L11a142

L11a141

L11a143

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a142 at Knotilus!

Link Presentations

[edit Notes on L11a142's Link Presentations]

Planar diagram presentation	X8192 X18,11,19,12 X10,4,11,3 X2,17,3,18 X12,5,13,6 X6718 X16,10,17,9 X20,16,21,15 X22,14,7,13 X14,22,15,21 X4,20,5,19
Gauss code	{1, -4, 3, -11, 5, -6}, {6, -1, 7, -3, 2, -5, 9, -10, 8, -7, 4, -2, 11, -8, 10, -9}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {2u^{2}v^{4}-6u^{2}v^{3}+6u^{2}v^{2}-2u^{2}v-2uv^{4}+8uv^{3}-13uv^{2}+8uv-2u-2v^{3}+6v^{2}-6v+2}{uv^{2}}}$ (db)
Jones polynomial	$4q^{9/2}-{\frac {4}{q^{9/2}}}-8q^{7/2}+{\frac {8}{q^{7/2}}}+13q^{5/2}-{\frac {14}{q^{5/2}}}-18q^{3/2}+{\frac {18}{q^{3/2}}}-q^{11/2}+{\frac {1}{q^{11/2}}}+20{\sqrt {q}}-{\frac {21}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$az^{7}+z^{7}a^{-1}-a^{3}z^{5}+3az^{5}+3z^{5}a^{-1}-z^{5}a^{-3}-2a^{3}z^{3}+2az^{3}+2z^{3}a^{-1}-2z^{3}a^{-3}-az+a^{3}z^{-1}-az^{-1}$ (db)
Kauffman polynomial	$-3z^{10}a^{-2}-3z^{10}-9az^{9}-15z^{9}a^{-1}-6z^{9}a^{-3}-13a^{2}z^{8}-2z^{8}a^{-2}-4z^{8}a^{-4}-11z^{8}-12a^{3}z^{7}+11az^{7}+44z^{7}a^{-1}+20z^{7}a^{-3}-z^{7}a^{-5}-8a^{4}z^{6}+23a^{2}z^{6}+31z^{6}a^{-2}+14z^{6}a^{-4}+48z^{6}-4a^{5}z^{5}+16a^{3}z^{5}+7az^{5}-35z^{5}a^{-1}-19z^{5}a^{-3}+3z^{5}a^{-5}-a^{6}z^{4}+6a^{4}z^{4}-14a^{2}z^{4}-37z^{4}a^{-2}-14z^{4}a^{-4}-44z^{4}+2a^{5}z^{3}-6a^{3}z^{3}-8az^{3}+8z^{3}a^{-1}+6z^{3}a^{-3}-2z^{3}a^{-5}+4a^{2}z^{2}+12z^{2}a^{-2}+4z^{2}a^{-4}+12z^{2}-a^{3}z-az-a^{2}+a^{3}z^{-1}+az^{-1}$ (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   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 6 χ 12                       1 1 10                     3   -3 8                   5 1   4 6                 8 3     -5 4               10 5       5 2             10 8         -2 0           11 10           1 -2         8 11             3 -4       6 10               -4 -6     3 9                 6 -8   1 5                   -4 -10   3                     3 -12 1                       -1

Integral Khovanov Homology (db, data source)

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

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.

L11a141 L11a143