L11n411

L11n410

L11n412

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n411 at Knotilus!

Link Presentations

[edit Notes on L11n411's Link Presentations]

Planar diagram presentation	X6172 X12,3,13,4 X13,22,14,19 X7,20,8,21 X19,10,20,11 X9,16,10,17 X17,14,18,15 X15,8,16,9 X21,18,22,5 X2536 X4,11,1,12
Gauss code	{1, -10, 2, -11}, {-5, 4, -9, 3}, {10, -1, -4, 8, -6, 5, 11, -2, -3, 7, -8, 6, -7, 9}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(w-1)(w+1)\left(2uvw^{2}-2uvw+uv+w^{3}-2w^{2}+2w\right)}{{\sqrt {u}}{\sqrt {v}}w^{5/2}}}$ (db)
Jones polynomial	$-q^{-13}+2q^{-12}-2q^{-11}+q^{-10}+q^{-8}+2q^{-7}-q^{-6}+2q^{-5}-q^{-4}+q^{-3}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$-a^{14}z^{-2}+4a^{12}z^{-2}+5a^{12}-z^{4}a^{10}-8z^{2}a^{10}-5a^{10}z^{-2}-12a^{10}+z^{6}a^{8}+5z^{4}a^{8}+6z^{2}a^{8}+2a^{8}z^{-2}+5a^{8}+z^{6}a^{6}+5z^{4}a^{6}+6z^{2}a^{6}+2a^{6}$ (db)
Kauffman polynomial	$a^{15}z^{7}-5a^{15}z^{5}+6a^{15}z^{3}-4a^{15}z+a^{15}z^{-1}+2a^{14}z^{8}-11a^{14}z^{6}+15a^{14}z^{4}-8a^{14}z^{2}-a^{14}z^{-2}+3a^{14}+a^{13}z^{9}-3a^{13}z^{7}-10a^{13}z^{5}+30a^{13}z^{3}-21a^{13}z+5a^{13}z^{-1}+4a^{12}z^{8}-26a^{12}z^{6}+47a^{12}z^{4}-35a^{12}z^{2}-4a^{12}z^{-2}+16a^{12}+a^{11}z^{9}-2a^{11}z^{7}-20a^{11}z^{5}+54a^{11}z^{3}-39a^{11}z+9a^{11}z^{-1}+3a^{10}z^{8}-21a^{10}z^{6}+44a^{10}z^{4}-43a^{10}z^{2}-5a^{10}z^{-2}+21a^{10}+3a^{9}z^{7}-19a^{9}z^{5}+32a^{9}z^{3}-22a^{9}z+5a^{9}z^{-1}+a^{8}z^{8}-5a^{8}z^{6}+7a^{8}z^{4}-10a^{8}z^{2}-2a^{8}z^{-2}+7a^{8}+a^{7}z^{7}-4a^{7}z^{5}+2a^{7}z^{3}+a^{6}z^{6}-5a^{6}z^{4}+6a^{6}z^{2}-2a^{6}$ (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   \ -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -5                       1 1 -7                     1 1 0 -9                 1 2     1 -11               1 1 1     1 -13             2 5 2       1 -15           2 2 3         3 -17         2 5 2           1 -19       2 2 1             1 -21     1 3 3               -1 -23   1 1                   0 -25   1                     1 -27 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-7$ $i=-5$ $i=-3$ $r=-11$ ${\mathbb {Z} }$ $r=-10$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-9$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-8$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }^{2}$ $r=-7$ ${\mathbb {Z} }^{3}$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-6$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-5$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-1$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=0$ ${\mathbb {Z} }$ ${\mathbb {Z} }$

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.

L11n410 L11n412