L11n262

L11n261

L11n263

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n262 at Knotilus!

Link Presentations

[edit Notes on L11n262's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X14,7,15,8 X8,13,5,14 X11,18,12,19 X19,22,20,9 X15,20,16,21 X21,16,22,17 X17,12,18,13 X2536 X4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -4}, {11, -2, -5, 9, 4, -3, -7, 8, -9, 5, -6, 7, -8, 6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {uvw^{3}-uvw+uv+uw^{3}-3uw^{2}+3uw-u+vw^{3}-3vw^{2}+3vw-v-w^{3}+w^{2}-1}{{\sqrt {u}}{\sqrt {v}}w^{3/2}}}$ (db)
Jones polynomial	$-q^{-13}+2q^{-12}-5q^{-11}+6q^{-10}-6q^{-9}+7q^{-8}-4q^{-7}+5q^{-6}-q^{-5}+q^{-3}$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$-a^{14}z^{-2}+3a^{12}z^{-2}+4a^{12}-5z^{2}a^{10}-2a^{10}z^{-2}-7a^{10}+z^{4}a^{8}-a^{8}z^{-2}-a^{8}+z^{6}a^{6}+6z^{4}a^{6}+8z^{2}a^{6}+a^{6}z^{-2}+4a^{6}$ (db)
Kauffman polynomial	$a^{15}z^{7}-5a^{15}z^{5}+9a^{15}z^{3}-7a^{15}z+2a^{15}z^{-1}+2a^{14}z^{8}-8a^{14}z^{6}+8a^{14}z^{4}-2a^{14}z^{2}-a^{14}z^{-2}+a^{14}+a^{13}z^{9}+2a^{13}z^{7}-24a^{13}z^{5}+39a^{13}z^{3}-27a^{13}z+8a^{13}z^{-1}+6a^{12}z^{8}-24a^{12}z^{6}+24a^{12}z^{4}-10a^{12}z^{2}-3a^{12}z^{-2}+5a^{12}+a^{11}z^{9}+4a^{11}z^{7}-33a^{11}z^{5}+52a^{11}z^{3}-34a^{11}z+10a^{11}z^{-1}+4a^{10}z^{8}-18a^{10}z^{6}+24a^{10}z^{4}-13a^{10}z^{2}-2a^{10}z^{-2}+4a^{10}+3a^{9}z^{7}-15a^{9}z^{5}+22a^{9}z^{3}-10a^{9}z+2a^{9}z^{-1}-a^{8}z^{6}+2a^{8}z^{4}+3a^{8}z^{2}+a^{8}z^{-2}-3a^{8}-a^{7}z^{5}+4a^{7}z-2a^{7}z^{-1}+a^{6}z^{6}-6a^{6}z^{4}+8a^{6}z^{2}+a^{6}z^{-2}-4a^{6}$ (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   \ -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -5                       1 1 -7                       1 1 -9                 2 1     -1 -11               4         4 -13             4 6 1       1 -15           4 1           3 -17         2 4 1           1 -19       4 4               0 -21     1 2                 1 -23   1 4                   -3 -25   1                     1 -27 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-7$ $i=-5$ $i=-3$ $r=-11$ ${\mathbb {Z} }$ $r=-10$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-9$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-8$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-7$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-6$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-5$ ${\mathbb {Z} }$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-4$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }^{4}$ $r=-3$ ${\mathbb {Z} }$ ${\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-2$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-1$ $r=0$ ${\mathbb {Z} }$ ${\mathbb {Z} }$

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.

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