L11a53

L11a52

L11a54

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a53 at Knotilus!

Link Presentations

[edit Notes on L11a53's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(u-1)(v-1)\left(v^{2}+1\right)\left(2v^{2}-3v+2\right)}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$-q^{11/2}+4q^{9/2}-7q^{7/2}+11q^{5/2}-16q^{3/2}+17{\sqrt {q}}-{\frac {18}{\sqrt {q}}}+{\frac {15}{q^{3/2}}}-{\frac {12}{q^{5/2}}}+{\frac {7}{q^{7/2}}}-{\frac {3}{q^{9/2}}}+{\frac {1}{q^{11/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$az^{7}+z^{7}a^{-1}-a^{3}z^{5}+4az^{5}+3z^{5}a^{-1}-z^{5}a^{-3}-3a^{3}z^{3}+7az^{3}-2z^{3}a^{-3}-3a^{3}z+8az-6za^{-1}+za^{-3}-a^{3}z^{-1}+4az^{-1}-4a^{-1}z^{-1}+a^{-3}z^{-1}$ (db)
Kauffman polynomial	$a^{6}z^{4}-a^{6}z^{2}+z^{7}a^{-5}+3a^{5}z^{5}-3z^{5}a^{-5}-2a^{5}z^{3}+2z^{3}a^{-5}+4z^{8}a^{-4}+6a^{4}z^{6}-15z^{6}a^{-4}-6a^{4}z^{4}+15z^{4}a^{-4}+4a^{4}z^{2}-z^{2}a^{-4}-a^{4}-a^{-4}+5z^{9}a^{-3}+9a^{3}z^{7}-17z^{7}a^{-3}-15a^{3}z^{5}+16z^{5}a^{-3}+15a^{3}z^{3}-3z^{3}a^{-3}-6a^{3}z-2za^{-3}+a^{3}z^{-1}+a^{-3}z^{-1}+2z^{10}a^{-2}+8a^{2}z^{8}+3z^{8}a^{-2}-8a^{2}z^{6}-23z^{6}a^{-2}-6a^{2}z^{4}+18z^{4}a^{-2}+12a^{2}z^{2}+3z^{2}a^{-2}-4a^{2}-4a^{-2}+5az^{9}+10z^{9}a^{-1}+2az^{7}-25z^{7}a^{-1}-26az^{5}+11z^{5}a^{-1}+29az^{3}+7z^{3}a^{-1}-15az-11za^{-1}+4az^{-1}+4a^{-1}z^{-1}+2z^{10}+7z^{8}-22z^{6}+4z^{4}+11z^{2}-7$ (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 12                       1 1 10                     3   -3 8                   4 1   3 6                 7 3     -4 4               9 4       5 2             8 7         -1 0           10 9           1 -2         7 10             3 -4       5 8               -3 -6     2 7                 5 -8   1 5                   -4 -10   2                     2 -12 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-2$ $i=0$ $r=-5$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-2$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=-1$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=0$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{10}$ $r=1$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=2$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=3$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=5$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=6$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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