L11a459

L11a458

L11a460

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a459 at Knotilus!

Link Presentations

[edit Notes on L11a459's Link Presentations]

Planar diagram presentation	X6172 X14,4,15,3 X16,5,17,6 X12,15,5,16 X22,20,13,19 X18,8,19,7 X10,14,11,13 X8,21,9,22 X20,9,21,10 X2,11,3,12 X4,18,1,17
Gauss code	{1, -10, 2, -11}, {3, -1, 6, -8, 9, -7, 10, -4}, {7, -2, 4, -3, 11, -6, 5, -9, 8, -5}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {t(1)t(3)^{3}t(2)^{3}-t(3)^{3}t(2)^{3}-2t(1)t(3)^{2}t(2)^{3}+2t(3)^{2}t(2)^{3}+t(1)t(3)t(2)^{3}-t(3)t(2)^{3}-2t(1)t(3)^{3}t(2)^{2}+2t(3)^{3}t(2)^{2}+5t(1)t(3)^{2}t(2)^{2}-5t(3)^{2}t(2)^{2}+t(1)t(2)^{2}-4t(1)t(3)t(2)^{2}+5t(3)t(2)^{2}-t(2)^{2}+t(1)t(3)^{3}t(2)-t(3)^{3}t(2)-5t(1)t(3)^{2}t(2)+4t(3)^{2}t(2)-2t(1)t(2)+5t(1)t(3)t(2)-5t(3)t(2)+2t(2)+t(1)t(3)^{2}-t(3)^{2}+t(1)-2t(1)t(3)+2t(3)-1}{{\sqrt {t(1)}}t(2)^{3/2}t(3)^{3/2}}}$ (db)
Jones polynomial	$q^{-6}-q^{5}-4q^{-5}+4q^{4}+9q^{-4}-8q^{3}-13q^{-3}+13q^{2}+19q^{-2}-18q-20q^{-1}+22$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a^{4}z^{4}+2a^{4}z^{2}+a^{4}z^{-2}+a^{4}-2a^{2}z^{6}-z^{6}a^{-2}-7a^{2}z^{4}-3z^{4}a^{-2}-7a^{2}z^{2}-2z^{2}a^{-2}-2a^{2}z^{-2}-3a^{2}+z^{8}+5z^{6}+9z^{4}+6z^{2}+z^{-2}+2$ (db)
Kauffman polynomial	$2a^{2}z^{10}+2z^{10}+6a^{3}z^{9}+12az^{9}+6z^{9}a^{-1}+7a^{4}z^{8}+11a^{2}z^{8}+8z^{8}a^{-2}+12z^{8}+4a^{5}z^{7}-9a^{3}z^{7}-24az^{7}-4z^{7}a^{-1}+7z^{7}a^{-3}+a^{6}z^{6}-17a^{4}z^{6}-39a^{2}z^{6}-12z^{6}a^{-2}+4z^{6}a^{-4}-37z^{6}-9a^{5}z^{5}-2a^{3}z^{5}+14az^{5}-5z^{5}a^{-1}-11z^{5}a^{-3}+z^{5}a^{-5}-2a^{6}z^{4}+12a^{4}z^{4}+40a^{2}z^{4}+7z^{4}a^{-2}-6z^{4}a^{-4}+39z^{4}+4a^{5}z^{3}+4a^{3}z^{3}+az^{3}+6z^{3}a^{-1}+4z^{3}a^{-3}-z^{3}a^{-5}+a^{6}z^{2}-7a^{4}z^{2}-18a^{2}z^{2}-3z^{2}a^{-2}+z^{2}a^{-4}-14z^{2}-3a^{3}z-3az+3a^{4}+5a^{2}+3+2a^{3}z^{-1}+2az^{-1}-a^{4}z^{-2}-2a^{2}z^{-2}-z^{-2}$ (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                   5 1   -4 5                 8 3     5 3               10 5       -5 1             12 8         4 -1           11 13           2 -3         8 9             -1 -5       5 11               6 -7     4 8                 -4 -9   1 6                   5 -11   3                     -3 -13 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-1$ $i=1$ $r=-6$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{4}$ $r=-3$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=-2$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=-1$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=0$ ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{12}$ $r=1$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=2$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=3$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=4$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=5$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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.

L11a458 L11a460