L11n204

L11n203

L11n205

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n204 at Knotilus!

Link Presentations

[edit Notes on L11n204's Link Presentations]

Planar diagram presentation	X10,1,11,2 X7,16,8,17 X11,18,12,19 X2,19,3,20 X3,12,4,13 X20,13,21,14 X14,5,15,6 X6,9,7,10 X15,22,16,9 X17,8,18,1 X21,4,22,5
Gauss code	{1, -4, -5, 11, 7, -8, -2, 10}, {8, -1, -3, 5, 6, -7, -9, 2, -10, 3, 4, -6, -11, 9}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {\left(t(1)t(2)^{2}+1\right)\left(t(1)^{2}t(2)^{3}+1\right)}{t(1)^{3/2}t(2)^{5/2}}}$ (db)
Jones polynomial	$-{\frac {1}{q^{9/2}}}-{\frac {1}{q^{13/2}}}+{\frac {1}{q^{21/2}}}-{\frac {1}{q^{23/2}}}$ (db)
Signature	-8 (db)
HOMFLY-PT polynomial	$a^{13}\left(-z^{3}\right)-3a^{13}z-a^{13}z^{-1}+a^{11}z^{7}+8a^{11}z^{5}+20a^{11}z^{3}+17a^{11}z+3a^{11}z^{-1}-a^{9}z^{9}-9a^{9}z^{7}-28a^{9}z^{5}-36a^{9}z^{3}-18a^{9}z-2a^{9}z^{-1}$ (db)
Kauffman polynomial	$-za^{15}-a^{14}+z^{3}a^{13}-3za^{13}+a^{13}z^{-1}-z^{8}a^{12}+8z^{6}a^{12}-20z^{4}a^{12}+17z^{2}a^{12}-3a^{12}-z^{9}a^{11}+9z^{7}a^{11}-28z^{5}a^{11}+37z^{3}a^{11}-20za^{11}+3a^{11}z^{-1}-z^{8}a^{10}+8z^{6}a^{10}-20z^{4}a^{10}+17z^{2}a^{10}-3a^{10}-z^{9}a^{9}+9z^{7}a^{9}-28z^{5}a^{9}+36z^{3}a^{9}-18za^{9}+2a^{9}z^{-1}$ (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   \ -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       0 -18     1 1           0 -20       1           -1 -22 1 1               0 -24 1                 1

Integral Khovanov Homology (db, data source)

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

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.

L11n203 L11n205