L11n222

L11n221

L11n223

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n222 at Knotilus!

Link Presentations

[edit Notes on L11n222's Link Presentations]

Planar diagram presentation	X10,1,11,2 X2,11,3,12 X12,3,13,4 X13,20,14,21 X5,14,6,15 X4,21,5,22 X16,9,17,10 X22,15,9,16 X17,6,18,7 X7,18,8,19 X19,8,20,1
Gauss code	{1, -2, 3, -6, -5, 9, -10, 11}, {7, -1, 2, -3, -4, 5, 8, -7, -9, 10, -11, 4, 6, -8}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(uv+1)\left(u^{2}v^{4}-uv^{3}+2uv^{2}-uv+1\right)}{u^{3/2}v^{5/2}}}$ (db)
Jones polynomial	$-{\frac {1}{q^{9/2}}}-{\frac {1}{q^{27/2}}}+{\frac {2}{q^{25/2}}}-{\frac {2}{q^{23/2}}}+{\frac {2}{q^{21/2}}}-{\frac {2}{q^{19/2}}}+{\frac {1}{q^{17/2}}}-{\frac {1}{q^{13/2}}}$ (db)
Signature	-7 (db)
HOMFLY-PT polynomial	$a^{13}\left(-z^{3}\right)-2a^{13}z-a^{13}z^{-1}+a^{11}z^{7}+8a^{11}z^{5}+19a^{11}z^{3}+15a^{11}z+3a^{11}z^{-1}-a^{9}z^{9}-9a^{9}z^{7}-28a^{9}z^{5}-36a^{9}z^{3}-17a^{9}z-2a^{9}z^{-1}$ (db)
Kauffman polynomial	$-z^{3}a^{17}+za^{17}-2z^{4}a^{16}+3z^{2}a^{16}-z^{5}a^{15}+za^{15}-2z^{4}a^{14}+2z^{2}a^{14}-a^{14}-z^{3}a^{13}-za^{13}+a^{13}z^{-1}-z^{8}a^{12}+8z^{6}a^{12}-19z^{4}a^{12}+14z^{2}a^{12}-3a^{12}-z^{9}a^{11}+9z^{7}a^{11}-27z^{5}a^{11}+34z^{3}a^{11}-18za^{11}+3a^{11}z^{-1}-z^{8}a^{10}+8z^{6}a^{10}-19z^{4}a^{10}+15z^{2}a^{10}-3a^{10}-z^{9}a^{9}+9z^{7}a^{9}-28z^{5}a^{9}+36z^{3}a^{9}-17za^{9}+2a^{9}z^{-1}$ (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   \ -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -8                     1 1 -10                     1 1 -12                 1     1 -14             1         1 -16           1 1 1       -1 -18         2 1           1 -20         1 1           0 -22     2 2               0 -24   1 1                 0 -26   1                   -1 -28 1                     1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-10$ $i=-8$ $i=-6$ $r=-10$ ${\mathbb {Z} }$ $r=-9$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-8$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }^{2}$ $r=-7$ ${\mathbb {Z} }^{2}$ $r=-6$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-5$ ${\mathbb {Z} }$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-1$ $r=0$ ${\mathbb {Z} }$ ${\mathbb {Z} }$

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.

L11n221 L11n223