L10a160

L10a159

L10a161

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a160 at Knotilus!

Link Presentations

[edit Notes on L10a160's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {u^{2}v^{2}w^{2}+u^{2}vw^{3}-2u^{2}vw^{2}+u^{2}vw-u^{2}w^{3}+u^{2}w^{2}-uv^{2}w^{2}+uv^{2}w-uvw^{3}+2uvw^{2}-2uvw+uv-uw^{2}+uw-v^{2}w+v^{2}-vw^{2}+2vw-v-w}{uvw^{3/2}}}$ (db)
Jones polynomial	$q^{10}-2q^{9}+4q^{8}-6q^{7}+8q^{6}-7q^{5}+8q^{4}-5q^{3}+4q^{2}-2q+1$ (db)
Signature	4 (db)
HOMFLY-PT polynomial	$-z^{6}a^{-4}-z^{6}a^{-6}+z^{4}a^{-2}-3z^{4}a^{-4}-4z^{4}a^{-6}+z^{4}a^{-8}+3z^{2}a^{-2}-6z^{2}a^{-6}+3z^{2}a^{-8}+a^{-2}+3a^{-4}-6a^{-6}+2a^{-8}+a^{-4}z^{-2}-2a^{-6}z^{-2}+a^{-8}z^{-2}$ (db)
Kauffman polynomial	$z^{4}a^{-12}-2z^{2}a^{-12}+2z^{5}a^{-11}-3z^{3}a^{-11}+3z^{6}a^{-10}-6z^{4}a^{-10}+5z^{2}a^{-10}-a^{-10}+3z^{7}a^{-9}-6z^{5}a^{-9}+6z^{3}a^{-9}+3z^{8}a^{-8}-10z^{6}a^{-8}+19z^{4}a^{-8}-14z^{2}a^{-8}-a^{-8}z^{-2}+5a^{-8}+z^{9}a^{-7}+z^{7}a^{-7}-9z^{5}a^{-7}+16z^{3}a^{-7}-9za^{-7}+2a^{-7}z^{-1}+5z^{8}a^{-6}-20z^{6}a^{-6}+35z^{4}a^{-6}-31z^{2}a^{-6}-2a^{-6}z^{-2}+11a^{-6}+z^{9}a^{-5}-8z^{5}a^{-5}+12z^{3}a^{-5}-9za^{-5}+2a^{-5}z^{-1}+2z^{8}a^{-4}-6z^{6}a^{-4}+5z^{4}a^{-4}-6z^{2}a^{-4}-a^{-4}z^{-2}+5a^{-4}+2z^{7}a^{-3}-7z^{5}a^{-3}+5z^{3}a^{-3}+z^{6}a^{-2}-4z^{4}a^{-2}+4z^{2}a^{-2}-a^{-2}$ (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             4 2       2 11           3 4         1 9         5 4           1 7       2 5             3 5     2 3               -1 3   1 3                 2 1   1                   -1 -1 1                     1

Integral Khovanov Homology (db, data source)

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

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.

L10a159 L10a161