L11a372

L11a371

L11a373

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a372 at Knotilus!

Link Presentations

[edit Notes on L11a372's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {t(2)^{3}t(1)^{4}-t(2)^{2}t(1)^{4}+t(2)^{4}t(1)^{3}-t(2)^{3}t(1)^{3}+t(2)^{2}t(1)^{3}-t(2)t(1)^{3}-t(2)^{4}t(1)^{2}+t(2)^{3}t(1)^{2}-t(2)^{2}t(1)^{2}+t(2)t(1)^{2}-t(1)^{2}-t(2)^{3}t(1)+t(2)^{2}t(1)-t(2)t(1)+t(1)-t(2)^{2}+t(2)}{t(1)^{2}t(2)^{2}}}$ (db)
Jones polynomial	$-{\frac {5}{q^{9/2}}}+{\frac {4}{q^{7/2}}}-{\frac {4}{q^{5/2}}}-q^{3/2}+{\frac {3}{q^{3/2}}}+{\frac {1}{q^{19/2}}}-{\frac {2}{q^{17/2}}}+{\frac {3}{q^{15/2}}}-{\frac {4}{q^{13/2}}}+{\frac {4}{q^{11/2}}}+{\sqrt {q}}-{\frac {2}{\sqrt {q}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$-z^{5}a^{7}-4z^{3}a^{7}-3za^{7}+z^{7}a^{5}+5z^{5}a^{5}+6z^{3}a^{5}+za^{5}+z^{7}a^{3}+6z^{5}a^{3}+11z^{3}a^{3}+7za^{3}+a^{3}z^{-1}-z^{5}a-5z^{3}a-6za-az^{-1}$ (db)
Kauffman polynomial	$-z^{2}a^{12}-2z^{3}a^{11}-3z^{4}a^{10}+2z^{2}a^{10}-4z^{5}a^{9}+7z^{3}a^{9}-2za^{9}-4z^{6}a^{8}+9z^{4}a^{8}-2z^{2}a^{8}-4z^{7}a^{7}+13z^{5}a^{7}-9z^{3}a^{7}+2za^{7}-3z^{8}a^{6}+11z^{6}a^{6}-8z^{4}a^{6}-2z^{9}a^{5}+8z^{7}a^{5}-5z^{5}a^{5}-5z^{3}a^{5}+2za^{5}-z^{10}a^{4}+4z^{8}a^{4}-z^{6}a^{4}-7z^{4}a^{4}+3z^{2}a^{4}-3z^{9}a^{3}+20z^{7}a^{3}-44z^{5}a^{3}+37z^{3}a^{3}-11za^{3}+a^{3}z^{-1}-z^{10}a^{2}+7z^{8}a^{2}-16z^{6}a^{2}+13z^{4}a^{2}-2z^{2}a^{2}-a^{2}-z^{9}a+8z^{7}a-22z^{5}a+24z^{3}a-9za+az^{-1}$ (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 4                       1 1 2                         0 0                   2 1   1 -2                 1       -1 -4               3 2       1 -6             2 2         0 -8           3 2           1 -10         2 3             1 -12       2 2               0 -14     1 2                 1 -16   1 2                   -1 -18   1                     1 -20 1                       -1

Integral Khovanov Homology (db, data source)

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

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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