L11a448

L11a447

L11a449

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a448 at Knotilus!

Link Presentations

[edit Notes on L11a448's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {t(1)t(3)^{3}t(2)^{3}-t(3)^{3}t(2)^{3}-t(1)t(3)^{2}t(2)^{3}+2t(3)^{2}t(2)^{3}-t(3)t(2)^{3}-2t(1)t(3)^{3}t(2)^{2}+t(3)^{3}t(2)^{2}+2t(1)t(3)^{2}t(2)^{2}-2t(3)^{2}t(2)^{2}-t(1)t(3)t(2)^{2}+2t(3)t(2)^{2}-t(2)^{2}+t(1)t(3)^{3}t(2)-2t(1)t(3)^{2}t(2)+t(3)^{2}t(2)-t(1)t(2)+2t(1)t(3)t(2)-2t(3)t(2)+2t(2)+t(1)t(3)^{2}+t(1)-2t(1)t(3)+t(3)-1}{{\sqrt {t(1)}}t(2)^{3/2}t(3)^{3/2}}}$ (db)
Jones polynomial	$-q^{9}+3q^{8}-5q^{7}+7q^{6}-9q^{5}+11q^{4}-9q^{3}+9q^{2}+q^{-2}-6q-2q^{-1}+5$ (db)
Signature	4 (db)
HOMFLY-PT polynomial	$-z^{6}a^{-6}-4z^{4}a^{-6}-4z^{2}a^{-6}-2a^{-6}+z^{8}a^{-4}+6z^{6}a^{-4}+13z^{4}a^{-4}+14z^{2}a^{-4}+a^{-4}z^{-2}+7a^{-4}-2z^{6}a^{-2}-10z^{4}a^{-2}-15z^{2}a^{-2}-2a^{-2}z^{-2}-9a^{-2}+z^{4}+4z^{2}+z^{-2}+4$ (db)
Kauffman polynomial	$z^{10}a^{-2}+z^{10}a^{-4}+2z^{9}a^{-1}+6z^{9}a^{-3}+4z^{9}a^{-5}+z^{8}a^{-2}+6z^{8}a^{-4}+6z^{8}a^{-6}+z^{8}-10z^{7}a^{-1}-25z^{7}a^{-3}-9z^{7}a^{-5}+6z^{7}a^{-7}-25z^{6}a^{-2}-39z^{6}a^{-4}-14z^{6}a^{-6}+6z^{6}a^{-8}-6z^{6}+14z^{5}a^{-1}+24z^{5}a^{-3}-3z^{5}a^{-5}-8z^{5}a^{-7}+5z^{5}a^{-9}+54z^{4}a^{-2}+56z^{4}a^{-4}+5z^{4}a^{-6}-7z^{4}a^{-8}+3z^{4}a^{-10}+13z^{4}-3z^{3}a^{-1}+4z^{3}a^{-3}+9z^{3}a^{-5}-3z^{3}a^{-7}-4z^{3}a^{-9}+z^{3}a^{-11}-41z^{2}a^{-2}-31z^{2}a^{-4}+2z^{2}a^{-8}-z^{2}a^{-10}-13z^{2}-5za^{-1}-8za^{-3}-3za^{-5}+za^{-7}+za^{-9}+13a^{-2}+9a^{-4}-a^{-8}+6+2a^{-1}z^{-1}+2a^{-3}z^{-1}-2a^{-2}z^{-2}-a^{-4}z^{-2}-z^{-2}$ (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   \ -4 -3 -2 -1 0 1 2 3 4 5 6 7 χ 19                       1 -1 17                     2   2 15                   3 1   -2 13                 4 2     2 11               6 4       -2 9             5 3         2 7           4 6           2 5         5 5             0 3       4 7               3 1     1 2                 -1 -1   1 4                   3 -3   1                     -1 -5 1                       1

Integral Khovanov Homology (db, data source)

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

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.

L11a447 L11a449