L11n235

L11n234

L11n236

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n235 at Knotilus!

Link Presentations

[edit Notes on L11n235's Link Presentations]

Planar diagram presentation	X10,1,11,2 X5,20,6,21 X14,3,15,4 X4,15,5,16 X19,22,20,9 X16,7,17,8 X18,12,19,11 X12,18,13,17 X2,9,3,10 X8,13,1,14 X21,6,22,7
Gauss code	{1, -9, 3, -4, -2, 11, 6, -10}, {9, -1, 7, -8, 10, -3, 4, -6, 8, -7, -5, 2, -11, 5}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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.

L11n234 L11n236