L11a399

L11a398

L11a400

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a399 at Knotilus!

Link Presentations

[edit Notes on L11a399's Link Presentations]

Planar diagram presentation	X6172 X12,6,13,5 X8493 X2,14,3,13 X14,7,15,8 X4,15,1,16 X18,22,19,21 X20,9,21,10 X10,19,5,20 X16,12,17,11 X22,18,11,17
Gauss code	{1, -4, 3, -6}, {2, -1, 5, -3, 8, -9}, {10, -2, 4, -5, 6, -10, 11, -7, 9, -8, 7, -11}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {2(t(1)-1)(t(2)-1)^{2}(t(3)-1)^{2}}{{\sqrt {t(1)}}t(2)t(3)}}$ (db)
Jones polynomial	$q^{6}-4q^{5}-q^{-5}+8q^{4}+4q^{-4}-13q^{3}-7q^{-3}+18q^{2}+14q^{-2}-20q-17q^{-1}+21$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$z^{4}a^{-4}-a^{4}z^{2}+z^{2}a^{-4}+a^{4}z^{-2}-z^{6}a^{-2}+2a^{2}z^{4}-2z^{4}a^{-2}+2a^{2}z^{2}-2z^{2}a^{-2}-2a^{2}z^{-2}-a^{2}-z^{6}-z^{4}+z^{-2}+1$ (db)
Kauffman polynomial	$z^{6}a^{-6}-2z^{4}a^{-6}+4z^{7}a^{-5}+a^{5}z^{5}-10z^{5}a^{-5}-a^{5}z^{3}+5z^{3}a^{-5}+7z^{8}a^{-4}+4a^{4}z^{6}-19z^{6}a^{-4}-7a^{4}z^{4}+15z^{4}a^{-4}+4a^{4}z^{2}-4z^{2}a^{-4}-a^{4}z^{-2}+a^{4}+6z^{9}a^{-3}+6a^{3}z^{7}-11z^{7}a^{-3}-7a^{3}z^{5}+2z^{5}a^{-3}+a^{3}z^{3}+3z^{3}a^{-3}-a^{3}z+2a^{3}z^{-1}+2z^{10}a^{-2}+6a^{2}z^{8}+9z^{8}a^{-2}-a^{2}z^{6}-32z^{6}a^{-2}-12a^{2}z^{4}+30z^{4}a^{-2}+8a^{2}z^{2}-8z^{2}a^{-2}-2a^{2}z^{-2}+a^{2}+5az^{9}+11z^{9}a^{-1}-az^{7}-22z^{7}a^{-1}-3az^{5}+17z^{5}a^{-1}-2az^{3}-6z^{3}a^{-1}-az+2az^{-1}+2z^{10}+8z^{8}-17z^{6}+8z^{4}-z^{-2}+1$ (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   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 6 χ 13                       1 1 11                     3   -3 9                   5 1   4 7                 8 3     -5 5               10 5       5 3             10 8         -2 1           11 10           1 -1         8 12             4 -3       6 9               -3 -5     3 10                 7 -7   1 4                   -3 -9   3                     3 -11 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-1$ $i=1$ $r=-5$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-2$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{6}$ $r=-1$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=0$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{11}$ $r=1$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=2$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=3$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=5$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=6$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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