L11n395

L11n394

L11n396

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n395 at Knotilus!

Link Presentations

[edit Notes on L11n395's Link Presentations]

Planar diagram presentation	X6172 X14,7,15,8 X15,1,16,4 X9,22,10,19 X3849 X21,17,22,16 X11,5,12,18 X5,21,6,20 X17,11,18,10 X19,12,20,13 X2,14,3,13
Gauss code	{1, -11, -5, 3}, {-10, 8, -6, 4}, {-8, -1, 2, 5, -4, 9, -7, 10, 11, -2, -3, 6, -9, 7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(u-1)(v-1)(w-1)\left(w^{2}-w+1\right)}{{\sqrt {u}}{\sqrt {v}}w^{3/2}}}$ (db)
Jones polynomial	$q^{5}-4q^{4}-q^{-4}+6q^{3}+3q^{-3}-6q^{2}-4q^{-2}+9q+7q^{-1}-7$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z^{6}-a^{2}z^{4}-2z^{4}a^{-2}+4z^{4}-2a^{2}z^{2}-4z^{2}a^{-2}+z^{2}a^{-4}+5z^{2}+a^{2}z^{-2}+a^{-2}z^{-2}-2z^{-2}$ (db)
Kauffman polynomial	$2az^{9}+2z^{9}a^{-1}+3a^{2}z^{8}+5z^{8}a^{-2}+8z^{8}+a^{3}z^{7}-5az^{7}-z^{7}a^{-1}+5z^{7}a^{-3}-14a^{2}z^{6}-18z^{6}a^{-2}+2z^{6}a^{-4}-34z^{6}-4a^{3}z^{5}-5az^{5}-17z^{5}a^{-1}-16z^{5}a^{-3}+19a^{2}z^{4}+22z^{4}a^{-2}-z^{4}a^{-4}+42z^{4}+4a^{3}z^{3}+13az^{3}+22z^{3}a^{-1}+17z^{3}a^{-3}+4z^{3}a^{-5}-8a^{2}z^{2}-13z^{2}a^{-2}-z^{2}a^{-4}+z^{2}a^{-6}-19z^{2}-a^{3}z-4az-6za^{-1}-4za^{-3}-za^{-5}+1-2az^{-1}-2a^{-1}z^{-1}+a^{2}z^{-2}+a^{-2}z^{-2}+2z^{-2}$ (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 4 χ 11                   1 1 9                 3   -3 7               4 2   2 5             3 3     0 3           6 4 1     3 1         5 7         2 -1       2 3 1         0 -3     2 5             3 -5   1 2               -1 -7   2                 2 -9 1                   -1

Integral Khovanov Homology (db, data source)

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

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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