L11n287

L11n286

L11n288

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n287 at Knotilus!

Link Presentations

[edit Notes on L11n287's Link Presentations]

Planar diagram presentation	X6172 X5,12,6,13 X8493 X2,14,3,13 X14,7,15,8 X4,15,1,16 X21,18,22,19 X9,21,10,20 X19,5,20,10 X11,16,12,17 X17,22,18,11
Gauss code	{1, -4, 3, -6}, {-2, -1, 5, -3, -8, 9}, {-10, 2, 4, -5, 6, 10, -11, 7, -9, 8, -7, 11}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(u-1)(v-1)(w-1)}{{\sqrt {u}}{\sqrt {v}}{\sqrt {w}}}}$ (db)
Jones polynomial	$q^{2}+2-q^{-1}+3q^{-2}-2q^{-3}+2q^{-4}-2q^{-5}+2q^{-6}-q^{-7}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a^{6}\left(-z^{2}\right)-a^{6}+a^{4}z^{4}+3a^{4}z^{2}+2a^{4}-a^{2}z^{2}+a^{2}z^{-2}+a^{-2}z^{-2}+a^{-2}-z^{2}-2z^{-2}-2$ (db)
Kauffman polynomial	$a^{5}z^{9}+a^{3}z^{9}+2a^{6}z^{8}+3a^{4}z^{8}+a^{2}z^{8}+a^{7}z^{7}-4a^{5}z^{7}-5a^{3}z^{7}-11a^{6}z^{6}-17a^{4}z^{6}-6a^{2}z^{6}-5a^{7}z^{5}+6a^{3}z^{5}+az^{5}+16a^{6}z^{4}+28a^{4}z^{4}+12a^{2}z^{4}+6a^{7}z^{3}+8a^{5}z^{3}-2a^{3}z^{3}-4az^{3}-7a^{6}z^{2}-18a^{4}z^{2}-10a^{2}z^{2}+z^{2}a^{-2}+2z^{2}-2a^{7}z-4a^{5}z+4az+2za^{-1}+2a^{6}+4a^{4}-2a^{-2}-3-2az^{-1}-2a^{-1}z^{-1}+a^{2}z^{-2}+a^{-2}z^{-2}+2z^{-2}$ (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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 5                   1 1 3                   1 1 1               3 1   2 -1             2 4 1   1 -3           2 2 2     2 -5         2 3         1 -7       1 2 1         0 -9     1 2 1           0 -11   1 1               0 -13   1                 1 -15 1                   -1

Integral Khovanov Homology (db, data source)

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

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.

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