L11a343

L11a342

L11a344

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a343 at Knotilus!

Link Presentations

[edit Notes on L11a343's Link Presentations]

Planar diagram presentation	X12,1,13,2 X16,8,17,7 X10,5,1,6 X6374 X4,9,5,10 X18,16,19,15 X22,19,11,20 X20,13,21,14 X14,21,15,22 X2,11,3,12 X8,18,9,17
Gauss code	{1, -10, 4, -5, 3, -4, 2, -11, 5, -3}, {10, -1, 8, -9, 6, -2, 11, -6, 7, -8, 9, -7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(t(1)-1)(t(2)-1)\left(t(2)^{2}t(1)^{2}-2t(2)t(1)^{2}+t(1)^{2}-2t(2)^{2}t(1)+3t(2)t(1)-2t(1)+t(2)^{2}-2t(2)+1\right)}{t(1)^{3/2}t(2)^{3/2}}}$ (db)
Jones polynomial	$q^{3/2}-4{\sqrt {q}}+{\frac {8}{\sqrt {q}}}-{\frac {14}{q^{3/2}}}+{\frac {16}{q^{5/2}}}-{\frac {20}{q^{7/2}}}+{\frac {19}{q^{9/2}}}-{\frac {16}{q^{11/2}}}+{\frac {12}{q^{13/2}}}-{\frac {6}{q^{15/2}}}+{\frac {3}{q^{17/2}}}-{\frac {1}{q^{19/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{9}z+a^{9}z^{-1}-3a^{7}z^{3}-7a^{7}z-5a^{7}z^{-1}+3a^{5}z^{5}+10a^{5}z^{3}+14a^{5}z+8a^{5}z^{-1}-a^{3}z^{7}-4a^{3}z^{5}-8a^{3}z^{3}-9a^{3}z-4a^{3}z^{-1}+az^{5}+2az^{3}+az$ (db)
Kauffman polynomial	$a^{11}z^{5}-2a^{11}z^{3}+a^{11}z+3a^{10}z^{6}-6a^{10}z^{4}+5a^{10}z^{2}-2a^{10}+4a^{9}z^{7}-3a^{9}z^{5}-2a^{9}z^{3}+a^{9}z+a^{9}z^{-1}+4a^{8}z^{8}+3a^{8}z^{6}-18a^{8}z^{4}+22a^{8}z^{2}-9a^{8}+3a^{7}z^{9}+6a^{7}z^{7}-17a^{7}z^{5}+16a^{7}z^{3}-9a^{7}z+5a^{7}z^{-1}+a^{6}z^{10}+10a^{6}z^{8}-13a^{6}z^{6}-12a^{6}z^{4}+29a^{6}z^{2}-14a^{6}+7a^{5}z^{9}-27a^{5}z^{5}+33a^{5}z^{3}-21a^{5}z+8a^{5}z^{-1}+a^{4}z^{10}+12a^{4}z^{8}-26a^{4}z^{6}+5a^{4}z^{4}+13a^{4}z^{2}-8a^{4}+4a^{3}z^{9}+2a^{3}z^{7}-24a^{3}z^{5}+25a^{3}z^{3}-14a^{3}z+4a^{3}z^{-1}+6a^{2}z^{8}-12a^{2}z^{6}+3a^{2}z^{4}+2a^{2}z^{2}+4az^{7}-10az^{5}+8az^{3}-2az+z^{6}-2z^{4}+z^{2}$ (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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 χ 4                       1 -1 2                     3   3 0                   5 1   -4 -2                 9 3     6 -4               9 7       -2 -6             11 7         4 -8           8 9           1 -10         8 11             -3 -12       4 8               4 -14     2 8                 -6 -16   1 4                   3 -18   2                     -2 -20 1                       1

Integral Khovanov Homology (db, data source)

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

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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