+v^2-v+1}{u v^2}$ (db)
Jones polynomial	$-\frac{1}{\sqrt{q}}+\frac{1}{q^{3/2}}-\frac{3}{q^{5/2}}+\frac{3}{q^{7/2}}-\frac{4}{q^{9/2}}+\frac{4}{q^{11/2}}-\frac{4}{q^{13/2}}+\frac{3}{q^{15/2}}-\frac{2}{q^{17/2}}+\frac{1}{q^{19/2}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$a^7 \left(-z^5\right)-4 a^7 z^3-4 a^7 z-a^7 z^{-1} +a^5 z^7+6 a^5 z^5+12 a^5 z^3+10 a^5 z+3 a^5 z^{-1} -a^3 z^5-5 a^3 z^3-7 a^3 z-2 a^3 z^{-1}$ (db)
Kauffman polynomial	$a^{12} z^2+2 a^{11} z^3+3 a^{10} z^4-2 a^{10} z^2+4 a^9 z^5-7 a^9 z^3+2 a^9 z+4 a^8 z^6-10 a^8 z^4+5 a^8 z^2-a^8+3 a^7 z^7-9 a^7 z^5+6 a^7 z^3-3 a^7 z+a^7 z^{-1} +a^6 z^8-11 a^6 z^4+13 a^6 z^2-3 a^6+4 a^5 z^7-19 a^5 z^5+27 a^5 z^3-14 a^5 z+3 a^5 z^{-1} +a^4 z^8-4 a^4 z^6+2 a^4 z^4+5 a^4 z^2-3 a^4+a^3 z^7-6 a^3 z^5+12 a^3 z^3-9 a^3 z+2 a^3 z^{-1}$ (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 1 2 χ 0                   1 1 -2                     0 -4               3 1   2 -6             1 1     0 -8           3 2       1 -10         2 2         0 -12       2 2           0 -14     1 2             1 -16   1 2               -1 -18   1                 1 -20 1                   -1

Integral Khovanov Homology (db, data source)

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

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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