L11a259

L11a258

L11a260

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a259 at Knotilus!

Link Presentations

[edit Notes on L11a259's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {t(1)^{2}t(2)^{5}-t(1)t(2)^{5}+t(1)^{3}t(2)^{4}-3t(1)^{2}t(2)^{4}+2t(1)t(2)^{4}-t(2)^{4}-t(1)^{3}t(2)^{3}+3t(1)^{2}t(2)^{3}-3t(1)t(2)^{3}+t(2)^{3}+t(1)^{3}t(2)^{2}-3t(1)^{2}t(2)^{2}+3t(1)t(2)^{2}-t(2)^{2}-t(1)^{3}t(2)+2t(1)^{2}t(2)-3t(1)t(2)+t(2)-t(1)^{2}+t(1)}{t(1)^{3/2}t(2)^{5/2}}}$ (db)
Jones polynomial	$-{\frac {9}{q^{9/2}}}+{\frac {7}{q^{7/2}}}-{\frac {5}{q^{5/2}}}+{\frac {2}{q^{3/2}}}+{\frac {1}{q^{23/2}}}-{\frac {2}{q^{21/2}}}+{\frac {4}{q^{19/2}}}-{\frac {7}{q^{17/2}}}+{\frac {9}{q^{15/2}}}-{\frac {11}{q^{13/2}}}+{\frac {10}{q^{11/2}}}-{\frac {1}{\sqrt {q}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$-z^{5}a^{9}-4z^{3}a^{9}-4za^{9}-a^{9}z^{-1}+z^{7}a^{7}+5z^{5}a^{7}+9z^{3}a^{7}+8za^{7}+3a^{7}z^{-1}+z^{7}a^{5}+4z^{5}a^{5}+3z^{3}a^{5}-2za^{5}-2a^{5}z^{-1}-z^{5}a^{3}-4z^{3}a^{3}-4za^{3}$ (db)
Kauffman polynomial	$a^{14}z^{4}-2a^{14}z^{2}+2a^{13}z^{5}-3a^{13}z^{3}+3a^{12}z^{6}-5a^{12}z^{4}+3a^{12}z^{2}+4a^{11}z^{7}-10a^{11}z^{5}+13a^{11}z^{3}-4a^{11}z+4a^{10}z^{8}-11a^{10}z^{6}+14a^{10}z^{4}-4a^{10}z^{2}+a^{10}+3a^{9}z^{9}-8a^{9}z^{7}+9a^{9}z^{5}-3a^{9}z^{3}+a^{9}z-a^{9}z^{-1}+a^{8}z^{10}+2a^{8}z^{8}-14a^{8}z^{6}+20a^{8}z^{4}-14a^{8}z^{2}+3a^{8}+5a^{7}z^{9}-19a^{7}z^{7}+29a^{7}z^{5}-30a^{7}z^{3}+15a^{7}z-3a^{7}z^{-1}+a^{6}z^{10}-8a^{6}z^{6}+8a^{6}z^{4}-6a^{6}z^{2}+3a^{6}+2a^{5}z^{9}-6a^{5}z^{7}+3a^{5}z^{5}-3a^{5}z^{3}+6a^{5}z-2a^{5}z^{-1}+2a^{4}z^{8}-8a^{4}z^{6}+8a^{4}z^{4}-a^{4}z^{2}+a^{3}z^{7}-5a^{3}z^{5}+8a^{3}z^{3}-4a^{3}z$ (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                   4 1   3 -6                 4 2     -2 -8               5 3       2 -10             5 4         -1 -12           6 5           1 -14         4 6             2 -16       3 5               -2 -18     1 4                 3 -20   1 3                   -2 -22   1                     1 -24 1                       -1

Integral Khovanov Homology (db, data source)

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

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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