L11a405

L11a404

L11a406

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a405 at Knotilus!

Link Presentations

[edit Notes on L11a405's Link Presentations]

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {-t(2)^{2}t(3)^{4}+t(1)t(3)^{4}-t(1)t(2)t(3)^{4}+2t(2)t(3)^{4}-t(3)^{4}-t(1)t(2)^{2}t(3)^{3}+2t(2)^{2}t(3)^{3}-2t(1)t(3)^{3}+4t(1)t(2)t(3)^{3}-4t(2)t(3)^{3}+t(3)^{3}+t(1)t(2)^{2}t(3)^{2}-2t(2)^{2}t(3)^{2}+2t(1)t(3)^{2}-4t(1)t(2)t(3)^{2}+4t(2)t(3)^{2}-t(3)^{2}-t(1)t(2)^{2}t(3)+2t(2)^{2}t(3)-2t(1)t(3)+4t(1)t(2)t(3)-4t(2)t(3)+t(3)+t(1)t(2)^{2}-t(2)^{2}+t(1)-2t(1)t(2)+t(2)}{{\sqrt {t(1)}}t(2)t(3)^{2}}}$ (db)
Jones polynomial	$1-3q^{-1}+7q^{-2}-10q^{-3}+16q^{-4}-16q^{-5}+18q^{-6}-15q^{-7}+11q^{-8}-7q^{-9}+3q^{-10}-q^{-11}$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$-z^{2}a^{10}-a^{10}z^{-2}-2a^{10}+3z^{4}a^{8}+9z^{2}a^{8}+4a^{8}z^{-2}+9a^{8}-2z^{6}a^{6}-8z^{4}a^{6}-14z^{2}a^{6}-5a^{6}z^{-2}-14a^{6}-z^{6}a^{4}-z^{4}a^{4}+5z^{2}a^{4}+2a^{4}z^{-2}+7a^{4}+z^{4}a^{2}+2z^{2}a^{2}$ (db)
Kauffman polynomial	$a^{13}z^{5}-2a^{13}z^{3}+a^{13}z+3a^{12}z^{6}-5a^{12}z^{4}+a^{12}z^{2}+5a^{11}z^{7}-8a^{11}z^{5}+4a^{11}z^{3}-4a^{11}z+a^{11}z^{-1}+6a^{10}z^{8}-11a^{10}z^{6}+11a^{10}z^{4}-10a^{10}z^{2}-a^{10}z^{-2}+5a^{10}+4a^{9}z^{9}-18a^{9}z^{5}+33a^{9}z^{3}-21a^{9}z+5a^{9}z^{-1}+a^{8}z^{10}+13a^{8}z^{8}-48a^{8}z^{6}+72a^{8}z^{4}-48a^{8}z^{2}-4a^{8}z^{-2}+18a^{8}+8a^{7}z^{9}-14a^{7}z^{7}-6a^{7}z^{5}+36a^{7}z^{3}-29a^{7}z+9a^{7}z^{-1}+a^{6}z^{10}+12a^{6}z^{8}-50a^{6}z^{6}+77a^{6}z^{4}-57a^{6}z^{2}-5a^{6}z^{-2}+21a^{6}+4a^{5}z^{9}-6a^{5}z^{7}-5a^{5}z^{5}+13a^{5}z^{3}-13a^{5}z+5a^{5}z^{-1}+5a^{4}z^{8}-15a^{4}z^{6}+18a^{4}z^{4}-18a^{4}z^{2}-2a^{4}z^{-2}+9a^{4}+3a^{3}z^{7}-8a^{3}z^{5}+4a^{3}z^{3}+a^{2}z^{6}-3a^{2}z^{4}+2a^{2}z^{2}$ (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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 1                       1 1 -1                     2   -2 -3                   5 1   4 -5                 6 3     -3 -7               10 4       6 -9             8 8         0 -11           10 8           2 -13         6 9             3 -15       5 9               -4 -17     2 6                 4 -19   1 5                   -4 -21   2                     2 -23 1                       -1

Integral Khovanov Homology (db, data source)

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

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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