L11a391

L11a390

L11a392

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a391 at Knotilus!

Link Presentations

[edit Notes on L11a391's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X16,7,17,8 X8,15,5,16 X18,11,19,12 X22,17,9,18 X20,13,21,14 X12,19,13,20 X14,21,15,22 X2536 X4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -4}, {11, -2, 5, -8, 7, -9, 4, -3, 6, -5, 8, -7, 9, -6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {t(1)t(3)^{5}+t(2)t(3)^{5}-t(3)^{5}-3t(1)t(3)^{4}+2t(1)t(2)t(3)^{4}-3t(2)t(3)^{4}+2t(3)^{4}+3t(1)t(3)^{3}-2t(1)t(2)t(3)^{3}+3t(2)t(3)^{3}-2t(3)^{3}-3t(1)t(3)^{2}+2t(1)t(2)t(3)^{2}-3t(2)t(3)^{2}+2t(3)^{2}+3t(1)t(3)-2t(1)t(2)t(3)+3t(2)t(3)-2t(3)-t(1)+t(1)t(2)-t(2)}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)^{5/2}}}$ (db)
Jones polynomial	$-q^{-14}+2q^{-13}-6q^{-12}+9q^{-11}-12q^{-10}+15q^{-9}-14q^{-8}+14q^{-7}-8q^{-6}+7q^{-5}-3q^{-4}+q^{-3}$ (db)
Signature	-6 (db)
HOMFLY-PT polynomial	$-2a^{14}z^{-2}-a^{14}+4a^{12}z^{2}+7a^{12}z^{-2}+12a^{12}-6a^{10}z^{4}-23a^{10}z^{2}-8a^{10}z^{-2}-24a^{10}+3a^{8}z^{6}+14a^{8}z^{4}+21a^{8}z^{2}+3a^{8}z^{-2}+13a^{8}+a^{6}z^{6}+3a^{6}z^{4}+a^{6}z^{2}$ (db)
Kauffman polynomial	$a^{17}z^{5}-3a^{17}z^{3}+3a^{17}z-a^{17}z^{-1}+2a^{16}z^{6}-3a^{16}z^{4}+a^{16}+3a^{15}z^{7}-3a^{15}z^{5}-2a^{15}z^{3}+3a^{15}z-a^{15}z^{-1}+3a^{14}z^{8}-2a^{14}z^{6}+2a^{14}z^{4}-9a^{14}z^{2}-2a^{14}z^{-2}+7a^{14}+2a^{13}z^{9}+3a^{13}z^{7}-13a^{13}z^{5}+22a^{13}z^{3}-21a^{13}z+7a^{13}z^{-1}+a^{12}z^{10}+5a^{12}z^{8}-18a^{12}z^{6}+33a^{12}z^{4}-35a^{12}z^{2}-7a^{12}z^{-2}+22a^{12}+6a^{11}z^{9}-11a^{11}z^{7}-8a^{11}z^{5}+46a^{11}z^{3}-45a^{11}z+15a^{11}z^{-1}+a^{10}z^{10}+8a^{10}z^{8}-38a^{10}z^{6}+62a^{10}z^{4}-51a^{10}z^{2}-8a^{10}z^{-2}+28a^{10}+4a^{9}z^{9}-8a^{9}z^{7}-7a^{9}z^{5}+27a^{9}z^{3}-24a^{9}z+8a^{9}z^{-1}+6a^{8}z^{8}-23a^{8}z^{6}+31a^{8}z^{4}-24a^{8}z^{2}-3a^{8}z^{-2}+13a^{8}+3a^{7}z^{7}-8a^{7}z^{5}+2a^{7}z^{3}+a^{6}z^{6}-3a^{6}z^{4}+a^{6}z^{2}$ (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   \ -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -5                       1 1 -7                     3 1 -2 -9                   4     4 -11                 4 3     -1 -13               10 4       6 -15             7 7         0 -17           8 7           1 -19         4 7             3 -21       5 8               -3 -23     1 4                 3 -25   1 5                   -4 -27   1                     1 -29 1                       -1

Integral Khovanov Homology (db, data source)

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

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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