L11a253

L11a252

L11a254

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a253 at Knotilus!

Link Presentations

[edit Notes on L11a253's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {t(1)^{3}t(2)^{5}-t(1)^{2}t(2)^{5}-2t(1)^{3}t(2)^{4}+3t(1)^{2}t(2)^{4}-t(1)t(2)^{4}+2t(1)^{3}t(2)^{3}-5t(1)^{2}t(2)^{3}+3t(1)t(2)^{3}-t(2)^{3}-t(1)^{3}t(2)^{2}+3t(1)^{2}t(2)^{2}-5t(1)t(2)^{2}+2t(2)^{2}-t(1)^{2}t(2)+3t(1)t(2)-2t(2)-t(1)+1}{t(1)^{3/2}t(2)^{5/2}}}$ (db)
Jones polynomial	${\frac {10}{q^{9/2}}}-{\frac {12}{q^{7/2}}}-q^{5/2}+{\frac {11}{q^{5/2}}}+3q^{3/2}-{\frac {10}{q^{3/2}}}+{\frac {1}{q^{17/2}}}-{\frac {3}{q^{15/2}}}+{\frac {5}{q^{13/2}}}-{\frac {8}{q^{11/2}}}-5{\sqrt {q}}+{\frac {7}{\sqrt {q}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{3}z^{9}-a^{5}z^{7}+7a^{3}z^{7}-az^{7}-5a^{5}z^{5}+17a^{3}z^{5}-5az^{5}-7a^{5}z^{3}+16a^{3}z^{3}-7az^{3}-2a^{5}z+3a^{3}z-3az+a^{5}z^{-1}-a^{3}z^{-1}$ (db)
Kauffman polynomial	$-z^{4}a^{10}+z^{2}a^{10}-3z^{5}a^{9}+4z^{3}a^{9}-za^{9}-4z^{6}a^{8}+4z^{4}a^{8}-4z^{7}a^{7}+3z^{5}a^{7}+2z^{3}a^{7}-2za^{7}-4z^{8}a^{6}+6z^{6}a^{6}-3z^{4}a^{6}-4z^{9}a^{5}+12z^{7}a^{5}-17z^{5}a^{5}+11z^{3}a^{5}-za^{5}-a^{5}z^{-1}-2z^{10}a^{4}+3z^{8}a^{4}+5z^{6}a^{4}-10z^{4}a^{4}+3z^{2}a^{4}+a^{4}-8z^{9}a^{3}+35z^{7}a^{3}-51z^{5}a^{3}+30z^{3}a^{3}-4za^{3}-a^{3}z^{-1}-2z^{10}a^{2}+4z^{8}a^{2}+8z^{6}a^{2}-16z^{4}a^{2}+6z^{2}a^{2}-4z^{9}a+18z^{7}a-24z^{5}a+14z^{3}a-4za-3z^{8}+13z^{6}-14z^{4}+4z^{2}-z^{7}a^{-1}+4z^{5}a^{-1}-3z^{3}a^{-1}$ (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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 6                       1 1 4                     2   -2 2                   3 1   2 0                 4 2     -2 -2               6 3       3 -4             6 5         -1 -6           6 5           1 -8         4 6             2 -10       4 6               -2 -12     2 5                 3 -14   1 3                   -2 -16   2                     2 -18 1                       -1

Integral Khovanov Homology (db, data source)

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

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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