L11n343

L11n342

L11n344

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n343 at Knotilus!

Link Presentations

[edit Notes on L11n343's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X11,19,12,18 X7,14,8,15 X13,8,14,9 X15,13,16,22 X17,21,18,20 X21,17,22,16 X19,5,20,12 X2536 X4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, -4, 5, 11, -2, -3, 9}, {-5, 4, -6, 8, -7, 3, -9, 7, -8, 6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(v-1)(w-1)\left(uv-2u+2v^{2}-v\right)}{{\sqrt {u}}v^{3/2}{\sqrt {w}}}}$ (db)
Jones polynomial	$-2q^{3}+5q^{2}-6q+8-8q^{-1}+8q^{-2}-5q^{-3}+4q^{-4}-q^{-5}+q^{-6}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a^{6}z^{-2}+a^{6}-2z^{2}a^{4}-2a^{4}z^{-2}-3a^{4}+z^{4}a^{2}+a^{2}z^{-2}+2z^{4}+4z^{2}+3-2z^{2}a^{-2}-a^{-2}$ (db)
Kauffman polynomial	$a^{3}z^{9}+az^{9}+a^{4}z^{8}+4a^{2}z^{8}+3z^{8}+a^{5}z^{7}-2a^{3}z^{7}+3z^{7}a^{-1}+a^{6}z^{6}-12a^{2}z^{6}+z^{6}a^{-2}-10z^{6}-2a^{5}z^{5}+4a^{3}z^{5}-6z^{5}a^{-1}-5a^{6}z^{4}-9a^{4}z^{4}+15a^{2}z^{4}+4z^{4}a^{-2}+23z^{4}-3a^{5}z^{3}-10a^{3}z^{3}-az^{3}+9z^{3}a^{-1}+3z^{3}a^{-3}+8a^{6}z^{2}+13a^{4}z^{2}-8a^{2}z^{2}-7z^{2}a^{-2}-20z^{2}+6a^{5}z+8a^{3}z-4za^{-1}-2za^{-3}-5a^{6}-8a^{4}-a^{2}+2a^{-2}+5-2a^{5}z^{-1}-2a^{3}z^{-1}+a^{6}z^{-2}+2a^{4}z^{-2}+a^{2}z^{-2}$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 χ 7                   2 -2 5                 3   3 3               3 2   -1 1             5 3     2 -1           5 5       0 -3         3 3         0 -5       2 5           3 -7     2 3             -1 -9   1 4               3 -11                     0 -13 1                   1

Integral Khovanov Homology (db, data source)

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

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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