L11a524

L11a523

L11a525

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a524 at Knotilus!

Link Presentations

[edit Notes on L11a524's Link Presentations]

Planar diagram presentation	X8192 X22,10,13,9 X18,8,19,7 X20,17,21,18 X6,22,1,21 X4,11,5,12 X16,6,17,5 X10,16,11,15 X12,20,7,19 X2,13,3,14 X14,3,15,4
Gauss code	{1, -10, 11, -6, 7, -5}, {3, -1, 2, -8, 6, -9}, {10, -11, 8, -7, 4, -3, 9, -4, 5, -2}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(v-1)\left(u^{2}vw^{3}-3u^{2}vw^{2}+2u^{2}vw-u^{2}w^{3}+2u^{2}w^{2}-u^{2}w-uvw^{3}+4uvw^{2}-5uvw+uv+uw^{3}-5uw^{2}+4uw-u-vw^{2}+2vw-v+2w^{2}-3w+1\right)}{uvw^{3/2}}}$ (db)
Jones polynomial	$q^{6}-4q^{5}-q^{-5}+10q^{4}+5q^{-4}-16q^{3}-11q^{-3}+24q^{2}+18q^{-2}-26q-24q^{-1}+28$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$z^{4}a^{-4}+2z^{2}a^{-4}+a^{-4}z^{-2}+a^{-4}-a^{2}z^{6}-2z^{6}a^{-2}-2a^{2}z^{4}-6z^{4}a^{-2}-5z^{2}a^{-2}-2a^{-2}z^{-2}+a^{2}-2a^{-2}+z^{8}+4z^{6}+5z^{4}+z^{2}+z^{-2}$ (db)
Kauffman polynomial	$z^{6}a^{-6}-2z^{4}a^{-6}+z^{2}a^{-6}+4z^{7}a^{-5}+a^{5}z^{5}-8z^{5}a^{-5}+4z^{3}a^{-5}+8z^{8}a^{-4}+5a^{4}z^{6}-18z^{6}a^{-4}-4a^{4}z^{4}+16z^{4}a^{-4}-10z^{2}a^{-4}-a^{-4}z^{-2}+4a^{-4}+8z^{9}a^{-3}+11a^{3}z^{7}-11z^{7}a^{-3}-14a^{3}z^{5}-z^{5}a^{-3}+4a^{3}z^{3}+6z^{3}a^{-3}-5za^{-3}+2a^{-3}z^{-1}+3z^{10}a^{-2}+14a^{2}z^{8}+16z^{8}a^{-2}-19a^{2}z^{6}-52z^{6}a^{-2}+6a^{2}z^{4}+50z^{4}a^{-2}-22z^{2}a^{-2}-2a^{-2}z^{-2}-a^{2}+6a^{-2}+10az^{9}+18z^{9}a^{-1}-2az^{7}-28z^{7}a^{-1}-20az^{5}+2z^{5}a^{-1}+13az^{3}+11z^{3}a^{-1}-5za^{-1}+2a^{-1}z^{-1}+3z^{10}+22z^{8}-57z^{6}+42z^{4}-11z^{2}-z^{-2}+2$ (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 13                       1 1 11                     3   -3 9                   7 1   6 7                 10 4     -6 5               14 6       8 3             13 11         -2 1           15 13           2 -1         10 14             4 -3       8 14               -6 -5     4 11                 7 -7   1 7                   -6 -9   4                     4 -11 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-1$ $i=1$ $r=-5$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-2$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{8}$ $r=-1$ ${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=0$ ${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{14}$ ${\mathbb {Z} }^{15}$ $r=1$ ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{13}$ ${\mathbb {Z} }^{13}$ $r=2$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{13}$ ${\mathbb {Z} }^{14}$ $r=3$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=4$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{7}$ $r=5$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=6$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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.

L11a523 L11a525