L11a398

L11a397

L11a399

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a398 at Knotilus!

Link Presentations

[edit Notes on L11a398's Link Presentations]

Planar diagram presentation	X6172 X12,6,13,5 X8493 X2,14,3,13 X14,7,15,8 X18,10,19,9 X22,17,11,18 X20,11,21,12 X16,21,17,22 X4,15,1,16 X10,20,5,19
Gauss code	{1, -4, 3, -10}, {2, -1, 5, -3, 6, -11}, {8, -2, 4, -5, 10, -9, 7, -6, 11, -8, 9, -7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(t(1)-1)(t(2)-1)(t(3)-1)\left(t(2)t(3)^{3}-2t(2)t(3)^{2}+2t(3)^{2}+2t(2)t(3)-2t(3)+1\right)}{{\sqrt {t(1)}}t(2)t(3)^{2}}}$ (db)
Jones polynomial	$q^{-6}-q^{5}-5q^{-5}+4q^{4}+10q^{-4}-9q^{3}-17q^{-3}+17q^{2}+23q^{-2}-21q-25q^{-1}+27$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$z^{8}-2a^{2}z^{6}-z^{6}a^{-2}+5z^{6}+a^{4}z^{4}-6a^{2}z^{4}-3z^{4}a^{-2}+10z^{4}+a^{4}z^{2}-4a^{2}z^{2}-3z^{2}a^{-2}+6z^{2}-a^{4}+4a^{2}+a^{-2}-4-a^{4}z^{-2}+4a^{2}z^{-2}+2a^{-2}z^{-2}-5z^{-2}$ (db)
Kauffman polynomial	$a^{6}z^{6}-a^{6}z^{4}+5a^{5}z^{7}-10a^{5}z^{5}+z^{5}a^{-5}+5a^{5}z^{3}-z^{3}a^{-5}+a^{5}z-a^{5}z^{-1}+9a^{4}z^{8}-19a^{4}z^{6}+4z^{6}a^{-4}+12a^{4}z^{4}-5z^{4}a^{-4}-4a^{4}z^{2}+2z^{2}a^{-4}+a^{4}z^{-2}+7a^{3}z^{9}-a^{3}z^{7}+8z^{7}a^{-3}-25a^{3}z^{5}-10z^{5}a^{-3}+17a^{3}z^{3}+4z^{3}a^{-3}+5a^{3}z-5a^{3}z^{-1}+2a^{2}z^{10}+22a^{2}z^{8}+11z^{8}a^{-2}-63a^{2}z^{6}-17z^{6}a^{-2}+53a^{2}z^{4}+14z^{4}a^{-2}-18a^{2}z^{2}-5z^{2}a^{-2}+4a^{2}z^{-2}+2a^{-2}z^{-2}-2a^{2}-3a^{-2}+15az^{9}+8z^{9}a^{-1}-15az^{7}-z^{7}a^{-1}-19az^{5}-15z^{5}a^{-1}+17az^{3}+10z^{3}a^{-1}+9az+5za^{-1}-9az^{-1}-5a^{-1}z^{-1}+2z^{10}+24z^{8}-64z^{6}+59z^{4}-21z^{2}+5z^{-2}-4$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 11                       1 -1 9                     3   3 7                   6 1   -5 5                 11 3     8 3               12 8       -4 1             15 9         6 -1           12 14           2 -3         11 13             -2 -5       6 12               6 -7     4 11                 -7 -9   1 6                   5 -11   4                     -4 -13 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-1$ $i=1$ $r=-6$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-3$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=-2$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=-1$ ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=0$ ${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{13}$ ${\mathbb {Z} }^{15}$ $r=1$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=2$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{11}$ $r=3$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=4$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=5$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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.

L11a397 L11a399