L11a140

L11a139

L11a141

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a140 at Knotilus!

Link Presentations

[edit Notes on L11a140's Link Presentations]

Planar diagram presentation	X8192 X10,4,11,3 X22,10,7,9 X2738 X16,11,17,12 X12,5,13,6 X4,18,5,17 X14,19,15,20 X20,13,21,14 X18,22,19,21 X6,15,1,16
Gauss code	{1, -4, 2, -7, 6, -11}, {4, -1, 3, -2, 5, -6, 9, -8, 11, -5, 7, -10, 8, -9, 10, -3}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {\left(t(1)t(2)^{2}-t(2)^{2}-3t(1)t(2)+2t(2)+t(1)-1\right)\left(t(1)t(2)^{2}-t(2)^{2}-2t(1)t(2)+3t(2)+t(1)-1\right)}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	$-q^{7/2}+5q^{5/2}-11q^{3/2}+17{\sqrt {q}}-{\frac {24}{\sqrt {q}}}+{\frac {26}{q^{3/2}}}-{\frac {26}{q^{5/2}}}+{\frac {22}{q^{7/2}}}-{\frac {16}{q^{9/2}}}+{\frac {9}{q^{11/2}}}-{\frac {4}{q^{13/2}}}+{\frac {1}{q^{15/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-za^{7}+3z^{3}a^{5}+3za^{5}-3z^{5}a^{3}-6z^{3}a^{3}-3za^{3}+a^{3}z^{-1}+z^{7}a+3z^{5}a+4z^{3}a-az^{-1}-z^{5}a^{-1}-z^{3}a^{-1}$ (db)
Kauffman polynomial	$a^{8}z^{6}-2a^{8}z^{4}+a^{8}z^{2}+4a^{7}z^{7}-9a^{7}z^{5}+7a^{7}z^{3}-2a^{7}z+7a^{6}z^{8}-14a^{6}z^{6}+9a^{6}z^{4}-2a^{6}z^{2}+6a^{5}z^{9}-a^{5}z^{7}-20a^{5}z^{5}+21a^{5}z^{3}-6a^{5}z+2a^{4}z^{10}+18a^{4}z^{8}-47a^{4}z^{6}+34a^{4}z^{4}-8a^{4}z^{2}+14a^{3}z^{9}-9a^{3}z^{7}-29a^{3}z^{5}+z^{5}a^{-3}+29a^{3}z^{3}-5a^{3}z-a^{3}z^{-1}+2a^{2}z^{10}+24a^{2}z^{8}-53a^{2}z^{6}+5z^{6}a^{-2}+31a^{2}z^{4}-4z^{4}a^{-2}-6a^{2}z^{2}+a^{2}+8az^{9}+7az^{7}+11z^{7}a^{-1}-34az^{5}-15z^{5}a^{-1}+21az^{3}+6z^{3}a^{-1}-az-az^{-1}+13z^{8}-16z^{6}+4z^{4}-z^{2}$ (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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 8                       1 1 6                     4   -4 4                   7 1   6 2                 10 4     -6 0               14 7       7 -2             13 11         -2 -4           13 13           0 -6         10 14             4 -8       6 12               -6 -10     3 10                 7 -12   1 6                   -5 -14   3                     3 -16 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-2$ $i=0$ $r=-7$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-4$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=-3$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=-2$ ${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{13}$ $r=-1$ ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{13}$ ${\mathbb {Z} }^{13}$ $r=0$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{13}$ ${\mathbb {Z} }^{14}$ $r=1$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=2$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=3$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=4$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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.

L11a139 L11a141