L11a423

L11a422

L11a424

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a423 at Knotilus!

Link Presentations

[edit Notes on L11a423's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {-t(1)t(3)^{4}+t(1)t(2)t(3)^{4}-2t(2)t(3)^{4}+t(3)^{4}+t(1)t(2)^{2}t(3)^{3}-3t(2)^{2}t(3)^{3}+3t(1)t(3)^{3}-4t(1)t(2)t(3)^{3}+6t(2)t(3)^{3}-2t(3)^{3}-2t(1)t(2)^{2}t(3)^{2}+4t(2)^{2}t(3)^{2}-4t(1)t(3)^{2}+7t(1)t(2)t(3)^{2}-7t(2)t(3)^{2}+2t(3)^{2}+2t(1)t(2)^{2}t(3)-3t(2)^{2}t(3)+3t(1)t(3)-6t(1)t(2)t(3)+4t(2)t(3)-t(3)-t(1)t(2)^{2}+t(2)^{2}+2t(1)t(2)-t(2)}{{\sqrt {t(1)}}t(2)t(3)^{2}}}$ (db)
Jones polynomial	$-q^{8}+4q^{7}-10q^{6}+16q^{5}-21q^{4}+25q^{3}+q^{-3}-23q^{2}-3q^{-2}+21q+9q^{-1}-14$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$-z^{4}a^{-6}-z^{2}a^{-6}-a^{-6}z^{-2}-2a^{-6}+z^{6}a^{-4}+2z^{4}a^{-4}+6z^{2}a^{-4}+4a^{-4}z^{-2}+9a^{-4}+z^{6}a^{-2}-z^{4}a^{-2}+a^{2}z^{2}-8z^{2}a^{-2}-5a^{-2}z^{-2}+a^{2}-11a^{-2}-2z^{4}-z^{2}+2z^{-2}+3$ (db)
Kauffman polynomial	$z^{5}a^{-9}-z^{3}a^{-9}+4z^{6}a^{-8}-4z^{4}a^{-8}+9z^{7}a^{-7}-14z^{5}a^{-7}+9z^{3}a^{-7}-4za^{-7}+a^{-7}z^{-1}+12z^{8}a^{-6}-22z^{6}a^{-6}+19z^{4}a^{-6}-9z^{2}a^{-6}-a^{-6}z^{-2}+3a^{-6}+8z^{9}a^{-5}-2z^{7}a^{-5}-20z^{5}a^{-5}+29z^{3}a^{-5}-19za^{-5}+5a^{-5}z^{-1}+2z^{10}a^{-4}+20z^{8}a^{-4}-54z^{6}a^{-4}+52z^{4}a^{-4}-29z^{2}a^{-4}-4a^{-4}z^{-2}+15a^{-4}+13z^{9}a^{-3}-14z^{7}a^{-3}-21z^{5}a^{-3}+45z^{3}a^{-3}-33za^{-3}+9a^{-3}z^{-1}+2z^{10}a^{-2}+14z^{8}a^{-2}+a^{2}z^{6}-43z^{6}a^{-2}-3a^{2}z^{4}+48z^{4}a^{-2}+3a^{2}z^{2}-38z^{2}a^{-2}-5a^{-2}z^{-2}-a^{2}+20a^{-2}+5z^{9}a^{-1}+3az^{7}-6az^{5}-22z^{5}a^{-1}+3az^{3}+29z^{3}a^{-1}-18za^{-1}+5a^{-1}z^{-1}+6z^{8}-14z^{6}+16z^{4}-15z^{2}-2z^{-2}+8$ (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                   7 1   -6 11                 9 3     6 9               12 7       -5 7             13 9         4 5           12 14           2 3         9 11             -2 1       6 13               7 -1     3 8                 -5 -3     6                   6 -5 1 3                     -2 -7 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=1$ $i=3$ $r=-4$ ${\mathbb {Z} }$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{3}$ $r=-2$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-1$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=0$ ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{9}$ $r=1$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=2$ ${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{13}$ $r=3$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=4$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=5$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=6$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=7$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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