L11n207

L11n206

L11n208

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n207 at Knotilus!

Link Presentations

[edit Notes on L11n207's Link Presentations]

Planar diagram presentation	X10,1,11,2 X2,11,3,12 X12,3,13,4 X5,14,6,15 X17,22,18,9 X19,5,20,4 X21,6,22,7 X7,17,8,16 X8,9,1,10 X13,18,14,19 X15,21,16,20
Gauss code	{1, -2, 3, 6, -4, 7, -8, -9}, {9, -1, 2, -3, -10, 4, -11, 8, -5, 10, -6, 11, -7, 5}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {u^{3}v^{3}-u^{3}v^{2}-2u^{2}v^{3}+6u^{2}v^{2}-4u^{2}v+u^{2}+uv^{3}-4uv^{2}+6uv-2u-v+1}{u^{3/2}v^{3/2}}}$ (db)
Jones polynomial	${\frac {10}{q^{9/2}}}-{\frac {10}{q^{7/2}}}+{\frac {6}{q^{5/2}}}-{\frac {4}{q^{3/2}}}-{\frac {1}{q^{19/2}}}+{\frac {3}{q^{17/2}}}-{\frac {6}{q^{15/2}}}+{\frac {9}{q^{13/2}}}-{\frac {10}{q^{11/2}}}+{\frac {1}{\sqrt {q}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$z^{5}a^{7}+3z^{3}a^{7}+3za^{7}-z^{7}a^{5}-5z^{5}a^{5}-9z^{3}a^{5}-5za^{5}+a^{5}z^{-1}+z^{5}a^{3}+2z^{3}a^{3}-a^{3}z^{-1}$ (db)
Kauffman polynomial	$-z^{5}a^{11}+2z^{3}a^{11}-za^{11}-3z^{6}a^{10}+6z^{4}a^{10}-3z^{2}a^{10}-4z^{7}a^{9}+6z^{5}a^{9}-za^{9}-3z^{8}a^{8}+2z^{6}a^{8}+2z^{4}a^{8}+z^{2}a^{8}-z^{9}a^{7}-3z^{7}a^{7}+2z^{5}a^{7}+6z^{3}a^{7}-4za^{7}-4z^{8}a^{6}+5z^{6}a^{6}-6z^{4}a^{6}+5z^{2}a^{6}-z^{9}a^{5}+z^{7}a^{5}-9z^{5}a^{5}+13z^{3}a^{5}-5za^{5}-a^{5}z^{-1}-z^{8}a^{4}-3z^{4}a^{4}+2z^{2}a^{4}+a^{4}-4z^{5}a^{3}+5z^{3}a^{3}-za^{3}-a^{3}z^{-1}-z^{4}a^{2}+z^{2}a^{2}$ (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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 χ 0                   1 -1 -2                 3   3 -4               4 2   -2 -6             6 2     4 -8           4 4       0 -10         6 6         0 -12       4 5           1 -14     2 5             -3 -16   1 4               3 -18   2                 -2 -20 1                   1

Integral Khovanov Homology (db, data source)

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

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.

L11n206 L11n208