L11a431

L11a430

L11a432

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a431 at Knotilus!

Link Presentations

[edit Notes on L11a431's Link Presentations]

Planar diagram presentation	X6172 X14,6,15,5 X8493 X2,16,3,15 X16,7,17,8 X18,10,19,9 X20,11,21,12 X22,20,13,19 X12,14,5,13 X4,17,1,18 X10,21,11,22
Gauss code	{1, -4, 3, -10}, {2, -1, 5, -3, 6, -11, 7, -9}, {9, -2, 4, -5, 10, -6, 8, -7, 11, -8}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(u-1)(v-1)(w-1)\left(v^{2}w^{2}-v^{2}w-vw^{2}+vw-v-w+1\right)}{{\sqrt {u}}v^{3/2}w^{3/2}}}$ (db)
Jones polynomial	$q^{7}-4q^{6}+7q^{5}-11q^{4}-q^{-4}+16q^{3}+4q^{-3}-17q^{2}-7q^{-2}+18q+12q^{-1}-14$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$-z^{8}a^{-2}-5z^{6}a^{-2}+z^{6}a^{-4}+2z^{6}-a^{2}z^{4}-8z^{4}a^{-2}+3z^{4}a^{-4}+7z^{4}-2a^{2}z^{2}-3z^{2}a^{-2}+z^{2}a^{-4}+4z^{2}+a^{2}+3a^{-2}-a^{-4}-3+a^{2}z^{-2}+a^{-2}z^{-2}-2z^{-2}$ (db)
Kauffman polynomial	$2z^{10}a^{-2}+2z^{10}+5az^{9}+11z^{9}a^{-1}+6z^{9}a^{-3}+4a^{2}z^{8}+9z^{8}a^{-2}+8z^{8}a^{-4}+5z^{8}+a^{3}z^{7}-16az^{7}-31z^{7}a^{-1}-6z^{7}a^{-3}+8z^{7}a^{-5}-15a^{2}z^{6}-36z^{6}a^{-2}-8z^{6}a^{-4}+7z^{6}a^{-6}-36z^{6}-3a^{3}z^{5}+11az^{5}+19z^{5}a^{-1}-5z^{5}a^{-3}-6z^{5}a^{-5}+4z^{5}a^{-7}+16a^{2}z^{4}+28z^{4}a^{-2}-4z^{4}a^{-4}-7z^{4}a^{-6}+z^{4}a^{-8}+40z^{4}+2a^{3}z^{3}+2z^{3}a^{-3}-3z^{3}a^{-5}-3z^{3}a^{-7}-5a^{2}z^{2}+z^{2}a^{-2}+5z^{2}a^{-4}+2z^{2}a^{-6}-7z^{2}+az+3za^{-1}+3za^{-3}+za^{-5}-2a^{2}-6a^{-2}-2a^{-4}-5-2az^{-1}-2a^{-1}z^{-1}+a^{2}z^{-2}+a^{-2}z^{-2}+2z^{-2}$ (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   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 6 χ 15                       1 1 13                     3   -3 11                   4 1   3 9                 7 3     -4 7               9 4       5 5             8 7         -1 3           10 9           1 1         8 12             4 -1       4 6               -2 -3     3 8                 5 -5   1 4                   -3 -7   3                     3 -9 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=1$ $i=3$ $r=-5$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-2$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-1$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=0$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{10}$ $r=1$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=2$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=3$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=5$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=6$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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.

L11a430 L11a432