L11n18

L11n17

L11n19

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n18 at Knotilus!

Link Presentations

[edit Notes on L11n18's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(u-1)(v-1)\left(2v^{2}-3v+2\right)}{{\sqrt {u}}v^{3/2}}}$ (db)
Jones polynomial	$-{\frac {2}{q^{3/2}}}+{\frac {3}{q^{5/2}}}-{\frac {7}{q^{7/2}}}+{\frac {8}{q^{9/2}}}-{\frac {10}{q^{11/2}}}+{\frac {9}{q^{13/2}}}-{\frac {7}{q^{15/2}}}+{\frac {6}{q^{17/2}}}-{\frac {3}{q^{19/2}}}+{\frac {1}{q^{21/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{9}\left(-z^{3}\right)-a^{9}z-a^{9}z^{-1}+a^{7}z^{5}+2a^{7}z^{3}+2a^{7}z+a^{7}z^{-1}+a^{5}z^{5}+2a^{5}z^{3}+3a^{5}z+2a^{5}z^{-1}-2a^{3}z^{3}-4a^{3}z-2a^{3}z^{-1}$ (db)
Kauffman polynomial	$-z^{6}a^{12}+3z^{4}a^{12}-3z^{2}a^{12}+a^{12}-3z^{7}a^{11}+9z^{5}a^{11}-6z^{3}a^{11}+za^{11}-3z^{8}a^{10}+6z^{6}a^{10}+3z^{4}a^{10}-5z^{2}a^{10}-z^{9}a^{9}-4z^{7}a^{9}+17z^{5}a^{9}-12z^{3}a^{9}+2za^{9}+a^{9}z^{-1}-5z^{8}a^{8}+10z^{6}a^{8}-3z^{4}a^{8}+2z^{2}a^{8}-3a^{8}-z^{9}a^{7}-3z^{7}a^{7}+11z^{5}a^{7}-11z^{3}a^{7}+2za^{7}+a^{7}z^{-1}-2z^{8}a^{6}+2z^{6}a^{6}-3z^{4}a^{6}+z^{2}a^{6}-2z^{7}a^{5}+3z^{5}a^{5}-8z^{3}a^{5}+6za^{5}-2a^{5}z^{-1}-z^{6}a^{4}-3z^{2}a^{4}+3a^{4}-3z^{3}a^{3}+5za^{3}-2a^{3}z^{-1}$ (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 χ -2                   2 2 -4                 3 2 -1 -6               4     4 -8             4 3     -1 -10           6 4       2 -12         3 4         1 -14       4 6           -2 -16     2 3             1 -18   1 4               -3 -20   2                 2 -22 1                   -1

Integral Khovanov Homology (db, data source)

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

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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