L11n458

L11n457

L11n459

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n458 at Knotilus!

Link L11n458.

A graph, L11n458.

A part of a knot and a part of a graph.

Link Presentations

[edit Notes on L11n458's Link Presentations]

Planar diagram presentation	X6172 X3,13,4,12 X7,21,8,20 X19,5,20,10 X13,19,14,22 X21,11,22,18 X17,15,18,14 X9,17,10,16 X15,9,16,8 X2536 X11,1,12,4
Gauss code	{1, -10, -2, 11}, {-4, 3, -6, 5}, {10, -1, -3, 9, -8, 4}, {-11, 2, -5, 7, -9, 8, -7, 6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {t(1)t(4)^{2}t(3)^{2}-t(4)^{2}t(3)^{2}+t(1)t(3)^{2}-t(1)t(2)t(3)^{2}-2t(1)t(4)t(3)^{2}+2t(1)t(2)t(4)t(3)^{2}+t(4)t(3)^{2}-t(1)t(4)^{2}t(3)-2t(2)t(4)^{2}t(3)+2t(4)^{2}t(3)-2t(1)t(3)+2t(1)t(2)t(3)-t(2)t(3)+3t(1)t(4)t(3)-3t(1)t(2)t(4)t(3)+3t(2)t(4)t(3)-3t(4)t(3)+t(2)t(4)^{2}-t(4)^{2}-t(1)t(2)+t(2)+t(1)t(2)t(4)-2t(2)t(4)+2t(4)}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)t(4)}}$ (db)
Jones polynomial	$11q^{9/2}-11q^{7/2}+5q^{5/2}-3q^{3/2}+q^{21/2}-4q^{19/2}+7q^{17/2}-11q^{15/2}+12q^{13/2}-15q^{11/2}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z^{3}a^{-9}-a^{-9}z^{-3}-a^{-9}z^{-1}-z^{5}a^{-7}+3a^{-7}z^{-3}+4za^{-7}+6a^{-7}z^{-1}-2z^{5}a^{-5}-6z^{3}a^{-5}-3a^{-5}z^{-3}-10za^{-5}-9a^{-5}z^{-1}+3z^{3}a^{-3}+a^{-3}z^{-3}+6za^{-3}+4a^{-3}z^{-1}$ (db)
Kauffman polynomial	$z^{6}a^{-12}-2z^{4}a^{-12}+z^{2}a^{-12}+4z^{7}a^{-11}-11z^{5}a^{-11}+8z^{3}a^{-11}-2za^{-11}+5z^{8}a^{-10}-11z^{6}a^{-10}+2z^{4}a^{-10}+2z^{2}a^{-10}+2z^{9}a^{-9}+7z^{7}a^{-9}-33z^{5}a^{-9}+32z^{3}a^{-9}-a^{-9}z^{-3}-16za^{-9}+5a^{-9}z^{-1}+11z^{8}a^{-8}-25z^{6}a^{-8}+9z^{4}a^{-8}+10z^{2}a^{-8}+3a^{-8}z^{-2}-10a^{-8}+2z^{9}a^{-7}+10z^{7}a^{-7}-41z^{5}a^{-7}+52z^{3}a^{-7}-3a^{-7}z^{-3}-31za^{-7}+12a^{-7}z^{-1}+6z^{8}a^{-6}-10z^{6}a^{-6}+2z^{4}a^{-6}+18z^{2}a^{-6}+6a^{-6}z^{-2}-19a^{-6}+7z^{7}a^{-5}-19z^{5}a^{-5}+34z^{3}a^{-5}-3a^{-5}z^{-3}-27za^{-5}+12a^{-5}z^{-1}+3z^{6}a^{-4}-3z^{4}a^{-4}+9z^{2}a^{-4}+3a^{-4}z^{-2}-10a^{-4}+6z^{3}a^{-3}-a^{-3}z^{-3}-10za^{-3}+5a^{-3}z^{-1}$ (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   \ 0 1 2 3 4 5 6 7 8 9 χ 22                   1 -1 20                 3   3 18               4 1   -3 16             7 3     4 14           7 6       -1 12         8 5         3 10       5 9           4 8     6 6             0 6   1 7               6 4 2 4                 -2 2 3                   3

Integral Khovanov Homology (db, data source)

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

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.

L11n457 L11n459