L9a45

L9a44

L9a46

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a45 at Knotilus!
	L9a45 is $9_{7}^{3}$ in the Rolfsen table of links.

Link Presentations

[edit Notes on L9a45's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Khovanov Homology

The coefficients of the monomials 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 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             3 2     1 3           3 2       -1 1         4 3         1 -1       3 6           3 -3     1 1             0 -5     3               3 -7 1 1                 0 -9 1                   1

Integral Khovanov Homology (db, data source)

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