L11a144

L11a143

L11a145

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a144 at Knotilus!

Link Presentations

[edit Notes on L11a144's Link Presentations]

Planar diagram presentation	X8192 X2,9,3,10 X10,3,11,4 X14,8,15,7 X20,14,21,13 X6,19,1,20 X18,11,19,12 X12,6,13,5 X22,16,7,15 X4,18,5,17 X16,22,17,21
Gauss code	{1, -2, 3, -10, 8, -6}, {4, -1, 2, -3, 7, -8, 5, -4, 9, -11, 10, -7, 6, -5, 11, -9}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {2u^{2}v^{4}-5u^{2}v^{3}+5u^{2}v^{2}-2u^{2}v-2uv^{4}+7uv^{3}-11uv^{2}+7uv-2u-2v^{3}+5v^{2}-5v+2}{uv^{2}}}$ (db)
Jones polynomial	$3q^{9/2}-{\frac {4}{q^{9/2}}}-6q^{7/2}+{\frac {8}{q^{7/2}}}+11q^{5/2}-{\frac {13}{q^{5/2}}}-15q^{3/2}+{\frac {16}{q^{3/2}}}-q^{11/2}+{\frac {1}{q^{11/2}}}+17{\sqrt {q}}-{\frac {19}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$az^{7}+z^{7}a^{-1}-a^{3}z^{5}+3az^{5}+4z^{5}a^{-1}-z^{5}a^{-3}-2a^{3}z^{3}+az^{3}+6z^{3}a^{-1}-3z^{3}a^{-3}-3az+4za^{-1}-2za^{-3}+a^{3}z^{-1}-az^{-1}$ (db)
Kauffman polynomial	$-2z^{10}a^{-2}-2z^{10}-6az^{9}-10z^{9}a^{-1}-4z^{9}a^{-3}-10a^{2}z^{8}-z^{8}a^{-2}-3z^{8}a^{-4}-8z^{8}-11a^{3}z^{7}+2az^{7}+28z^{7}a^{-1}+14z^{7}a^{-3}-z^{7}a^{-5}-8a^{4}z^{6}+13a^{2}z^{6}+18z^{6}a^{-2}+12z^{6}a^{-4}+27z^{6}-4a^{5}z^{5}+15a^{3}z^{5}+14az^{5}-25z^{5}a^{-1}-16z^{5}a^{-3}+4z^{5}a^{-5}-a^{6}z^{4}+7a^{4}z^{4}-2a^{2}z^{4}-22z^{4}a^{-2}-15z^{4}a^{-4}-17z^{4}+2a^{5}z^{3}-7a^{3}z^{3}-8az^{3}+14z^{3}a^{-1}+9z^{3}a^{-3}-4z^{3}a^{-5}-a^{4}z^{2}-a^{2}z^{2}+9z^{2}a^{-2}+6z^{2}a^{-4}+3z^{2}-2az-5za^{-1}-3za^{-3}-a^{2}+a^{3}z^{-1}+az^{-1}$ (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                     2   -2 8                   4 1   3 6                 7 2     -5 4               8 4       4 2             9 7         -2 0           10 8           2 -2         7 10             3 -4       6 9               -3 -6     3 8                 5 -8   1 5                   -4 -10   3                     3 -12 1                       -1

Integral Khovanov Homology (db, data source)

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

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.

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