L11a357

L11a356

L11a358

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a357 at Knotilus!

Link Presentations

[edit Notes on L11a357's Link Presentations]

Planar diagram presentation	X12,1,13,2 X18,7,19,8 X14,3,15,4 X20,10,21,9 X22,19,11,20 X6,15,7,16 X16,5,17,6 X4,17,5,18 X8,22,9,21 X2,11,3,12 X10,13,1,14
Gauss code	{1, -10, 3, -8, 7, -6, 2, -9, 4, -11}, {10, -1, 11, -3, 6, -7, 8, -2, 5, -4, 9, -5}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(u+v-1)(uv-u-v)\left(u^{2}v+uv^{2}-uv+u+v\right)}{u^{2}v^{2}}}$ (db)
Jones polynomial	${\frac {15}{q^{9/2}}}-{\frac {13}{q^{7/2}}}+{\frac {10}{q^{5/2}}}-{\frac {7}{q^{3/2}}}+{\frac {1}{q^{21/2}}}-{\frac {2}{q^{19/2}}}+{\frac {4}{q^{17/2}}}-{\frac {9}{q^{15/2}}}+{\frac {11}{q^{13/2}}}-{\frac {14}{q^{11/2}}}-{\sqrt {q}}+{\frac {3}{\sqrt {q}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{9}\left(-z^{3}\right)-2a^{9}z+a^{7}z^{5}+2a^{7}z^{3}+2a^{7}z+a^{7}z^{-1}+2a^{5}z^{5}+4a^{5}z^{3}+a^{5}z-a^{5}z^{-1}+a^{3}z^{5}-3a^{3}z-az^{3}-az$ (db)
Kauffman polynomial	$a^{12}z^{6}-4a^{12}z^{4}+4a^{12}z^{2}+2a^{11}z^{7}-7a^{11}z^{5}+8a^{11}z^{3}-4a^{11}z+2a^{10}z^{8}-3a^{10}z^{6}-3a^{10}z^{4}+4a^{10}z^{2}+2a^{9}z^{9}-3a^{9}z^{7}+a^{9}z^{5}+2a^{9}z^{3}-6a^{9}z+a^{8}z^{10}+a^{8}z^{8}-a^{8}z^{6}-4a^{8}z^{4}+4a^{8}z^{2}+5a^{7}z^{9}-11a^{7}z^{7}+15a^{7}z^{5}-11a^{7}z^{3}+4a^{7}z-a^{7}z^{-1}+a^{6}z^{10}+4a^{6}z^{8}-8a^{6}z^{6}+5a^{6}z^{4}+a^{6}z^{2}+a^{6}+3a^{5}z^{9}-a^{5}z^{7}-3a^{5}z^{5}+2a^{5}z^{3}+2a^{5}z-a^{5}z^{-1}+5a^{4}z^{8}-8a^{4}z^{6}+5a^{4}z^{4}-2a^{4}z^{2}+5a^{3}z^{7}-9a^{3}z^{5}+5a^{3}z^{3}-3a^{3}z+3a^{2}z^{6}-5a^{2}z^{4}+a^{2}z^{2}+az^{5}-2az^{3}+az$ (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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 2                       1 1 0                     2   -2 -2                   5 1   4 -4                 6 3     -3 -6               7 4       3 -8             8 6         -2 -10           6 7           -1 -12         5 8             3 -14       4 6               -2 -16     1 6                 5 -18   1 3                   -2 -20   1                     1 -22 1                       -1

Integral Khovanov Homology (db, data source)

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

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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