L9a47

L9a46

L9a48

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a47 at Knotilus!
	L9a47 is $9_{2}^{3}$ in the Rolfsen table of links.

Link Presentations

[edit Notes on L9a47's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {-t(3)^{2}t(2)^{2}+t(1)t(2)^{2}-t(1)t(3)t(2)^{2}+2t(3)t(2)^{2}-t(2)^{2}-t(1)t(3)^{2}t(2)+2t(3)^{2}t(2)-2t(1)t(2)+4t(1)t(3)t(2)-4t(3)t(2)+t(2)+t(1)t(3)^{2}-t(3)^{2}+t(1)-2t(1)t(3)+t(3)}{{\sqrt {t(1)}}t(2)t(3)}}$ (db)
Jones polynomial	$-q^{-8}+3q^{-7}-5q^{-6}+8q^{-5}-8q^{-4}+10q^{-3}-7q^{-2}+q+6q^{-1}-3$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$-a^{8}+3z^{2}a^{6}+a^{6}z^{-2}+3a^{6}-2z^{4}a^{4}-4z^{2}a^{4}-2a^{4}z^{-2}-5a^{4}-z^{4}a^{2}+z^{2}a^{2}+a^{2}z^{-2}+3a^{2}+z^{2}$ (db)
Kauffman polynomial	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z^5 a^9-2 z^3 a^9+z a^9+3 z^6 a^8-7 z^4 a^8+5 z^2 a^8-2 a^8+3 z^7 a^7-3 z^5 a^7-4 z^3 a^7+3 z a^7+z^8 a^6+8 z^6 a^6-24 z^4 a^6+21 z^2 a^6+a^6 z^{-2} -8 a^6+7 z^7 a^5-10 z^5 a^5+5 z a^5-2 a^5 z^{-1} +z^8 a^4+10 z^6 a^4-26 z^4 a^4+24 z^2 a^4+2 a^4 z^{-2} -9 a^4+4 z^7 a^3-3 z^5 a^3-z^3 a^3+3 z a^3-2 a^3 z^{-1} +5 z^6 a^2-8 z^4 a^2+7 z^2 a^2+a^2 z^{-2} -4 a^2+3 z^5 a-3 z^3 a+z^4-z^2} (db)

Khovanov Homology

The coefficients of the monomials $t^{r}q^{j}$ 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 3                   1 1 1                 2   -2 -1               4 1   3 -3             4 3     -1 -5           6 3       3 -7         4 6         2 -9       4 4           0 -11     2 5             3 -13   1 3               -2 -15   2                 2 -17 1                   -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-3$ $i=-1$ $r=-7$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-4$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{4}$ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-3} ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-2$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{6}$ $r=-1$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=0$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{4}$ $r=1$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=2$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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.

L9a46 L9a48