L11n389

L11n388

L11n390

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n389 at Knotilus!

Link Presentations

[edit Notes on L11n389's Link Presentations]

Planar diagram presentation	X6172 X16,7,17,8 X4,17,1,18 X5,12,6,13 X3849 X13,22,14,19 X9,20,10,21 X19,10,20,11 X21,14,22,15 X11,18,12,5 X15,2,16,3
Gauss code	{1, 11, -5, -3}, {-8, 7, -9, 6}, {-4, -1, 2, 5, -7, 8, -10, 4, -6, 9, -11, -2, 3, 10}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(t(3)-1)\left(t(1)t(3)^{4}-t(1)t(3)^{3}+2t(1)t(2)t(3)^{3}+t(1)t(3)^{2}-t(1)t(2)t(3)^{2}+t(2)t(3)^{2}-t(3)^{2}-t(2)t(3)+2t(3)+t(2)\right)}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)^{5/2}}}$ (db)
Jones polynomial	$q^{-3}-2q^{-4}+5q^{-5}-6q^{-6}+9q^{-7}-7q^{-8}+8q^{-9}-6q^{-10}+3q^{-11}-q^{-12}$ (db)
Signature	-6 (db)
HOMFLY-PT polynomial	$-a^{14}z^{-2}+z^{2}a^{12}+4a^{12}z^{-2}+6a^{12}-3z^{4}a^{10}-13z^{2}a^{10}-5a^{10}z^{-2}-15a^{10}+2z^{6}a^{8}+9z^{4}a^{8}+12z^{2}a^{8}+2a^{8}z^{-2}+8a^{8}+z^{6}a^{6}+4z^{4}a^{6}+4z^{2}a^{6}+a^{6}$ (db)
Kauffman polynomial	$z^{3}a^{15}-2za^{15}+a^{15}z^{-1}+3z^{4}a^{14}-3z^{2}a^{14}-a^{14}z^{-2}+2a^{14}+2z^{7}a^{13}-7z^{5}a^{13}+18z^{3}a^{13}-15za^{13}+5a^{13}z^{-1}+3z^{8}a^{12}-12z^{6}a^{12}+26z^{4}a^{12}-25z^{2}a^{12}-4a^{12}z^{-2}+14a^{12}+z^{9}a^{11}+3z^{7}a^{11}-22z^{5}a^{11}+44z^{3}a^{11}-33za^{11}+9a^{11}z^{-1}+6z^{8}a^{10}-24z^{6}a^{10}+42z^{4}a^{10}-42z^{2}a^{10}-5a^{10}z^{-2}+21a^{10}+z^{9}a^{9}+3z^{7}a^{9}-21z^{5}a^{9}+30z^{3}a^{9}-20za^{9}+5a^{9}z^{-1}+3z^{8}a^{8}-11z^{6}a^{8}+15z^{4}a^{8}-16z^{2}a^{8}-2a^{8}z^{-2}+9a^{8}+2z^{7}a^{7}-6z^{5}a^{7}+3z^{3}a^{7}+z^{6}a^{6}-4z^{4}a^{6}+4z^{2}a^{6}-a^{6}$ (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 χ -5                   1 1 -7                 2 1 -1 -9               3     3 -11             3 2     -1 -13           6 3       3 -15         3 5         2 -17       5 4           1 -19     1 3             2 -21   2 5               -3 -23   2                 2 -25 1                   -1

Integral Khovanov Homology (db, data source)

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

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.

L11n388 L11n390