L11n382

L11n381

L11n383

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n382 at Knotilus!

Link Presentations

[edit Notes on L11n382's Link Presentations]

Planar diagram presentation	X6172 X14,7,15,8 X4,15,1,16 X5,10,6,11 X8493 X22,18,19,17 X20,12,21,11 X12,20,13,19 X18,22,5,21 X9,16,10,17 X2,14,3,13
Gauss code	{1, -11, 5, -3}, {8, -7, 9, -6}, {-4, -1, 2, -5, -10, 4, 7, -8, 11, -2, 3, 10, 6, -9}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(t(1)-1)(t(3)-1)^{2}(t(2)+t(3))}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)^{3/2}}}$ (db)
Jones polynomial	$2q^{4}-2q^{3}-2q^{-3}+6q^{2}+4q^{-2}-5q-5q^{-1}+6$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$2a^{-4}z^{-2}+2a^{-4}-2a^{2}z^{2}-4z^{2}a^{-2}-a^{2}z^{-2}-5a^{-2}z^{-2}-2a^{2}-8a^{-2}+2z^{4}+6z^{2}+4z^{-2}+8$ (db)
Kauffman polynomial	$3z^{4}a^{-4}-9z^{2}a^{-4}-2a^{-4}z^{-2}+8a^{-4}+z^{7}a^{-3}-3z^{5}a^{-3}+3a^{3}z^{3}+9z^{3}a^{-3}-3a^{3}z-13za^{-3}+a^{3}z^{-1}+5a^{-3}z^{-1}+z^{8}a^{-2}+a^{2}z^{6}-3z^{6}a^{-2}+2a^{2}z^{4}+11z^{4}a^{-2}-4a^{2}z^{2}-20z^{2}a^{-2}-a^{2}z^{-2}-5a^{-2}z^{-2}+2a^{2}+16a^{-2}+2az^{7}+3z^{7}a^{-1}-5az^{5}-8z^{5}a^{-1}+13az^{3}+19z^{3}a^{-1}-14az-24za^{-1}+5az^{-1}+9a^{-1}z^{-1}+z^{8}-2z^{6}+10z^{4}-15z^{2}-4z^{-2}+11$ (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   \ -3 -2 -1 0 1 2 3 4 χ 9               2 2 7             2 2 0 5           4     4 3         1 2     1 1       5 4       1 -1     2 3         1 -3   2 3           -1 -5   2             2 -7 2               -2

Integral Khovanov Homology (db, data source)

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

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.

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