L11n437

L11n436

L11n438

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n437 at Knotilus!

Link Presentations

[edit Notes on L11n437's Link Presentations]

Planar diagram presentation	X8192 X12,3,13,4 X13,20,14,21 X9,18,10,19 X21,10,22,11 X19,14,20,7 X5,17,6,16 X17,22,18,15 X2738 X4,11,5,12 X15,1,16,6
Gauss code	{1, -9, 2, -10, -7, 11}, {9, -1, -4, 5, 10, -2, -3, 6}, {-11, 7, -8, 4, -6, 3, -5, 8}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	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 -\frac{(u w+v) \left(u v^2 w-u v^2+u-v^2 w+w-1\right)}{u v^{3/2} w}} (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 1- q^{-1} + q^{-2} + q^{-4} + q^{-5} + q^{-7} - q^{-8} + q^{-9} } (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{8}z^{2}+a^{8}z^{-2}+a^{8}-2a^{6}z^{-2}-3a^{6}-a^{4}z^{6}-5a^{4}z^{4}-5a^{4}z^{2}+a^{4}z^{-2}+a^{2}z^{4}+4a^{2}z^{2}+2a^{2}$ (db)
Kauffman polynomial	$a^{10}z^{6}-5a^{10}z^{4}+5a^{10}z^{2}+a^{9}z^{7}-5a^{9}z^{5}+5a^{9}z^{3}+a^{8}z^{8}-7a^{8}z^{6}+16a^{8}z^{4}-17a^{8}z^{2}-a^{8}z^{-2}+7a^{8}+a^{7}z^{7}-7a^{7}z^{5}+13a^{7}z^{3}-9a^{7}z+2a^{7}z^{-1}+a^{6}z^{8}-8a^{6}z^{6}+21a^{6}z^{4}-25a^{6}z^{2}-2a^{6}z^{-2}+11a^{6}+a^{5}z^{7}-7a^{5}z^{5}+13a^{5}z^{3}-9a^{5}z+2a^{5}z^{-1}+a^{4}z^{6}-5a^{4}z^{4}+3a^{4}z^{2}-a^{4}z^{-2}+3a^{4}+a^{3}z^{7}-5a^{3}z^{5}+5a^{3}z^{3}+a^{2}z^{6}-5a^{2}z^{4}+6a^{2}z^{2}-2a^{2}$ (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 $\chi$ (fixed $j$, alternation over $r$).

\   r    \    j   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 1                     1 1 -1                       0 -3               1 2 1   0 -5             1 1 1     1 -7           2 4 1       1 -9         1 1 2         2 -11       1 3 1           1 -13     1 1 1             1 -15     1 1               0 -17 1 1                   0 -19 1                     1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-5$ $i=-3$ $i=-1$ $r=-8$ ${\mathbb {Z} }$ ${\mathbb {Z} }$ $r=-7$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ 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}} 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} ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }^{4}$ ${\mathbb {Z} }$ $r=-1$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=0$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }^{2}$ $r=1$ ${\mathbb {Z} }$ $r=2$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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