L11a468

L11a467

L11a469

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a468 at Knotilus!

Link Presentations

[edit Notes on L11a468's Link Presentations]

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(t(3)-1)\left(t(2)^{2}t(3)^{3}-t(2)t(3)^{3}+2t(1)t(2)^{2}t(3)^{2}-2t(2)^{2}t(3)^{2}-2t(1)t(2)t(3)^{2}+2t(2)t(3)^{2}-t(3)^{2}-t(1)t(2)^{2}t(3)-2t(1)t(3)+2t(1)t(2)t(3)-2t(2)t(3)+2t(3)+t(1)-t(1)t(2)\right)}{{\sqrt {t(1)}}t(2)t(3)^{2}}}$ (db)
Jones polynomial	$1-2q^{-1}+5q^{-2}-8q^{-3}+12q^{-4}-13q^{-5}+15q^{-6}-12q^{-7}+10q^{-8}-6q^{-9}+3q^{-10}-q^{-11}$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$a^{10}\left(-z^{2}\right)-2a^{10}+3a^{8}z^{4}+9a^{8}z^{2}+a^{8}z^{-2}+6a^{8}-2a^{6}z^{6}-8a^{6}z^{4}-11a^{6}z^{2}-2a^{6}z^{-2}-8a^{6}-a^{4}z^{6}-2a^{4}z^{4}+2a^{4}z^{2}+a^{4}z^{-2}+3a^{4}+a^{2}z^{4}+3a^{2}z^{2}+a^{2}$ (db)
Kauffman polynomial	$z^{5}a^{13}-2z^{3}a^{13}+za^{13}+3z^{6}a^{12}-6z^{4}a^{12}+3z^{2}a^{12}-a^{12}+4z^{7}a^{11}-5z^{5}a^{11}-2z^{3}a^{11}+za^{11}+4z^{8}a^{10}-4z^{6}a^{10}-2z^{4}a^{10}+z^{2}a^{10}+3z^{9}a^{9}-2z^{7}a^{9}-4z^{5}a^{9}+8z^{3}a^{9}-3za^{9}+z^{10}a^{8}+6z^{8}a^{8}-25z^{6}a^{8}+42z^{4}a^{8}-29z^{2}a^{8}-a^{8}z^{-2}+10a^{8}+6z^{9}a^{7}-16z^{7}a^{7}+14z^{5}a^{7}+7z^{3}a^{7}-10za^{7}+2a^{7}z^{-1}+z^{10}a^{6}+5z^{8}a^{6}-28z^{6}a^{6}+51z^{4}a^{6}-40z^{2}a^{6}-2a^{6}z^{-2}+14a^{6}+3z^{9}a^{5}-8z^{7}a^{5}+6z^{5}a^{5}+2z^{3}a^{5}-7za^{5}+2a^{5}z^{-1}+3z^{8}a^{4}-9z^{6}a^{4}+9z^{4}a^{4}-9z^{2}a^{4}-a^{4}z^{-2}+5a^{4}+2z^{7}a^{3}-6z^{5}a^{3}+3z^{3}a^{3}+z^{6}a^{2}-4z^{4}a^{2}+4z^{2}a^{2}-a^{2}$ (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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 1                       1 1 -1                     1   -1 -3                   4 1   3 -5                 5 2     -3 -7               7 3       4 -9             7 6         -1 -11           8 6           2 -13         5 8             3 -15       5 7               -2 -17     2 6                 4 -19   1 4                   -3 -21   2                     2 -23 1                       -1

Integral Khovanov Homology (db, data source)

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

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.

L11a467 L11a469