L11a370

L11a369

L11a371

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a370 at Knotilus!

Link Presentations

[edit Notes on L11a370's Link Presentations]

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

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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