L10a144

L10a143

L10a145

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a144 at Knotilus!

Link Presentations

[edit Notes on L10a144's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {t(1)t(3)^{2}t(2)^{3}-t(3)^{2}t(2)^{3}+t(3)t(2)^{3}+t(1)t(3)^{3}t(2)^{2}-t(3)^{3}t(2)^{2}-2t(1)t(3)^{2}t(2)^{2}+2t(3)^{2}t(2)^{2}+t(1)t(3)t(2)^{2}-2t(3)t(2)^{2}+t(2)^{2}-t(1)t(3)^{3}t(2)+2t(1)t(3)^{2}t(2)-t(3)^{2}t(2)+t(1)t(2)-2t(1)t(3)t(2)+2t(3)t(2)-t(2)-t(1)t(3)^{2}+t(1)t(3)-t(3)}{{\sqrt {t(1)}}t(2)^{3/2}t(3)^{3/2}}}$ (db)
Jones polynomial	$q^{10}-2q^{9}+4q^{8}-6q^{7}+9q^{6}-8q^{5}+8q^{4}-6q^{3}+5q^{2}-2q+1$ (db)
Signature	4 (db)
HOMFLY-PT polynomial	$-z^{6}a^{-4}-z^{6}a^{-6}+z^{4}a^{-2}-3z^{4}a^{-4}-4z^{4}a^{-6}+z^{4}a^{-8}+3z^{2}a^{-2}-z^{2}a^{-4}-6z^{2}a^{-6}+3z^{2}a^{-8}+2a^{-2}+a^{-4}-5a^{-6}+2a^{-8}+a^{-4}z^{-2}-2a^{-6}z^{-2}+a^{-8}z^{-2}$ (db)
Kauffman polynomial	$z^{9}a^{-5}+z^{9}a^{-7}+2z^{8}a^{-4}+5z^{8}a^{-6}+3z^{8}a^{-8}+2z^{7}a^{-3}+z^{7}a^{-5}+2z^{7}a^{-7}+3z^{7}a^{-9}+z^{6}a^{-2}-4z^{6}a^{-4}-16z^{6}a^{-6}-8z^{6}a^{-8}+3z^{6}a^{-10}-6z^{5}a^{-3}-9z^{5}a^{-5}-10z^{5}a^{-7}-5z^{5}a^{-9}+2z^{5}a^{-11}-4z^{4}a^{-2}-z^{4}a^{-4}+22z^{4}a^{-6}+12z^{4}a^{-8}-6z^{4}a^{-10}+z^{4}a^{-12}+3z^{3}a^{-3}+10z^{3}a^{-5}+14z^{3}a^{-7}+4z^{3}a^{-9}-3z^{3}a^{-11}+5z^{2}a^{-2}-z^{2}a^{-4}-19z^{2}a^{-6}-5z^{2}a^{-8}+6z^{2}a^{-10}-2z^{2}a^{-12}+za^{-3}-8za^{-5}-8za^{-7}+za^{-9}-2a^{-2}+3a^{-4}+9a^{-6}+3a^{-8}-2a^{-10}+2a^{-5}z^{-1}+2a^{-7}z^{-1}-a^{-4}z^{-2}-2a^{-6}z^{-2}-a^{-8}z^{-2}$ (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   \ -2 -1 0 1 2 3 4 5 6 7 8 χ 21                     1 1 19                   2 1 -1 17                 2     2 15               4 2     -2 13             5 2       3 11           4 5         1 9         4 4           0 7       2 4             2 5     3 4               -1 3   1 4                 3 1   1                   -1 -1 1                     1

Integral Khovanov Homology (db, data source)

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

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.

L10a143 L10a145