L11n409

L11n408

L11n410

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n409 at Knotilus!

Link Presentations

[edit Notes on L11n409's Link Presentations]

Planar diagram presentation	X6172 X12,4,13,3 X7,16,8,17 X22,17,19,18 X20,12,21,11 X10,20,11,19 X18,21,5,22 X9,14,10,15 X15,8,16,9 X2536 X4,14,1,13
Gauss code	{1, -10, 2, -11}, {6, -5, 7, -4}, {10, -1, -3, 9, -8, -6, 5, -2, 11, 8, -9, 3, 4, -7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(t(1)-1)(t(2)-1)(t(3)-1)\left(t(3)^{2}-3t(3)+1\right)}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)^{3/2}}}$ (db)
Jones polynomial	$q^{4}-4q^{3}+9q^{2}-11q+14-13q^{-1}+13q^{-2}-8q^{-3}+5q^{-4}-2q^{-5}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$-z^{6}+3a^{2}z^{4}+z^{4}a^{-2}-3z^{4}-2a^{4}z^{2}+7a^{2}z^{2}+z^{2}a^{-2}-6z^{2}-2a^{4}+6a^{2}+2a^{-2}-6+a^{2}z^{-2}+a^{-2}z^{-2}-2z^{-2}$ (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 a^3 z^9+a z^9+a^4 z^8+6 a^2 z^8+5 z^8+3 a^3 z^7+13 a z^7+10 z^7 a^{-1} +2 a^4 z^6-a^2 z^6+9 z^6 a^{-2} +6 z^6+3 a^5 z^5-5 a^3 z^5-25 a z^5-13 z^5 a^{-1} +4 z^5 a^{-3} -9 a^4 z^4-25 a^2 z^4-11 z^4 a^{-2} +z^4 a^{-4} -28 z^4-6 a^5 z^3-5 a^3 z^3+5 a z^3+3 z^3 a^{-1} -z^3 a^{-3} +10 a^4 z^2+30 a^2 z^2+6 z^2 a^{-2} +26 z^2+2 a^5 z+6 a^3 z+6 a z+2 z a^{-1} -4 a^4-12 a^2-4 a^{-2} -11-2 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 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 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 \chi} (fixed $j$, alternation over $r$).

\   r    \    j   \ -5 -4 -3 -2 -1 0 1 2 3 4 χ 9                   1 1 7                 3   -3 5               6 1   5 3             5 3     -2 1           9 6       3 -1         8 9         1 -3       5 5           0 -5     3 8             5 -7   2 5               -3 -9   3                 3 -11 2                   -2

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-1$ $i=1$ $r=-5$ ${\mathbb {Z} }^{2}$ $r=-4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-3$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-2$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=-1$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=0$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{9}$ $r=1$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=2$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=3$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=4$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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.

L11n408 L11n410