L11a472

L11a471

L11a473

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a472 at Knotilus!

Link Presentations

[edit Notes on L11a472's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(u-1)(w-1)^{2}\left(v^{2}w^{2}-v^{2}w-vw^{2}+vw-v-w+1\right)}{{\sqrt {u}}vw^{2}}}$ (db)
Jones polynomial	$q^{7}-4q^{6}+8q^{5}-12q^{4}-q^{-4}+17q^{3}+3q^{-3}-17q^{2}-6q^{-2}+18q+11q^{-1}-14$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$-z^{8}a^{-2}-5z^{6}a^{-2}+z^{6}a^{-4}+2z^{6}-a^{2}z^{4}-9z^{4}a^{-2}+3z^{4}a^{-4}+8z^{4}-3a^{2}z^{2}-8z^{2}a^{-2}+2z^{2}a^{-4}+9z^{2}-a^{2}-4a^{-2}+a^{-4}+4-2a^{-2}z^{-2}+a^{-4}z^{-2}+z^{-2}$ (db)
Kauffman polynomial	$z^{4}a^{-8}+4z^{5}a^{-7}-2z^{3}a^{-7}+8z^{6}a^{-6}-8z^{4}a^{-6}+2z^{2}a^{-6}+10z^{7}a^{-5}-11z^{5}a^{-5}+z^{3}a^{-5}+10z^{8}a^{-4}-15z^{6}a^{-4}+6z^{4}a^{-4}-3z^{2}a^{-4}-a^{-4}z^{-2}+3a^{-4}+7z^{9}a^{-3}+a^{3}z^{7}-10z^{7}a^{-3}-4a^{3}z^{5}+z^{5}a^{-3}+4a^{3}z^{3}+z^{3}a^{-3}-a^{3}z-2za^{-3}+2a^{-3}z^{-1}+2z^{10}a^{-2}+3a^{2}z^{8}+10z^{8}a^{-2}-12a^{2}z^{6}-43z^{6}a^{-2}+15a^{2}z^{4}+48z^{4}a^{-2}-9a^{2}z^{2}-23z^{2}a^{-2}-2a^{-2}z^{-2}+2a^{2}+7a^{-2}+4az^{9}+11z^{9}a^{-1}-13az^{7}-34z^{7}a^{-1}+9az^{5}+29z^{5}a^{-1}+2az^{3}-4z^{3}a^{-1}-3az-4za^{-1}+2a^{-1}z^{-1}+2z^{10}+3z^{8}-32z^{6}+48z^{4}-27z^{2}-z^{-2}+7$ (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                   5 1   4 9                 7 3     -4 7               10 5       5 5             9 9         0 3           9 8           1 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)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=1$ $i=3$ $r=-5$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-2$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-1$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=0$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{9}$ $r=1$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=2$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{10}$ $r=3$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=5$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=6$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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.

L11a471 L11a473