L11a238

L11a237

L11a239

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a238 at Knotilus!

Link Presentations

[edit Notes on L11a238's Link Presentations]

Planar diagram presentation	X8192 X12,4,13,3 X22,12,7,11 X16,9,17,10 X14,22,15,21 X10,15,11,16 X18,6,19,5 X20,18,21,17 X2738 X4,14,5,13 X6,20,1,19
Gauss code	{1, -9, 2, -10, 7, -11}, {9, -1, 4, -6, 3, -2, 10, -5, 6, -4, 8, -7, 11, -8, 5, -3}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {t(1)^{2}t(2)^{4}-2t(1)t(2)^{4}+t(2)^{4}-5t(1)^{2}t(2)^{3}+8t(1)t(2)^{3}-4t(2)^{3}+7t(1)^{2}t(2)^{2}-11t(1)t(2)^{2}+7t(2)^{2}-4t(1)^{2}t(2)+8t(1)t(2)-5t(2)+t(1)^{2}-2t(1)+1}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	$q^{17/2}-4q^{15/2}+9q^{13/2}-15q^{11/2}+19q^{9/2}-22q^{7/2}+21q^{5/2}-18q^{3/2}+13{\sqrt {q}}-{\frac {8}{\sqrt {q}}}+{\frac {3}{q^{3/2}}}-{\frac {1}{q^{5/2}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z^{3}a^{-7}+za^{-7}-2z^{5}a^{-5}-4z^{3}a^{-5}-2za^{-5}-a^{-5}z^{-1}+z^{7}a^{-3}+3z^{5}a^{-3}+4z^{3}a^{-3}+4za^{-3}+2a^{-3}z^{-1}-2z^{5}a^{-1}+az^{3}-5z^{3}a^{-1}+2az-4za^{-1}+az^{-1}-2a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{4}a^{-10}+4z^{5}a^{-9}-z^{3}a^{-9}+9z^{6}a^{-8}-7z^{4}a^{-8}+2z^{2}a^{-8}+14z^{7}a^{-7}-20z^{5}a^{-7}+12z^{3}a^{-7}-3za^{-7}+14z^{8}a^{-6}-21z^{6}a^{-6}+8z^{4}a^{-6}-z^{2}a^{-6}+8z^{9}a^{-5}+z^{7}a^{-5}-33z^{5}a^{-5}+29z^{3}a^{-5}-9za^{-5}+a^{-5}z^{-1}+2z^{10}a^{-4}+18z^{8}a^{-4}-55z^{6}a^{-4}+39z^{4}a^{-4}-8z^{2}a^{-4}+12z^{9}a^{-3}-22z^{7}a^{-3}-10z^{5}a^{-3}+28z^{3}a^{-3}-13za^{-3}+2a^{-3}z^{-1}+2z^{10}a^{-2}+7z^{8}a^{-2}-35z^{6}a^{-2}+34z^{4}a^{-2}-9z^{2}a^{-2}+a^{-2}+4z^{9}a^{-1}+az^{7}-8z^{7}a^{-1}-4az^{5}-5z^{5}a^{-1}+6az^{3}+18z^{3}a^{-1}-4az-11za^{-1}+az^{-1}+2a^{-1}z^{-1}+3z^{8}-10z^{6}+11z^{4}-4z^{2}$ (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   \ -4 -3 -2 -1 0 1 2 3 4 5 6 7 χ 18                       1 -1 16                     3   3 14                   6 1   -5 12                 9 3     6 10               10 6       -4 8             12 9         3 6           10 11           1 4         8 11             -3 2       6 11               5 0     2 7                 -5 -2   1 6                   5 -4   2                     -2 -6 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=2$ $i=4$ $r=-4$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-1$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=0$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{8}$ $r=1$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=2$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{12}$ $r=3$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=4$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=5$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=6$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=7$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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