L11a453

L11a452

L11a454

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a453 at Knotilus!

Link Presentations

[edit Notes on L11a453's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(vw-v+1)(vw-w+1)(uvw-uw+u-vw+v-1)}{{\sqrt {u}}v^{3/2}w^{3/2}}}$ (db)
Jones polynomial	$q^{7}-3q^{6}+7q^{5}-12q^{4}+16q^{3}-16q^{2}+18q-14+11q^{-1}-6q^{-2}+3q^{-3}-q^{-4}$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z^{6}a^{-4}+4z^{4}a^{-4}+6z^{2}a^{-4}+a^{-4}z^{-2}+3a^{-4}-z^{8}a^{-2}-6z^{6}a^{-2}-a^{2}z^{4}-15z^{4}a^{-2}-3a^{2}z^{2}-18z^{2}a^{-2}-2a^{-2}z^{-2}-2a^{2}-9a^{-2}+2z^{6}+9z^{4}+14z^{2}+z^{-2}+8$ (db)
Kauffman polynomial	$z^{4}a^{-8}-z^{2}a^{-8}+3z^{5}a^{-7}-2z^{3}a^{-7}+6z^{6}a^{-6}-6z^{4}a^{-6}+4z^{2}a^{-6}-a^{-6}+9z^{7}a^{-5}-14z^{5}a^{-5}+11z^{3}a^{-5}-za^{-5}+9z^{8}a^{-4}-15z^{6}a^{-4}+11z^{4}a^{-4}-6z^{2}a^{-4}-a^{-4}z^{-2}+3a^{-4}+5z^{9}a^{-3}+a^{3}z^{7}+z^{7}a^{-3}-4a^{3}z^{5}-22z^{5}a^{-3}+5a^{3}z^{3}+22z^{3}a^{-3}-2a^{3}z-8za^{-3}+2a^{-3}z^{-1}+z^{10}a^{-2}+3a^{2}z^{8}+15z^{8}a^{-2}-12a^{2}z^{6}-51z^{6}a^{-2}+16a^{2}z^{4}+58z^{4}a^{-2}-9a^{2}z^{2}-37z^{2}a^{-2}-2a^{-2}z^{-2}+3a^{2}+11a^{-2}+3az^{9}+8z^{9}a^{-1}-6az^{7}-15z^{7}a^{-1}-6az^{5}-7z^{5}a^{-1}+16az^{3}+20z^{3}a^{-1}-7az-12za^{-1}+2a^{-1}z^{-1}+z^{10}+9z^{8}-42z^{6}+56z^{4}-35z^{2}-z^{-2}+11$ (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                     2   -2 11                   5 1   4 9                 8 3     -5 7               8 4       4 5             8 8         0 3           10 8           2 1         7 11             4 -1       4 7               -3 -3     2 7                 5 -5   1 4                   -3 -7   2                     2 -9 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=1$ $i=3$ $r=-5$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-2$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-1$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=0$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{10}$ $r=1$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=2$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=3$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{5}$ $r=5$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=6$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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.

L11a452 L11a454