L11a54

L11a53

L11a55

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a54 at Knotilus!

Link Presentations

[edit Notes on L11a54's Link Presentations]

Planar diagram presentation	X6172 X20,7,21,8 X4,21,1,22 X14,6,15,5 X10,4,11,3 X16,12,17,11 X12,16,13,15 X22,14,5,13 X18,9,19,10 X2,18,3,17 X8,19,9,20
Gauss code	{1, -10, 5, -3}, {4, -1, 2, -11, 9, -5, 6, -7, 8, -4, 7, -6, 10, -9, 11, -2, 3, -8}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {2(u-1)(v-1)\left(2v^{2}-3v+2\right)}{{\sqrt {u}}v^{3/2}}}$ (db)
Jones polynomial	$-8q^{9/2}+{\frac {1}{q^{9/2}}}+12q^{7/2}-{\frac {3}{q^{7/2}}}-16q^{5/2}+{\frac {6}{q^{5/2}}}+18q^{3/2}-{\frac {11}{q^{3/2}}}-q^{13/2}+4q^{11/2}-18{\sqrt {q}}+{\frac {14}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$az^{5}+2z^{5}a^{-1}+z^{5}a^{-3}-a^{3}z^{3}+az^{3}+3z^{3}a^{-1}-z^{3}a^{-5}-a^{3}z+az+za^{-1}-za^{-3}+az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{5}a^{-7}-z^{3}a^{-7}+4z^{6}a^{-6}-6z^{4}a^{-6}+7z^{7}a^{-5}-12z^{5}a^{-5}+3z^{3}a^{-5}+8z^{8}a^{-4}+a^{4}z^{6}-16z^{6}a^{-4}-3a^{4}z^{4}+12z^{4}a^{-4}+2a^{4}z^{2}-4z^{2}a^{-4}+6z^{9}a^{-3}+3a^{3}z^{7}-11z^{7}a^{-3}-9a^{3}z^{5}+11z^{5}a^{-3}+7a^{3}z^{3}-6z^{3}a^{-3}-2a^{3}z+2za^{-3}+2z^{10}a^{-2}+4a^{2}z^{8}+6z^{8}a^{-2}-9a^{2}z^{6}-24z^{6}a^{-2}+3a^{2}z^{4}+29z^{4}a^{-2}-8z^{2}a^{-2}+4az^{9}+10z^{9}a^{-1}-8az^{7}-29z^{7}a^{-1}+4az^{5}+37z^{5}a^{-1}-az^{3}-18z^{3}a^{-1}-2az+2za^{-1}+az^{-1}+a^{-1}z^{-1}+2z^{10}+2z^{8}-14z^{6}+17z^{4}-6z^{2}-1$ (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   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 6 χ 14                       1 1 12                     3   -3 10                   5 1   4 8                 7 3     -4 6               9 5       4 4             9 7         -2 2           9 9           0 0         7 11             4 -2       4 7               -3 -4     2 7                 5 -6   1 4                   -3 -8   2                     2 -10 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=0$ $i=2$ $r=-5$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-2$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-1$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=0$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{9}$ $r=1$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=2$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=3$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=5$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=6$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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