L11n317

L11n316

L11n318

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n317 at Knotilus!

Link Presentations

[edit Notes on L11n317's Link Presentations]

Planar diagram presentation	X6172 X12,4,13,3 X16,7,17,8 X9,20,10,21 X11,18,12,19 X19,22,20,11 X8,15,9,16 X21,10,22,5 X14,18,15,17 X2536 X4,14,1,13
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -7, -4, 8}, {-5, -2, 11, -9, 7, -3, 9, 5, -6, 4, -8, 6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {uv^{2}w^{2}-2uv^{2}w+uv^{2}-3uvw^{2}+5uvw-2uv+uw^{2}-3uw+u-v^{2}w^{2}+3v^{2}w-v^{2}+2vw^{2}-5vw+3v-w^{2}+2w-1}{{\sqrt {u}}vw}}$ (db)
Jones polynomial	$-q^{2}+4q-7+11q^{-1}-12q^{-2}+14q^{-3}-11q^{-4}+9q^{-5}-5q^{-6}+2q^{-7}$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$2a^{6}z^{2}+a^{6}z^{-2}+2a^{6}-3a^{4}z^{4}-7a^{4}z^{2}-2a^{4}z^{-2}-6a^{4}+a^{2}z^{6}+3a^{2}z^{4}+5a^{2}z^{2}+a^{2}z^{-2}+4a^{2}-z^{4}-z^{2}$ (db)
Kauffman polynomial	$3a^{8}z^{4}-4a^{8}z^{2}+a^{8}+a^{7}z^{7}+3a^{7}z^{5}-4a^{7}z^{3}+2a^{6}z^{8}+a^{6}z^{6}-3a^{6}z^{4}+4a^{6}z^{2}+a^{6}z^{-2}-3a^{6}+a^{5}z^{9}+5a^{5}z^{7}-5a^{5}z^{5}-4a^{5}z^{3}+6a^{5}z-2a^{5}z^{-1}+6a^{4}z^{8}-2a^{4}z^{6}-13a^{4}z^{4}+16a^{4}z^{2}+2a^{4}z^{-2}-8a^{4}+a^{3}z^{9}+10a^{3}z^{7}-19a^{3}z^{5}+4a^{3}z^{3}+6a^{3}z-2a^{3}z^{-1}+4a^{2}z^{8}+a^{2}z^{6}-14a^{2}z^{4}+11a^{2}z^{2}+a^{2}z^{-2}-5a^{2}+6az^{7}-10az^{5}+z^{5}a^{-1}+3az^{3}-z^{3}a^{-1}+4z^{6}-7z^{4}+3z^{2}$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 χ 5                   1 -1 3                 3   3 1               4 1   -3 -1             7 3     4 -3           7 6       -1 -5         7 5         2 -7       5 8           3 -9     4 6             -2 -11   1 5               4 -13 1 4                 -3 -15 2                   2

Integral Khovanov Homology (db, data source)

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

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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