L11a362

L11a361

L11a363

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a362 at Knotilus!

Link Presentations

[edit Notes on L11a362's Link Presentations]

Planar diagram presentation	X12,1,13,2 X14,4,15,3 X22,14,11,13 X2,11,3,12 X4,22,5,21 X20,10,21,9 X16,6,17,5 X18,8,19,7 X6,18,7,17 X8,20,9,19 X10,16,1,15
Gauss code	{1, -4, 2, -5, 7, -9, 8, -10, 6, -11}, {4, -1, 3, -2, 11, -7, 9, -8, 10, -6, 5, -3}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {t(2)^{3}t(1)^{4}-t(2)^{2}t(1)^{4}+t(2)^{4}t(1)^{3}-4t(2)^{3}t(1)^{3}+3t(2)^{2}t(1)^{3}-t(2)t(1)^{3}-t(2)^{4}t(1)^{2}+3t(2)^{3}t(1)^{2}-5t(2)^{2}t(1)^{2}+3t(2)t(1)^{2}-t(1)^{2}-t(2)^{3}t(1)+3t(2)^{2}t(1)-4t(2)t(1)+t(1)-t(2)^{2}+t(2)}{t(1)^{2}t(2)^{2}}}$ (db)
Jones polynomial	$q^{23/2}-3q^{21/2}+5q^{19/2}-8q^{17/2}+10q^{15/2}-11q^{13/2}+10q^{11/2}-9q^{9/2}+6q^{7/2}-4q^{5/2}+2q^{3/2}-{\sqrt {q}}$ (db)
Signature	5 (db)
HOMFLY-PT polynomial	$-z^{7}a^{-5}-z^{7}a^{-7}+z^{5}a^{-3}-4z^{5}a^{-5}-4z^{5}a^{-7}+z^{5}a^{-9}+4z^{3}a^{-3}-3z^{3}a^{-5}-4z^{3}a^{-7}+3z^{3}a^{-9}+3za^{-3}+za^{-5}-2za^{-7}+za^{-9}+a^{-5}z^{-1}-a^{-7}z^{-1}$ (db)
Kauffman polynomial	$z^{4}a^{-14}-z^{2}a^{-14}+3z^{5}a^{-13}-4z^{3}a^{-13}+za^{-13}+4z^{6}a^{-12}-4z^{4}a^{-12}+4z^{7}a^{-11}-3z^{5}a^{-11}-2z^{3}a^{-11}+2za^{-11}+4z^{8}a^{-10}-7z^{6}a^{-10}+6z^{4}a^{-10}-z^{2}a^{-10}+3z^{9}a^{-9}-7z^{7}a^{-9}+9z^{5}a^{-9}-6z^{3}a^{-9}+za^{-9}+z^{10}a^{-8}+2z^{8}a^{-8}-13z^{6}a^{-8}+16z^{4}a^{-8}-6z^{2}a^{-8}+5z^{9}a^{-7}-19z^{7}a^{-7}+25z^{5}a^{-7}-17z^{3}a^{-7}+6za^{-7}-a^{-7}z^{-1}+z^{10}a^{-6}-11z^{6}a^{-6}+16z^{4}a^{-6}-8z^{2}a^{-6}+a^{-6}+2z^{9}a^{-5}-7z^{7}a^{-5}+5z^{5}a^{-5}-2z^{3}a^{-5}+3za^{-5}-a^{-5}z^{-1}+2z^{8}a^{-4}-9z^{6}a^{-4}+11z^{4}a^{-4}-4z^{2}a^{-4}+z^{7}a^{-3}-5z^{5}a^{-3}+7z^{3}a^{-3}-3za^{-3}$ (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   \ -2 -1 0 1 2 3 4 5 6 7 8 9 χ 24                       1 -1 22                     2   2 20                   3 1   -2 18                 5 2     3 16               6 4       -2 14             5 4         1 12           5 6           1 10         4 5             -1 8       2 5               3 6     2 4                 -2 4   1 3                   2 2   1                     -1 0 1                       1

Integral Khovanov Homology (db, data source)

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

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.

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