L11a406

L11a405

L11a407

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a406 at Knotilus!

Link Presentations

[edit Notes on L11a406's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {2uv^{2}w-2uv^{2}+2uvw^{2}-7uvw+4uv-2uw^{2}+4uw-2u+2v^{2}w^{2}-4v^{2}w+2v^{2}-4vw^{2}+7vw-2v+2w^{2}-2w}{{\sqrt {u}}vw}}$ (db)
Jones polynomial	$-q^{-10}+3q^{-9}-6q^{-8}+10q^{-7}-13q^{-6}+16q^{-5}-15q^{-4}+15q^{-3}-10q^{-2}+q+7q^{-1}-3$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$-a^{10}+3a^{8}z^{2}+2a^{8}-2a^{6}z^{4}-a^{6}z^{2}+a^{6}z^{-2}-3a^{4}z^{4}-4a^{4}z^{2}-2a^{4}z^{-2}-4a^{4}-a^{2}z^{4}+2a^{2}z^{2}+a^{2}z^{-2}+3a^{2}+z^{2}$ (db)
Kauffman polynomial	$z^{7}a^{11}-4z^{5}a^{11}+5z^{3}a^{11}-2za^{11}+3z^{8}a^{10}-12z^{6}a^{10}+15z^{4}a^{10}-7z^{2}a^{10}+2a^{10}+3z^{9}a^{9}-7z^{7}a^{9}-3z^{5}a^{9}+12z^{3}a^{9}-4za^{9}+z^{10}a^{8}+7z^{8}a^{8}-34z^{6}a^{8}+42z^{4}a^{8}-21z^{2}a^{8}+4a^{8}+7z^{9}a^{7}-15z^{7}a^{7}+3z^{5}a^{7}+2z^{3}a^{7}+z^{10}a^{6}+10z^{8}a^{6}-30z^{6}a^{6}+23z^{4}a^{6}-7z^{2}a^{6}+a^{6}z^{-2}-2a^{6}+4z^{9}a^{5}-8z^{5}a^{5}-2z^{3}a^{5}+6za^{5}-2a^{5}z^{-1}+6z^{8}a^{4}-2z^{6}a^{4}-13z^{4}a^{4}+16z^{2}a^{4}+2a^{4}z^{-2}-7a^{4}+7z^{7}a^{3}-7z^{5}a^{3}+z^{3}a^{3}+4za^{3}-2a^{3}z^{-1}+6z^{6}a^{2}-8z^{4}a^{2}+8z^{2}a^{2}+a^{2}z^{-2}-4a^{2}+3z^{5}a-2z^{3}a+z^{4}-z^{2}$ (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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 3                       1 1 1                     2   -2 -1                   5 1   4 -3                 6 3     -3 -5               9 4       5 -7             8 8         0 -9           8 7           1 -11         6 9             3 -13       4 7               -3 -15     2 6                 4 -17   1 4                   -3 -19   2                     2 -21 1                       -1

Integral Khovanov Homology (db, data source)

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

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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