L11a430

L11a429

L11a431

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a430 at Knotilus!

Link Presentations

[edit Notes on L11a430's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(u-1)(v-1)(w-1)(vw-v+1)(vw-w+1)}{{\sqrt {u}}v^{3/2}w^{3/2}}}$ (db)
Jones polynomial	$-q^{5}+4q^{4}-8q^{3}+14q^{2}-19q+24-22q^{-1}+21q^{-2}-15q^{-3}+10q^{-4}-5q^{-5}+q^{-6}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a^{4}z^{4}+a^{4}z^{2}-a^{4}-2a^{2}z^{6}-z^{6}a^{-2}-6a^{2}z^{4}-3z^{4}a^{-2}-3a^{2}z^{2}-2z^{2}a^{-2}+a^{2}z^{-2}+a^{-2}z^{-2}+3a^{2}+a^{-2}+z^{8}+5z^{6}+9z^{4}+4z^{2}-2z^{-2}-3$ (db)
Kauffman polynomial	$2a^{2}z^{10}+2z^{10}+7a^{3}z^{9}+13az^{9}+6z^{9}a^{-1}+9a^{4}z^{8}+16a^{2}z^{8}+8z^{8}a^{-2}+15z^{8}+5a^{5}z^{7}-7a^{3}z^{7}-19az^{7}+7z^{7}a^{-3}+a^{6}z^{6}-21a^{4}z^{6}-48a^{2}z^{6}-9z^{6}a^{-2}+4z^{6}a^{-4}-39z^{6}-10a^{5}z^{5}-11a^{3}z^{5}-2az^{5}-12z^{5}a^{-1}-10z^{5}a^{-3}+z^{5}a^{-5}-a^{6}z^{4}+12a^{4}z^{4}+36a^{2}z^{4}+2z^{4}a^{-2}-6z^{4}a^{-4}+31z^{4}+3a^{5}z^{3}+7a^{3}z^{3}+8az^{3}+9z^{3}a^{-1}+4z^{3}a^{-3}-z^{3}a^{-5}-a^{4}z^{2}-3a^{2}z^{2}+z^{2}a^{-2}+2z^{2}a^{-4}-3z^{2}+a^{5}z+3a^{3}z+3az+za^{-1}-2a^{4}-6a^{2}-2a^{-2}-5-2az^{-1}-2a^{-1}z^{-1}+a^{2}z^{-2}+a^{-2}z^{-2}+2z^{-2}$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 11                       1 -1 9                     3   3 7                   5 1   -4 5                 9 3     6 3               10 5       -5 1             14 9         5 -1           12 14           2 -3         9 10             -1 -5       6 12               6 -7     4 9                 -5 -9   1 6                   5 -11   4                     -4 -13 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-1$ $i=1$ $r=-6$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-3$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=-2$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=-1$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=0$ ${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{14}$ $r=1$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=2$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=3$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=4$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=5$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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.

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