L11a420

L11a419

L11a421

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a420 at Knotilus!

Link Presentations

[edit Notes on L11a420's Link Presentations]

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

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {t(1)t(2)^{2}t(3)^{4}-t(2)^{2}t(3)^{4}-t(1)t(2)t(3)^{4}-2t(1)t(2)^{2}t(3)^{3}+3t(2)^{2}t(3)^{3}-t(1)t(3)^{3}+3t(1)t(2)t(3)^{3}-2t(2)t(3)^{3}+2t(1)t(2)^{2}t(3)^{2}-3t(2)^{2}t(3)^{2}+3t(1)t(3)^{2}-4t(1)t(2)t(3)^{2}+4t(2)t(3)^{2}-2t(3)^{2}+t(2)^{2}t(3)-3t(1)t(3)+2t(1)t(2)t(3)-3t(2)t(3)+2t(3)+t(1)+t(2)-1}{{\sqrt {t(1)}}t(2)t(3)^{2}}}$ (db)
Jones polynomial	$q^{7}-3q^{6}+7q^{5}-10q^{4}-q^{-4}+14q^{3}+2q^{-3}-14q^{2}-5q^{-2}+15q+9q^{-1}-11$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$-z^{8}a^{-2}-6z^{6}a^{-2}+z^{6}a^{-4}+2z^{6}-a^{2}z^{4}-15z^{4}a^{-2}+4z^{4}a^{-4}+10z^{4}-4a^{2}z^{2}-21z^{2}a^{-2}+6z^{2}a^{-4}+18z^{2}-4a^{2}-16a^{-2}+5a^{-4}+15-a^{2}z^{-2}-5a^{-2}z^{-2}+2a^{-4}z^{-2}+4z^{-2}$ (db)
Kauffman polynomial	$z^{10}a^{-2}+z^{10}+2az^{9}+6z^{9}a^{-1}+4z^{9}a^{-3}+2a^{2}z^{8}+9z^{8}a^{-2}+7z^{8}a^{-4}+4z^{8}+a^{3}z^{7}-3az^{7}-12z^{7}a^{-1}-z^{7}a^{-3}+7z^{7}a^{-5}-8a^{2}z^{6}-35z^{6}a^{-2}-12z^{6}a^{-4}+6z^{6}a^{-6}-25z^{6}-5a^{3}z^{5}-12az^{5}-11z^{5}a^{-1}-15z^{5}a^{-3}-8z^{5}a^{-5}+3z^{5}a^{-7}+10a^{2}z^{4}+47z^{4}a^{-2}+11z^{4}a^{-4}-8z^{4}a^{-6}+z^{4}a^{-8}+37z^{4}+8a^{3}z^{3}+28az^{3}+39z^{3}a^{-1}+23z^{3}a^{-3}+2z^{3}a^{-5}-2z^{3}a^{-7}-7a^{2}z^{2}-39z^{2}a^{-2}-9z^{2}a^{-4}+6z^{2}a^{-6}-z^{2}a^{-8}-30z^{2}-5a^{3}z-21az-33za^{-1}-16za^{-3}+za^{-5}+4a^{2}+20a^{-2}+6a^{-4}-2a^{-6}+17+a^{3}z^{-1}+5az^{-1}+9a^{-1}z^{-1}+5a^{-3}z^{-1}-a^{2}z^{-2}-5a^{-2}z^{-2}-2a^{-4}z^{-2}-4z^{-2}$ (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 5 6 χ 15                       1 1 13                     3 1 -2 11                   4     4 9                 6 3     -3 7               8 4       4 5             7 7         0 3           8 7           1 1         5 9             4 -1       4 6               -2 -3     1 5                 4 -5   1 4                   -3 -7   1                     1 -9 1                       -1

Integral Khovanov Homology (db, data source)

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

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.

L11a419 L11a421