L11a470

L11a469

L11a471

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a470 at Knotilus!

Link Presentations

[edit Notes on L11a470's Link Presentations]

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(w-1)\left(2uv^{2}w^{2}-2uv^{2}w-3uvw^{2}+3uvw-uv+uw^{2}-2uw+u+v^{2}w^{3}-2v^{2}w^{2}+v^{2}w-vw^{3}+3vw^{2}-3vw-2w^{2}+2w\right)}{{\sqrt {u}}vw^{2}}}$ (db)
Jones polynomial	$1-3q^{-1}+8q^{-2}-12q^{-3}+17q^{-4}-19q^{-5}+20q^{-6}-16q^{-7}+13q^{-8}-7q^{-9}+3q^{-10}-q^{-11}$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$-z^{2}a^{10}-2a^{10}+3z^{4}a^{8}+8z^{2}a^{8}+a^{8}z^{-2}+6a^{8}-2z^{6}a^{6}-7z^{4}a^{6}-10z^{2}a^{6}-2a^{6}z^{-2}-8a^{6}-z^{6}a^{4}-z^{4}a^{4}+3z^{2}a^{4}+a^{4}z^{-2}+3a^{4}+z^{4}a^{2}+2z^{2}a^{2}+a^{2}$ (db)
Kauffman polynomial	$a^{13}z^{5}-2a^{13}z^{3}+a^{13}z+3a^{12}z^{6}-5a^{12}z^{4}+3a^{12}z^{2}-a^{12}+5a^{11}z^{7}-6a^{11}z^{5}+a^{11}z^{3}+a^{11}z+6a^{10}z^{8}-7a^{10}z^{6}+4a^{10}z^{4}-2a^{10}z^{2}+4a^{9}z^{9}+3a^{9}z^{7}-15a^{9}z^{5}+14a^{9}z^{3}-3a^{9}z+a^{8}z^{10}+14a^{8}z^{8}-38a^{8}z^{6}+46a^{8}z^{4}-31a^{8}z^{2}-a^{8}z^{-2}+10a^{8}+8a^{7}z^{9}-7a^{7}z^{7}-12a^{7}z^{5}+19a^{7}z^{3}-10a^{7}z+2a^{7}z^{-1}+a^{6}z^{10}+13a^{6}z^{8}-40a^{6}z^{6}+48a^{6}z^{4}-36a^{6}z^{2}-2a^{6}z^{-2}+14a^{6}+4a^{5}z^{9}-2a^{5}z^{7}-11a^{5}z^{5}+12a^{5}z^{3}-7a^{5}z+2a^{5}z^{-1}+5a^{4}z^{8}-11a^{4}z^{6}+8a^{4}z^{4}-7a^{4}z^{2}-a^{4}z^{-2}+5a^{4}+3a^{3}z^{7}-7a^{3}z^{5}+4a^{3}z^{3}+a^{2}z^{6}-3a^{2}z^{4}+3a^{2}z^{2}-a^{2}$ (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                   6 1   5 -5                 7 3     -4 -7               10 5       5 -9             10 8         -2 -11           10 9           1 -13         7 11             4 -15       6 9               -3 -17     2 8                 6 -19   1 5                   -4 -21   2                     2 -23 1                       -1

Integral Khovanov Homology (db, data source)

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

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.

L11a469 L11a471