L11a297

L11a296

L11a298

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a297 at Knotilus!

Link Presentations

[edit Notes on L11a297's Link Presentations]

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

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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.

L11a296 L11a298