L11n381

L11n380

L11n382

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n381 at Knotilus!

Link Presentations

[edit Notes on L11n381's Link Presentations]

Planar diagram presentation	X6172 X14,7,15,8 X4,15,1,16 X5,10,6,11 X8493 X22,18,19,17 X11,20,12,21 X19,12,20,13 X18,22,5,21 X9,16,10,17 X2,14,3,13
Gauss code	{1, -11, 5, -3}, {-8, 7, 9, -6}, {-4, -1, 2, -5, -10, 4, -7, 8, 11, -2, 3, 10, 6, -9}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	0 (db)
Jones polynomial	$-q^{-7}+q^{-6}-q^{-5}+q^{-3}+q^{-2}+q+2q^{-1}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{6}\left(-z^{2}\right)-a^{6}z^{-2}-2a^{6}+a^{4}z^{4}+5a^{4}z^{2}+4a^{4}z^{-2}+7a^{4}-a^{2}z^{4}-5a^{2}z^{2}-5a^{2}z^{-2}-8a^{2}+z^{2}+2z^{-2}+3$ (db)
Kauffman polynomial	$a^{6}z^{8}+a^{4}z^{8}+a^{7}z^{7}+2a^{5}z^{7}+a^{3}z^{7}-6a^{6}z^{6}-7a^{4}z^{6}-a^{2}z^{6}-6a^{7}z^{5}-14a^{5}z^{5}-8a^{3}z^{5}+9a^{6}z^{4}+15a^{4}z^{4}+7a^{2}z^{4}+z^{4}+10a^{7}z^{3}+28a^{5}z^{3}+20a^{3}z^{3}+2az^{3}-4a^{6}z^{2}-15a^{4}z^{2}-15a^{2}z^{2}-4z^{2}-6a^{7}z-19a^{5}z-21a^{3}z-8az+3a^{6}+10a^{4}+11a^{2}+5+a^{7}z^{-1}+5a^{5}z^{-1}+9a^{3}z^{-1}+5az^{-1}-a^{6}z^{-2}-4a^{4}z^{-2}-5a^{2}z^{-2}-2z^{-2}$ (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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 3                   1 1 1               1     1 -1             1 4 1   2 -3           1 1 3     3 -5         1 3         2 -7       1 1 1         1 -9       1 2           -1 -11   1 1               0 -13                     0 -15 1                   -1

Integral Khovanov Homology (db, data source)

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

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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