L11a138

L11a137

L11a139

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a138 at Knotilus!

Link Presentations

[edit Notes on L11a138's Link Presentations]

Planar diagram presentation	X6172 X2,11,3,12 X14,3,15,4 X12,5,13,6 X22,13,5,14 X4,21,1,22 X20,16,21,15 X16,8,17,7 X8,20,9,19 X18,10,19,9 X10,18,11,17
Gauss code	{1, -2, 3, -6}, {4, -1, 8, -9, 10, -11, 2, -4, 5, -3, 7, -8, 11, -10, 9, -7, 6, -5}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {4t(1)t(2)^{3}-4t(2)^{3}-12t(1)t(2)^{2}+13t(2)^{2}+13t(1)t(2)-12t(2)-4t(1)+4}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones polynomial	$q^{9/2}-4q^{7/2}+8q^{5/2}-13q^{3/2}+18{\sqrt {q}}-{\frac {21}{\sqrt {q}}}+{\frac {21}{q^{3/2}}}-{\frac {19}{q^{5/2}}}+{\frac {13}{q^{7/2}}}-{\frac {9}{q^{9/2}}}+{\frac {4}{q^{11/2}}}-{\frac {1}{q^{13/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{5}z^{3}+a^{5}z^{-1}-a^{3}z^{5}+z^{3}a^{-3}-a^{3}z-a^{3}z^{-1}-2az^{5}-z^{5}a^{-1}-2az^{3}-az$ (db)
Kauffman polynomial	$a^{7}z^{5}-a^{7}z^{3}+4a^{6}z^{6}-5a^{6}z^{4}+8a^{5}z^{7}-14a^{5}z^{5}+8a^{5}z^{3}-3a^{5}z+a^{5}z^{-1}+9a^{4}z^{8}-15a^{4}z^{6}+z^{6}a^{-4}+10a^{4}z^{4}-2z^{4}a^{-4}-3a^{4}z^{2}-a^{4}+6a^{3}z^{9}-2a^{3}z^{7}+4z^{7}a^{-3}-10a^{3}z^{5}-10z^{5}a^{-3}+9a^{3}z^{3}+5z^{3}a^{-3}-2a^{3}z+a^{3}z^{-1}+2a^{2}z^{10}+12a^{2}z^{8}+7z^{8}a^{-2}-35a^{2}z^{6}-19z^{6}a^{-2}+34a^{2}z^{4}+15z^{4}a^{-2}-11a^{2}z^{2}-4z^{2}a^{-2}+12az^{9}+6z^{9}a^{-1}-25az^{7}-11z^{7}a^{-1}+18az^{5}+3z^{5}a^{-1}-4az^{3}+z^{3}a^{-1}+az+2z^{10}+10z^{8}-36z^{6}+36z^{4}-12z^{2}$ (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                 8 3     5 2               10 5       -5 0             11 8         3 -2           11 11           0 -4         8 10             -2 -6       5 11               6 -8     4 8                 -4 -10   1 6                   5 -12   3                     -3 -14 1                       1

Integral Khovanov Homology (db, data source)

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

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.

L11a137 L11a139