L11a528

L11a527

L11a529

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a528 at Knotilus!

Link Presentations

[edit Notes on L11a528's Link Presentations]

Planar diagram presentation	X8192 X14,3,15,4 X22,18,13,17 X16,11,17,12 X12,15,7,16 X10,22,11,21 X20,10,21,9 X18,6,19,5 X2738 X4,13,5,14 X6,20,1,19
Gauss code	{1, -9, 2, -10, 8, -11}, {9, -1, 7, -6, 4, -5}, {10, -2, 5, -4, 3, -8, 11, -7, 6, -3}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(t(3)-1)\left(t(2)t(3)^{2}t(1)^{2}-t(3)^{2}t(1)^{2}+t(2)t(1)^{2}+t(2)^{2}t(3)t(1)^{2}-t(2)t(3)t(1)^{2}+t(3)t(1)^{2}-t(1)^{2}+t(2)^{2}t(1)+t(2)^{2}t(3)^{2}t(1)-2t(2)t(3)^{2}t(1)+t(3)^{2}t(1)-2t(2)t(1)-t(2)^{2}t(3)t(1)-t(3)t(1)+t(1)-t(2)^{2}-t(2)^{2}t(3)^{2}+t(2)t(3)^{2}+t(2)+t(2)^{2}t(3)-t(2)t(3)+t(3)\right)}{t(1)t(2)t(3)^{3/2}}}$ (db)
Jones polynomial	$-q^{6}+3q^{5}-6q^{4}+10q^{3}-13q^{2}+15q-14+14q^{-1}-9q^{-2}+7q^{-3}-3q^{-4}+q^{-5}$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z^{6}a^{-2}+2z^{6}-3a^{2}z^{4}+2z^{4}a^{-2}-z^{4}a^{-4}+8z^{4}+a^{4}z^{2}-9a^{2}z^{2}-z^{2}a^{-2}-2z^{2}a^{-4}+11z^{2}+2a^{4}-7a^{2}-2a^{-2}+7+a^{4}z^{-2}-2a^{2}z^{-2}+z^{-2}$ (db)
Kauffman polynomial	$2a^{2}z^{10}+2z^{10}+3a^{3}z^{9}+11az^{9}+8z^{9}a^{-1}+a^{4}z^{8}-a^{2}z^{8}+13z^{8}a^{-2}+11z^{8}-14a^{3}z^{7}-44az^{7}-17z^{7}a^{-1}+13z^{7}a^{-3}-5a^{4}z^{6}-28a^{2}z^{6}-36z^{6}a^{-2}+10z^{6}a^{-4}-69z^{6}+20a^{3}z^{5}+46az^{5}-8z^{5}a^{-1}-28z^{5}a^{-3}+6z^{5}a^{-5}+10a^{4}z^{4}+63a^{2}z^{4}+28z^{4}a^{-2}-14z^{4}a^{-4}+3z^{4}a^{-6}+98z^{4}-7a^{3}z^{3}-5az^{3}+20z^{3}a^{-1}+14z^{3}a^{-3}-3z^{3}a^{-5}+z^{3}a^{-7}-10a^{4}z^{2}-46a^{2}z^{2}-14z^{2}a^{-2}+4z^{2}a^{-4}-54z^{2}-3a^{3}z-7az-6za^{-1}-2za^{-3}+5a^{4}+13a^{2}+4a^{-2}+13+2a^{3}z^{-1}+2az^{-1}-a^{4}z^{-2}-2a^{2}z^{-2}-z^{-2}$ (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 13                       1 -1 11                     2   2 9                   4 1   -3 7                 6 2     4 5               7 4       -3 3             8 6         2 1           8 9           1 -1         6 6             0 -3       5 10               5 -5     2 4                 -2 -7   1 5                   4 -9   2                     -2 -11 1                       1

Integral Khovanov Homology (db, data source)

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

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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