L11n205

L11n204

L11n206

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n205 at Knotilus!

Link Presentations

[edit Notes on L11n205's Link Presentations]

Planar diagram presentation	X10,1,11,2 X7,16,8,17 X18,12,19,11 X2,19,3,20 X3,12,4,13 X13,21,14,20 X14,5,15,6 X6,9,7,10 X22,16,9,15 X17,8,18,1 X21,4,22,5
Gauss code	{1, -4, -5, 11, 7, -8, -2, 10}, {8, -1, 3, 5, -6, -7, 9, 2, -10, -3, 4, 6, -11, -9}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {\left(uv^{2}+1\right)\left(u^{2}+v^{3}\right)}{u^{3/2}v^{5/2}}}$ (db)
Jones polynomial	$-{\frac {1}{q^{7/2}}}-{\frac {1}{q^{17/2}}}$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$z^{3}a^{7}+4za^{7}+2a^{7}z^{-1}-z^{5}a^{5}-6z^{3}a^{5}-9za^{5}-3a^{5}z^{-1}+za^{3}+a^{3}z^{-1}$ (db)
Kauffman polynomial	$a^{9}z^{7}-7a^{9}z^{5}+14a^{9}z^{3}-7a^{9}z-a^{7}z^{3}+4a^{7}z-2a^{7}z^{-1}-a^{6}z^{6}+6a^{6}z^{4}-9a^{6}z^{2}+3a^{6}-a^{5}z^{7}+7a^{5}z^{5}-15a^{5}z^{3}+12a^{5}z-3a^{5}z^{-1}-a^{4}z^{6}+6a^{4}z^{4}-9a^{4}z^{2}+3a^{4}+a^{3}z-a^{3}z^{-1}+a^{2}$ (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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 χ -2               1 1 0 -4               1 1 0 -6           1 2     1 -8             1     1 -10       1 2 1       0 -12                   0 -14     1 1           0 -16 1                 1 -18 1                 1

Integral Khovanov Homology (db, data source)

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

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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