L11a521

L11a520

L11a522

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a521 at Knotilus!

Link Presentations

[edit Notes on L11a521's Link Presentations]

Planar diagram presentation	X8192 X14,4,15,3 X10,14,11,13 X22,10,13,9 X20,17,21,18 X18,8,19,7 X12,20,7,19 X16,6,17,5 X2,11,3,12 X4,16,5,15 X6,22,1,21
Gauss code	{1, -9, 2, -10, 8, -11}, {6, -1, 4, -3, 9, -7}, {3, -2, 10, -8, 5, -6, 7, -5, 11, -4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {u^{2}v^{2}w^{3}-2u^{2}v^{2}w^{2}-2u^{2}vw^{3}+4u^{2}vw^{2}-2u^{2}vw+u^{2}w^{3}-2u^{2}w^{2}+u^{2}w-uv^{2}w^{3}+3uv^{2}w^{2}-2uv^{2}w+2uvw^{3}-5uvw^{2}+5uvw-2uv+2uw^{2}-3uw+u-v^{2}w^{2}+2v^{2}w-v^{2}+2vw^{2}-4vw+2v+2w-1}{uvw^{3/2}}}$ (db)
Jones polynomial	$q^{10}-3q^{9}+7q^{8}-11q^{7}+16q^{6}-17q^{5}+18q^{4}-15q^{3}+12q^{2}-7q+4-q^{-1}$ (db)
Signature	4 (db)
HOMFLY-PT polynomial	$z^{4}a^{-8}+3z^{2}a^{-8}+a^{-8}z^{-2}+2a^{-8}-2z^{6}a^{-6}-8z^{4}a^{-6}-10z^{2}a^{-6}-2a^{-6}z^{-2}-6a^{-6}+z^{8}a^{-4}+5z^{6}a^{-4}+9z^{4}a^{-4}+8z^{2}a^{-4}+a^{-4}z^{-2}+3a^{-4}-z^{6}a^{-2}-3z^{4}a^{-2}-z^{2}a^{-2}+a^{-2}$ (db)
Kauffman polynomial	$z^{4}a^{-12}-z^{2}a^{-12}+3z^{5}a^{-11}-2z^{3}a^{-11}+6z^{6}a^{-10}-7z^{4}a^{-10}+5z^{2}a^{-10}-a^{-10}+8z^{7}a^{-9}-10z^{5}a^{-9}+5z^{3}a^{-9}+9z^{8}a^{-8}-17z^{6}a^{-8}+17z^{4}a^{-8}-12z^{2}a^{-8}-a^{-8}z^{-2}+5a^{-8}+6z^{9}a^{-7}-5z^{7}a^{-7}-12z^{5}a^{-7}+17z^{3}a^{-7}-9za^{-7}+2a^{-7}z^{-1}+2z^{10}a^{-6}+10z^{8}a^{-6}-43z^{6}a^{-6}+50z^{4}a^{-6}-31z^{2}a^{-6}-2a^{-6}z^{-2}+11a^{-6}+11z^{9}a^{-5}-30z^{7}a^{-5}+16z^{5}a^{-5}+7z^{3}a^{-5}-9za^{-5}+2a^{-5}z^{-1}+2z^{10}a^{-4}+5z^{8}a^{-4}-35z^{6}a^{-4}+41z^{4}a^{-4}-17z^{2}a^{-4}-a^{-4}z^{-2}+5a^{-4}+5z^{9}a^{-3}-16z^{7}a^{-3}+12z^{5}a^{-3}-z^{3}a^{-3}+4z^{8}a^{-2}-15z^{6}a^{-2}+16z^{4}a^{-2}-4z^{2}a^{-2}-a^{-2}+z^{7}a^{-1}-3z^{5}a^{-1}+2z^{3}a^{-1}$ (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   \ -3 -2 -1 0 1 2 3 4 5 6 7 8 χ 21                       1 1 19                     3 1 -2 17                   4     4 15                 7 3     -4 13               9 4       5 11             8 7         -1 9           10 9           1 7         7 10             3 5       5 8               -3 3     3 8                 5 1   1 4                   -3 -1   3                     3 -3 1                       -1

Integral Khovanov Homology (db, data source)

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

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.

L11a520 L11a522