L11n438

L11n437

L11n439

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n438 at Knotilus!

Link Presentations

[edit Notes on L11n438's Link Presentations]

Planar diagram presentation	X6172 X2536 X11,19,12,18 X3,11,4,10 X9,1,10,4 X7,15,8,14 X13,5,14,8 X19,13,20,22 X15,21,16,20 X21,17,22,16 X17,9,18,12
Gauss code	{1, -2, -4, 5}, {2, -1, -6, 7}, {-5, 4, -3, 11}, {-7, 6, -9, 10, -11, 3, -8, 9, -10, 8}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {t(1)t(4)^{3}-t(1)t(2)t(4)^{3}+t(3)t(4)^{3}-t(4)^{3}-2t(1)t(4)^{2}+3t(1)t(2)t(4)^{2}-t(2)t(4)^{2}+t(1)t(3)t(4)^{2}-2t(1)t(2)t(3)t(4)^{2}+2t(2)t(3)t(4)^{2}-3t(3)t(4)^{2}+2t(4)^{2}+2t(1)t(4)-3t(1)t(2)t(4)+t(2)t(4)-t(1)t(3)t(4)+2t(1)t(2)t(3)t(4)-2t(2)t(3)t(4)+3t(3)t(4)-2t(4)+t(1)t(2)-t(1)t(2)t(3)+t(2)t(3)-t(3)}{{\sqrt {t(1)}}{\sqrt {t(2)}}{\sqrt {t(3)}}t(4)^{3/2}}}$ (db)
Jones polynomial	$11q^{9/2}-13q^{7/2}+6q^{5/2}-4q^{3/2}+q^{21/2}-2q^{19/2}+6q^{17/2}-10q^{15/2}+12q^{13/2}-15q^{11/2}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$-3z^{5}a^{-5}+4z^{3}a^{-3}-12z^{3}a^{-5}+6z^{3}a^{-7}+9za^{-3}-22za^{-5}+17za^{-7}-4za^{-9}+7a^{-3}z^{-1}-20a^{-5}z^{-1}+20a^{-7}z^{-1}-8a^{-9}z^{-1}+a^{-11}z^{-1}+2a^{-3}z^{-3}-7a^{-5}z^{-3}+9a^{-7}z^{-3}-5a^{-9}z^{-3}+a^{-11}z^{-3}$ (db)
Kauffman polynomial	$z^{6}a^{-12}-4z^{4}a^{-12}+6z^{2}a^{-12}+a^{-12}z^{-2}-4a^{-12}+2z^{7}a^{-11}-5z^{5}a^{-11}+4z^{3}a^{-11}-a^{-11}z^{-3}-2za^{-11}+2a^{-11}z^{-1}+2z^{8}a^{-10}+2z^{6}a^{-10}-21z^{4}a^{-10}+34z^{2}a^{-10}+7a^{-10}z^{-2}-24a^{-10}+z^{9}a^{-9}+8z^{7}a^{-9}-25z^{5}a^{-9}+22z^{3}a^{-9}-5a^{-9}z^{-3}-16za^{-9}+12a^{-9}z^{-1}+7z^{8}a^{-8}-46z^{4}a^{-8}+75z^{2}a^{-8}+18a^{-8}z^{-2}-58a^{-8}+z^{9}a^{-7}+16z^{7}a^{-7}-46z^{5}a^{-7}+50z^{3}a^{-7}-9a^{-7}z^{-3}-37za^{-7}+24a^{-7}z^{-1}+5z^{8}a^{-6}+5z^{6}a^{-6}-41z^{4}a^{-6}+73z^{2}a^{-6}+19a^{-6}z^{-2}-60a^{-6}+10z^{7}a^{-5}-26z^{5}a^{-5}+42z^{3}a^{-5}-7a^{-5}z^{-3}-39za^{-5}+23a^{-5}z^{-1}+6z^{6}a^{-4}-12z^{4}a^{-4}+26z^{2}a^{-4}+7a^{-4}z^{-2}-23a^{-4}+10z^{3}a^{-3}-2a^{-3}z^{-3}-16za^{-3}+9a^{-3}z^{-1}$ (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   \ 0 1 2 3 4 5 6 7 8 9 χ 22                   1 -1 20                 1   1 18               5 1   -4 16             5 1     4 14           7 5       -2 12         8 5         3 10       7 11           4 8     6 4             2 6     7               7 4 4 6                 -2 2 4                   4

Integral Khovanov Homology (db, data source)

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

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.

L11n437 L11n439