L11a355

L11a354

L11a356

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a355 at Knotilus!

Link Presentations

[edit Notes on L11a355's Link Presentations]

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(t(1)+t(2)-1)(t(2)t(1)-t(1)-t(2))\left(t(1)^{2}t(2)^{2}+t(1)t(2)+1\right)}{t(1)^{2}t(2)^{2}}}$ (db)
Jones polynomial	$-{\frac {1}{\sqrt {q}}}+{\frac {2}{q^{3/2}}}-{\frac {4}{q^{5/2}}}+{\frac {5}{q^{7/2}}}-{\frac {7}{q^{9/2}}}+{\frac {8}{q^{11/2}}}-{\frac {8}{q^{13/2}}}+{\frac {7}{q^{15/2}}}-{\frac {6}{q^{17/2}}}+{\frac {3}{q^{19/2}}}-{\frac {2}{q^{21/2}}}+{\frac {1}{q^{23/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$-z^{5}a^{9}-4z^{3}a^{9}-3za^{9}+z^{7}a^{7}+5z^{5}a^{7}+8z^{3}a^{7}+6za^{7}+a^{7}z^{-1}+z^{7}a^{5}+4z^{5}a^{5}+2z^{3}a^{5}-3za^{5}-a^{5}z^{-1}-z^{5}a^{3}-4z^{3}a^{3}-3za^{3}$ (db)
Kauffman polynomial	$-z^{4}a^{14}+2z^{2}a^{14}-2z^{5}a^{13}+4z^{3}a^{13}-2za^{13}-2z^{6}a^{12}+2z^{4}a^{12}-2z^{7}a^{11}+2z^{5}a^{11}-3za^{11}-2z^{8}a^{10}+4z^{6}a^{10}-5z^{4}a^{10}+z^{2}a^{10}-2z^{9}a^{9}+7z^{7}a^{9}-13z^{5}a^{9}+10z^{3}a^{9}-2za^{9}-z^{10}a^{8}+2z^{8}a^{8}-3z^{4}a^{8}+4z^{2}a^{8}-4z^{9}a^{7}+18z^{7}a^{7}-30z^{5}a^{7}+26z^{3}a^{7}-9za^{7}+a^{7}z^{-1}-z^{10}a^{6}+2z^{8}a^{6}+3z^{6}a^{6}-5z^{4}a^{6}+3z^{2}a^{6}-a^{6}-2z^{9}a^{5}+8z^{7}a^{5}-8z^{5}a^{5}+5z^{3}a^{5}-5za^{5}+a^{5}z^{-1}-2z^{8}a^{4}+9z^{6}a^{4}-10z^{4}a^{4}+2z^{2}a^{4}-z^{7}a^{3}+5z^{5}a^{3}-7z^{3}a^{3}+3za^{3}$ (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 0                       1 1 -2                     1   -1 -4                   3 1   2 -6                 3 2     -1 -8               4 2       2 -10             4 3         -1 -12           4 4           0 -14         3 4             1 -16       3 4               -1 -18     1 4                 3 -20   1 2                   -1 -22   1                     1 -24 1                       -1

Integral Khovanov Homology (db, data source)

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

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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