L11n388

L11n387

L11n389

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n388 at Knotilus!

Link Presentations

[edit Notes on L11n388's Link Presentations]

Planar diagram presentation	X6172 X14,7,15,8 X4,15,1,16 X5,12,6,13 X8493 X22,11,19,12 X18,22,5,21 X20,10,21,9 X10,17,11,18 X16,19,17,20 X2,14,3,13
Gauss code	{1, -11, 5, -3}, {10, -8, 7, -6}, {-4, -1, 2, -5, 8, -9, 6, 4, 11, -2, 3, -10, 9, -7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(u-1)(v-1)(w-1)^{3}}{{\sqrt {u}}{\sqrt {v}}w^{3/2}}}$ (db)
Jones polynomial	$-3q^{-6}+6q^{-5}-9q^{-4}+12q^{-3}-q^{2}-10q^{-2}+5q+11q^{-1}-7$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$-a^{6}z^{-2}-2a^{6}-a^{4}z^{4}+2a^{4}z^{2}+4a^{4}z^{-2}+7a^{4}+a^{2}z^{6}+2a^{2}z^{4}-2a^{2}z^{2}-5a^{2}z^{-2}-8a^{2}-z^{4}+2z^{-2}+3$ (db)
Kauffman polynomial	$3a^{4}z^{8}+3a^{2}z^{8}+6a^{5}z^{7}+13a^{3}z^{7}+7az^{7}+3a^{6}z^{6}+3a^{4}z^{6}+5a^{2}z^{6}+5z^{6}-13a^{5}z^{5}-27a^{3}z^{5}-13az^{5}+z^{5}a^{-1}-8a^{4}z^{4}-16a^{2}z^{4}-8z^{4}+6a^{7}z^{3}+24a^{5}z^{3}+24a^{3}z^{3}+6az^{3}-3a^{4}z^{2}-3a^{2}z^{2}-6a^{7}z-19a^{5}z-21a^{3}z-8az+3a^{6}+10a^{4}+11a^{2}+5+a^{7}z^{-1}+5a^{5}z^{-1}+9a^{3}z^{-1}+5az^{-1}-a^{6}z^{-2}-4a^{4}z^{-2}-5a^{2}z^{-2}-2z^{-2}$ (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   \ -5 -4 -3 -2 -1 0 1 2 3 χ 5                 1 -1 3               4   4 1             3 1   -2 -1           8 4     4 -3         6 7       1 -5       6 4         2 -7     3 6           3 -9   3 6             -3 -11   3               3 -13 3                 -3

Integral Khovanov Homology (db, data source)

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