L11a519

L11a518

L11a520

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a519 at Knotilus!

Link Presentations

[edit Notes on L11a519's Link Presentations]

Planar diagram presentation	X8192 X16,6,17,5 X22,20,13,19 X18,10,19,9 X10,21,11,22 X20,11,21,12 X2,13,3,14 X14,3,15,4 X4758 X12,16,7,15 X6,18,1,17
Gauss code	{1, -7, 8, -9, 2, -11}, {9, -1, 4, -5, 6, -10}, {7, -8, 10, -2, 11, -4, 3, -6, 5, -3}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(t(2)-1)^{2}(t(3)-1)(t(3)t(1)-t(1)+1)(t(1)t(3)-t(3)+1)}{t(1)t(2)t(3)^{3/2}}}$ (db)
Jones polynomial	$q^{-6}-q^{5}-4q^{-5}+4q^{4}+9q^{-4}-9q^{3}-14q^{-3}+15q^{2}+21q^{-2}-20q-22q^{-1}+24$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a^{4}z^{4}+2a^{4}z^{2}+a^{4}z^{-2}+a^{4}-2a^{2}z^{6}-z^{6}a^{-2}-7a^{2}z^{4}-3z^{4}a^{-2}-8a^{2}z^{2}-3z^{2}a^{-2}-2a^{2}z^{-2}-4a^{2}-a^{-2}+z^{8}+5z^{6}+10z^{4}+9z^{2}+z^{-2}+4$ (db)
Kauffman polynomial	$2a^{2}z^{10}+2z^{10}+6a^{3}z^{9}+13az^{9}+7z^{9}a^{-1}+7a^{4}z^{8}+14a^{2}z^{8}+10z^{8}a^{-2}+17z^{8}+4a^{5}z^{7}-6a^{3}z^{7}-20az^{7}-2z^{7}a^{-1}+8z^{7}a^{-3}+a^{6}z^{6}-16a^{4}z^{6}-44a^{2}z^{6}-16z^{6}a^{-2}+4z^{6}a^{-4}-47z^{6}-9a^{5}z^{5}-7a^{3}z^{5}+3az^{5}-12z^{5}a^{-1}-12z^{5}a^{-3}+z^{5}a^{-5}-2a^{6}z^{4}+12a^{4}z^{4}+45a^{2}z^{4}+12z^{4}a^{-2}-5z^{4}a^{-4}+48z^{4}+5a^{5}z^{3}+6a^{3}z^{3}+6az^{3}+12z^{3}a^{-1}+6z^{3}a^{-3}-z^{3}a^{-5}+a^{6}z^{2}-6a^{4}z^{2}-25a^{2}z^{2}-6z^{2}a^{-2}+z^{2}a^{-4}-25z^{2}-2a^{3}z-4az-3za^{-1}-za^{-3}+3a^{4}+7a^{2}+2a^{-2}+7+2a^{3}z^{-1}+2az^{-1}-a^{4}z^{-2}-2a^{2}z^{-2}-z^{-2}$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 11                       1 -1 9                     3   3 7                   6 1   -5 5                 9 3     6 3               11 6       -5 1             13 9         4 -1           11 13           2 -3         10 11             -1 -5       6 13               7 -7     3 8                 -5 -9   1 6                   5 -11   3                     -3 -13 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-1$ $i=1$ $r=-6$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-3$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=-2$ ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{10}$ $r=-1$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=0$ ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{13}$ $r=1$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=2$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=3$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=4$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=5$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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