L11a387

L11a386

L11a388

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a387 at Knotilus!

Link Presentations

[edit Notes on L11a387's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {2uvw^{4}-3uvw^{3}+3uvw^{2}-2uvw+uv+uw^{5}-3uw^{4}+4uw^{3}-4uw^{2}+3uw-u+vw^{5}-3vw^{4}+4vw^{3}-4vw^{2}+3vw-v-w^{5}+2w^{4}-3w^{3}+3w^{2}-2w}{{\sqrt {u}}{\sqrt {v}}w^{5/2}}}$ (db)
Jones polynomial	$q^{-3}-3q^{-4}+8q^{-5}-11q^{-6}+17q^{-7}-16q^{-8}+18q^{-9}-15q^{-10}+10q^{-11}-6q^{-12}+2q^{-13}-q^{-14}$ (db)
Signature	-6 (db)
HOMFLY-PT polynomial	$-2a^{14}z^{-2}-a^{14}+4a^{12}z^{2}+7a^{12}z^{-2}+12a^{12}-6a^{10}z^{4}-22a^{10}z^{2}-8a^{10}z^{-2}-24a^{10}+3a^{8}z^{6}+13a^{8}z^{4}+19a^{8}z^{2}+3a^{8}z^{-2}+13a^{8}+a^{6}z^{6}+3a^{6}z^{4}+2a^{6}z^{2}$ (db)
Kauffman polynomial	$a^{17}z^{5}-3a^{17}z^{3}+3a^{17}z-a^{17}z^{-1}+2a^{16}z^{6}-3a^{16}z^{4}+a^{16}+3a^{15}z^{7}-2a^{15}z^{5}-3a^{15}z^{3}+3a^{15}z-a^{15}z^{-1}+4a^{14}z^{8}-5a^{14}z^{6}+7a^{14}z^{4}-9a^{14}z^{2}-2a^{14}z^{-2}+7a^{14}+3a^{13}z^{9}+2a^{13}z^{7}-14a^{13}z^{5}+26a^{13}z^{3}-21a^{13}z+7a^{13}z^{-1}+a^{12}z^{10}+10a^{12}z^{8}-31a^{12}z^{6}+45a^{12}z^{4}-38a^{12}z^{2}-7a^{12}z^{-2}+22a^{12}+7a^{11}z^{9}-6a^{11}z^{7}-23a^{11}z^{5}+54a^{11}z^{3}-45a^{11}z+15a^{11}z^{-1}+a^{10}z^{10}+12a^{10}z^{8}-44a^{10}z^{6}+64a^{10}z^{4}-56a^{10}z^{2}-8a^{10}z^{-2}+28a^{10}+4a^{9}z^{9}-2a^{9}z^{7}-19a^{9}z^{5}+31a^{9}z^{3}-24a^{9}z+8a^{9}z^{-1}+6a^{8}z^{8}-19a^{8}z^{6}+26a^{8}z^{4}-25a^{8}z^{2}-3a^{8}z^{-2}+13a^{8}+3a^{7}z^{7}-7a^{7}z^{5}+3a^{7}z^{3}+a^{6}z^{6}-3a^{6}z^{4}+2a^{6}z^{2}$ (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                   5     5 -11                 6 3     -3 -13               11 5       6 -15             8 9         1 -17           10 8           2 -19         5 8             3 -21       5 10               -5 -23     1 5                 4 -25   1 5                   -4 -27   1                     1 -29 1                       -1

Integral Khovanov Homology (db, data source)

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

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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