L11a407

L11a406

L11a408

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a407 at Knotilus!

Link Presentations

[edit Notes on L11a407's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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