L11a422

L11a421

L11a423

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a422 at Knotilus!

Link Presentations

[edit Notes on L11a422's Link Presentations]

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {-t(2)^{2}t(3)^{4}+t(1)t(3)^{4}-t(1)t(2)t(3)^{4}+2t(2)t(3)^{4}-t(3)^{4}-t(1)t(2)^{2}t(3)^{3}+2t(2)^{2}t(3)^{3}-3t(1)t(3)^{3}+5t(1)t(2)t(3)^{3}-5t(2)t(3)^{3}+2t(3)^{3}+2t(1)t(2)^{2}t(3)^{2}-3t(2)^{2}t(3)^{2}+3t(1)t(3)^{2}-6t(1)t(2)t(3)^{2}+6t(2)t(3)^{2}-2t(3)^{2}-2t(1)t(2)^{2}t(3)+3t(2)^{2}t(3)-2t(1)t(3)+5t(1)t(2)t(3)-5t(2)t(3)+t(3)+t(1)t(2)^{2}-t(2)^{2}+t(1)-2t(1)t(2)+t(2)}{{\sqrt {t(1)}}t(2)t(3)^{2}}}$ (db)
Jones polynomial	$1-4q^{-1}+10q^{-2}-14q^{-3}+21q^{-4}-22q^{-5}+23q^{-6}-19q^{-7}+14q^{-8}-8q^{-9}+3q^{-10}-q^{-11}$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$-z^{2}a^{10}-a^{10}z^{-2}-2a^{10}+3z^{4}a^{8}+8z^{2}a^{8}+4a^{8}z^{-2}+9a^{8}-2z^{6}a^{6}-7z^{4}a^{6}-13z^{2}a^{6}-5a^{6}z^{-2}-14a^{6}-z^{6}a^{4}+6z^{2}a^{4}+2a^{4}z^{-2}+7a^{4}+z^{4}a^{2}+z^{2}a^{2}$ (db)
Kauffman polynomial	$z^{5}a^{13}-2z^{3}a^{13}+za^{13}+3z^{6}a^{12}-4z^{4}a^{12}+z^{2}a^{12}+6z^{7}a^{11}-9z^{5}a^{11}+7z^{3}a^{11}-4za^{11}+a^{11}z^{-1}+8z^{8}a^{10}-13z^{6}a^{10}+14z^{4}a^{10}-11z^{2}a^{10}-a^{10}z^{-2}+5a^{10}+6z^{9}a^{9}-20z^{5}a^{9}+33z^{3}a^{9}-21za^{9}+5a^{9}z^{-1}+2z^{10}a^{8}+16z^{8}a^{8}-51z^{6}a^{8}+65z^{4}a^{8}-44z^{2}a^{8}-4a^{8}z^{-2}+18a^{8}+13z^{9}a^{7}-20z^{7}a^{7}-7z^{5}a^{7}+34z^{3}a^{7}-29za^{7}+9a^{7}z^{-1}+2z^{10}a^{6}+16z^{8}a^{6}-56z^{6}a^{6}+67z^{4}a^{6}-47z^{2}a^{6}-5a^{6}z^{-2}+21a^{6}+7z^{9}a^{5}-10z^{7}a^{5}-5z^{5}a^{5}+13z^{3}a^{5}-13za^{5}+5a^{5}z^{-1}+8z^{8}a^{4}-20z^{6}a^{4}+18z^{4}a^{4}-14z^{2}a^{4}-2a^{4}z^{-2}+9a^{4}+4z^{7}a^{3}-8z^{5}a^{3}+3z^{3}a^{3}+z^{6}a^{2}-2z^{4}a^{2}+z^{2}a^{2}$ (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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 1                       1 1 -1                     3   -3 -3                   7 1   6 -5                 8 4     -4 -7               13 6       7 -9             11 10         -1 -11           12 11           1 -13         8 12             4 -15       6 11               -5 -17     2 8                 6 -19   1 6                   -5 -21   2                     2 -23 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-5$ $i=-3$ $r=-9$ ${\mathbb {Z} }$ $r=-8$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-7$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-6$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=-5$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=-4$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{12}$ $r=-3$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=-2$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{13}$ $r=-1$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=0$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{7}$ $r=1$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=2$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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