L11n368

L11n367

L11n369

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n368 at Knotilus!

Link Presentations

[edit Notes on L11n368's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X7,14,8,15 X15,20,16,21 X11,19,12,18 X17,13,18,12 X19,22,20,17 X21,16,22,5 X13,8,14,9 X2536 X4,9,1,10
Gauss code	{1, -10, 2, -11}, {-6, 5, -7, 4, -8, 7}, {10, -1, -3, 9, 11, -2, -5, 6, -9, 3, -4, 8}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(t(1)+1)(t(2)-1)(t(3)-1)^{2}(t(3)+1)}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)^{3/2}}}$ (db)
Jones polynomial	$1-q^{-1}+3q^{-2}-q^{-3}+q^{-4}+2q^{-7}-2q^{-8}+2q^{-9}-q^{-10}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-a^{10}+2a^{8}z^{2}+2a^{8}+a^{6}z^{2}+a^{6}z^{-2}+a^{6}-a^{4}z^{6}-5a^{4}z^{4}-7a^{4}z^{2}-2a^{4}z^{-2}-6a^{4}+a^{2}z^{4}+4a^{2}z^{2}+a^{2}z^{-2}+4a^{2}$ (db)
Kauffman polynomial	$a^{11}z^{7}-5a^{11}z^{5}+6a^{11}z^{3}-2a^{11}z+2a^{10}z^{8}-11a^{10}z^{6}+16a^{10}z^{4}-9a^{10}z^{2}+2a^{10}+a^{9}z^{9}-4a^{9}z^{7}-2a^{9}z^{5}+11a^{9}z^{3}-4a^{9}z+3a^{8}z^{8}-20a^{8}z^{6}+36a^{8}z^{4}-22a^{8}z^{2}+5a^{8}+a^{7}z^{9}-6a^{7}z^{7}+7a^{7}z^{5}-a^{7}z^{3}+a^{6}z^{8}-7a^{6}z^{6}+11a^{6}z^{4}-6a^{6}z^{2}+a^{6}z^{-2}-a^{6}+a^{5}z^{5}-8a^{5}z^{3}+8a^{5}z-2a^{5}z^{-1}+3a^{4}z^{6}-14a^{4}z^{4}+15a^{4}z^{2}+2a^{4}z^{-2}-8a^{4}+a^{3}z^{7}-3a^{3}z^{5}-2a^{3}z^{3}+6a^{3}z-2a^{3}z^{-1}+a^{2}z^{6}-5a^{2}z^{4}+8a^{2}z^{2}+a^{2}z^{-2}-5a^{2}$ (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                         0 -3                   3 1   2 -5               1 1 2     2 -7             1 2 1       0 -9           2 2 1         1 -11         2 4 2           0 -13       1 1 2             2 -15     1 2 1               0 -17   1 1                   0 -19   1                     1 -21 1                       -1

Integral Khovanov Homology (db, data source)

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

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.

L11n367 L11n369