L11a122

L11a121

L11a123

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a122 at Knotilus!

Link Presentations

[edit Notes on L11a122's Link Presentations]

Planar diagram presentation	X6172 X14,3,15,4 X18,8,19,7 X22,20,5,19 X20,9,21,10 X8,21,9,22 X16,12,17,11 X12,16,13,15 X10,18,11,17 X2536 X4,13,1,14
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -6, 5, -9, 7, -8, 11, -2, 8, -7, 9, -3, 4, -5, 6, -4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {2uv^{3}-8uv^{2}+10uv-3u-3v^{3}+10v^{2}-8v+2}{{\sqrt {u}}v^{3/2}}}$ (db)
Jones polynomial	$q^{9/2}-4q^{7/2}+7q^{5/2}-10q^{3/2}+13{\sqrt {q}}-{\frac {15}{\sqrt {q}}}+{\frac {14}{q^{3/2}}}-{\frac {12}{q^{5/2}}}+{\frac {8}{q^{7/2}}}-{\frac {5}{q^{9/2}}}+{\frac {2}{q^{11/2}}}-{\frac {1}{q^{13/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{7}z^{-1}-3a^{5}z-2a^{5}z^{-1}+3a^{3}z^{3}+z^{3}a^{-3}+3a^{3}z+2a^{3}z^{-1}-az^{5}-z^{5}a^{-1}-z^{3}a^{-1}-az-az^{-1}-za^{-1}$ (db)
Kauffman polynomial	$a^{7}z^{5}-3a^{7}z^{3}+3a^{7}z-a^{7}z^{-1}+2a^{6}z^{6}-4a^{6}z^{4}+2a^{6}z^{2}+2a^{5}z^{7}+a^{5}z^{5}-9a^{5}z^{3}+8a^{5}z-2a^{5}z^{-1}+2a^{4}z^{8}+2a^{4}z^{6}+z^{6}a^{-4}-7a^{4}z^{4}-2z^{4}a^{-4}+4a^{4}z^{2}-a^{4}+2a^{3}z^{9}-a^{3}z^{7}+4z^{7}a^{-3}+6a^{3}z^{5}-11z^{5}a^{-3}-13a^{3}z^{3}+5z^{3}a^{-3}+9a^{3}z-2a^{3}z^{-1}+a^{2}z^{10}+3a^{2}z^{8}+6z^{8}a^{-2}-7a^{2}z^{6}-18z^{6}a^{-2}+8a^{2}z^{4}+13z^{4}a^{-2}-3a^{2}z^{2}-2z^{2}a^{-2}+6az^{9}+4z^{9}a^{-1}-14az^{7}-7z^{7}a^{-1}+14az^{5}-3z^{5}a^{-1}-6az^{3}+6z^{3}a^{-1}+3az-za^{-1}-az^{-1}+z^{10}+7z^{8}-26z^{6}+26z^{4}-7z^{2}$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 10                       1 -1 8                     3   3 6                   4 1   -3 4                 6 3     3 2               7 4       -3 0             8 6         2 -2           7 8           1 -4         5 7             -2 -6       3 7               4 -8     2 5                 -3 -10   1 4                   3 -12   1                     -1 -14 1                       1

Integral Khovanov Homology (db, data source)

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

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.

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