L11a390

L11a389

L11a391

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a390 at Knotilus!

Link Presentations

[edit Notes on L11a390's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X16,7,17,8 X8,15,5,16 X18,12,19,11 X20,13,21,14 X22,20,9,19 X12,21,13,22 X14,18,15,17 X2536 X4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -4}, {11, -2, 5, -8, 6, -9, 4, -3, 9, -5, 7, -6, 8, -7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {uvw^{3}-4uvw^{2}+6uvw-2uv-2uw^{3}+7uw^{2}-7uw+2u-2vw^{3}+7vw^{2}-7vw+2v+2w^{3}-6w^{2}+4w-1}{{\sqrt {u}}{\sqrt {v}}w^{3/2}}}$ (db)
Jones polynomial	$q^{-9}-2q^{-8}+6q^{-7}-10q^{-6}+16q^{-5}-19q^{-4}+21q^{-3}-q^{2}-18q^{-2}+5q+15q^{-1}-10$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$a^{10}z^{-2}-3a^{8}z^{-2}-4a^{8}+6a^{6}z^{2}+4a^{6}z^{-2}+8a^{6}-4a^{4}z^{4}-7a^{4}z^{2}-3a^{4}z^{-2}-7a^{4}+a^{2}z^{6}+2a^{2}z^{4}+4a^{2}z^{2}+a^{2}z^{-2}+3a^{2}-z^{4}$ (db)
Kauffman polynomial	$a^{10}z^{6}-4a^{10}z^{4}+6a^{10}z^{2}+a^{10}z^{-2}-4a^{10}+2a^{9}z^{7}-5a^{9}z^{5}+3a^{9}z^{3}+a^{9}z-a^{9}z^{-1}+2a^{8}z^{8}+2a^{8}z^{6}-18a^{8}z^{4}+25a^{8}z^{2}+3a^{8}z^{-2}-14a^{8}+2a^{7}z^{9}+3a^{7}z^{7}-11a^{7}z^{5}+6a^{7}z^{3}+a^{7}z-a^{7}z^{-1}+a^{6}z^{10}+6a^{6}z^{8}-5a^{6}z^{6}-19a^{6}z^{4}+35a^{6}z^{2}+4a^{6}z^{-2}-21a^{6}+7a^{5}z^{9}-3a^{5}z^{7}-12a^{5}z^{5}+10a^{5}z^{3}+a^{5}z-a^{5}z^{-1}+a^{4}z^{10}+14a^{4}z^{8}-23a^{4}z^{6}-3a^{4}z^{4}+23a^{4}z^{2}+3a^{4}z^{-2}-14a^{4}+5a^{3}z^{9}+6a^{3}z^{7}-22a^{3}z^{5}+10a^{3}z^{3}+a^{3}z-a^{3}z^{-1}+10a^{2}z^{8}-12a^{2}z^{6}-3a^{2}z^{4}+7a^{2}z^{2}+a^{2}z^{-2}-4a^{2}+10az^{7}-15az^{5}+z^{5}a^{-1}+3az^{3}+5z^{6}-5z^{4}$ (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 1 2 3 χ 5                       1 -1 3                     4   4 1                   6 1   -5 -1                 9 4     5 -3               10 7       -3 -5             11 8         3 -7           8 10           2 -9         8 11             -3 -11       5 11               6 -13     1 5                 -4 -15   1 5                   4 -17   1                     -1 -19 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-3$ $i=-1$ $r=-8$ ${\mathbb {Z} }$ $r=-7$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=-4$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{8}$ $r=-3$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=-2$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=-1$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=0$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{9}$ $r=1$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=2$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=3$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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.

L11a389 L11a391