L11n215

L11n214

L11n216

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n215 at Knotilus!

Link Presentations

[edit Notes on L11n215's Link Presentations]

Planar diagram presentation	X10,1,11,2 X12,3,13,4 X14,5,15,6 X9,18,10,19 X17,22,18,9 X21,1,22,8 X20,15,21,16 X7,16,8,17 X6,20,7,19 X4,11,5,12 X2,13,3,14
Gauss code	{1, -11, 2, -10, 3, -9, -8, 6}, {-4, -1, 10, -2, 11, -3, 7, 8, -5, 4, 9, -7, -6, 5}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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.

L11n214 L11n216