L11n355

L11n354

L11n356

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n355 at Knotilus!

Link Presentations

[edit Notes on L11n355's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X18,12,19,11 X7,16,8,17 X15,8,16,9 X20,13,21,14 X22,20,15,19 X12,21,13,22 X14,18,5,17 X2536 X4,9,1,10
Gauss code	{1, -10, 2, -11}, {-5, 4, 9, -3, 7, -6, 8, -7}, {10, -1, -4, 5, 11, -2, 3, -8, 6, -9}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(w-1)\left(uv^{2}w-uv^{2}-2uvw+3uv+uw-u-v^{2}w+v^{2}+3vw-2v-w+1\right)}{{\sqrt {u}}vw}}$ (db)
Jones polynomial	$2q^{-7}-4q^{-6}+9q^{-5}-10q^{-4}+13q^{-3}-q^{2}-12q^{-2}+4q+10q^{-1}-7$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$a^{8}z^{-2}+z^{2}a^{6}-2a^{6}z^{-2}-2a^{6}-2z^{4}a^{4}-2z^{2}a^{4}+a^{4}z^{-2}+a^{4}+z^{6}a^{2}+3z^{4}a^{2}+4z^{2}a^{2}+a^{2}-z^{4}-z^{2}$ (db)
Kauffman polynomial	$3a^{8}z^{4}-6a^{8}z^{2}-a^{8}z^{-2}+4a^{8}+a^{7}z^{7}+a^{7}z^{5}+a^{7}z^{3}-5a^{7}z+2a^{7}z^{-1}+2a^{6}z^{8}-2a^{6}z^{6}+7a^{6}z^{4}-10a^{6}z^{2}-2a^{6}z^{-2}+6a^{6}+a^{5}z^{9}+4a^{5}z^{7}-7a^{5}z^{5}+7a^{5}z^{3}-5a^{5}z+2a^{5}z^{-1}+6a^{4}z^{8}-7a^{4}z^{6}+2a^{4}z^{4}-a^{4}z^{2}-a^{4}z^{-2}+2a^{4}+a^{3}z^{9}+9a^{3}z^{7}-20a^{3}z^{5}+11a^{3}z^{3}+4a^{2}z^{8}-a^{2}z^{6}-9a^{2}z^{4}+5a^{2}z^{2}-a^{2}+6az^{7}-11az^{5}+z^{5}a^{-1}+4az^{3}-z^{3}a^{-1}+4z^{6}-7z^{4}+2z^{2}$ (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 χ 5                   1 -1 3                 3   3 1               4 1   -3 -1             6 3     3 -3           7 5       -2 -5         6 5         1 -7       5 8           3 -9     4 5             -1 -11   1 6               5 -13 1 3                 -2 -15 2                   2

Integral Khovanov Homology (db, data source)

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

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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