L11a38

L11a37

L11a39

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a38 at Knotilus!

Link Presentations

[edit Notes on L11a38's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(t(1)-1)(t(2)-1)^{5}}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$q^{9/2}-4q^{7/2}+8q^{5/2}-14q^{3/2}+18{\sqrt {q}}-{\frac {21}{\sqrt {q}}}+{\frac {20}{q^{3/2}}}-{\frac {18}{q^{5/2}}}+{\frac {13}{q^{7/2}}}-{\frac {7}{q^{9/2}}}+{\frac {3}{q^{11/2}}}-{\frac {1}{q^{13/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{5}z^{3}+2a^{5}z+a^{5}z^{-1}-2a^{3}z^{5}-6a^{3}z^{3}+z^{3}a^{-3}-8a^{3}z+za^{-3}-3a^{3}z^{-1}+az^{7}+4az^{5}-2z^{5}a^{-1}+9az^{3}-5z^{3}a^{-1}+10az-5za^{-1}+4az^{-1}-2a^{-1}z^{-1}$ (db)
Kauffman polynomial	$a^{7}z^{5}-2a^{7}z^{3}+a^{7}z+3a^{6}z^{6}-5a^{6}z^{4}+3a^{6}z^{2}-a^{6}+5a^{5}z^{7}-6a^{5}z^{5}+3a^{5}z^{3}-2a^{5}z+a^{5}z^{-1}+6a^{4}z^{8}-5a^{4}z^{6}+z^{6}a^{-4}-2a^{4}z^{4}-2z^{4}a^{-4}+7a^{4}z^{2}+z^{2}a^{-4}-3a^{4}+4a^{3}z^{9}+6a^{3}z^{7}+4z^{7}a^{-3}-24a^{3}z^{5}-10z^{5}a^{-3}+28a^{3}z^{3}+8z^{3}a^{-3}-15a^{3}z-2za^{-3}+3a^{3}z^{-1}+a^{2}z^{10}+15a^{2}z^{8}+6z^{8}a^{-2}-32a^{2}z^{6}-12z^{6}a^{-2}+19a^{2}z^{4}+5z^{4}a^{-2}+2a^{2}z^{2}-3a^{2}+8az^{9}+4z^{9}a^{-1}+az^{7}+4z^{7}a^{-1}-36az^{5}-29z^{5}a^{-1}+43az^{3}+28z^{3}a^{-1}-21az-11za^{-1}+4az^{-1}+2a^{-1}z^{-1}+z^{10}+15z^{8}-37z^{6}+23z^{4}-3z^{2}-2$ (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                   5 1   -4 4                 9 3     6 2               9 5       -4 0             12 9         3 -2           10 11           1 -4         8 10             -2 -6       5 10               5 -8     2 8                 -6 -10   1 5                   4 -12   2                     -2 -14 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-2$ $i=0$ $r=-6$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-3$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=-2$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=-1$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=0$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{12}$ $r=1$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=2$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=3$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=4$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=5$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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.

L11a37 L11a39