L11a404

L11a403

L11a405

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a404 at Knotilus!

Link Presentations

[edit Notes on L11a404's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {uv^{2}w^{3}-2uv^{2}w^{2}+2uv^{2}w+uvw^{4}-4uvw^{3}+7uvw^{2}-5uvw+uv+2uw^{3}-4uw^{2}+3uw-u+v^{2}w^{4}-3v^{2}w^{3}+4v^{2}w^{2}-2v^{2}w-vw^{4}+5vw^{3}-7vw^{2}+4vw-v-2w^{3}+2w^{2}-w}{{\sqrt {u}}vw^{2}}}$ (db)
Jones polynomial	$q^{-9}-3q^{-8}+8q^{-7}-12q^{-6}+19q^{-5}-20q^{-4}+22q^{-3}-q^{2}-19q^{-2}+4q+14q^{-1}-9$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$a^{8}z^{2}+2a^{8}z^{-2}+2a^{8}-3a^{6}z^{4}-9a^{6}z^{2}-5a^{6}z^{-2}-12a^{6}+2a^{4}z^{6}+8a^{4}z^{4}+16a^{4}z^{2}+4a^{4}z^{-2}+14a^{4}+a^{2}z^{6}+a^{2}z^{4}-2a^{2}z^{2}-a^{2}z^{-2}-4a^{2}-z^{4}-z^{2}$ (db)
Kauffman polynomial	$z^{6}a^{10}-3z^{4}a^{10}+3z^{2}a^{10}-a^{10}+3z^{7}a^{9}-7z^{5}a^{9}+4z^{3}a^{9}+5z^{8}a^{8}-11z^{6}a^{8}+10z^{4}a^{8}-11z^{2}a^{8}-2a^{8}z^{-2}+8a^{8}+4z^{9}a^{7}-18z^{5}a^{7}+25z^{3}a^{7}-18za^{7}+5a^{7}z^{-1}+z^{10}a^{6}+16z^{8}a^{6}-50z^{6}a^{6}+62z^{4}a^{6}-45z^{2}a^{6}-5a^{6}z^{-2}+20a^{6}+9z^{9}a^{5}-3z^{7}a^{5}-34z^{5}a^{5}+53z^{3}a^{5}-33za^{5}+9a^{5}z^{-1}+z^{10}a^{4}+20z^{8}a^{4}-55z^{6}a^{4}+62z^{4}a^{4}-38z^{2}a^{4}-4a^{4}z^{-2}+15a^{4}+5z^{9}a^{3}+8z^{7}a^{3}-37z^{5}a^{3}+41z^{3}a^{3}-19za^{3}+5a^{3}z^{-1}+9z^{8}a^{2}-13z^{6}a^{2}+8z^{4}a^{2}-6z^{2}a^{2}-a^{2}z^{-2}+3a^{2}+8z^{7}a-13z^{5}a+8z^{3}a-4za+az^{-1}+4z^{6}-5z^{4}+z^{2}+z^{5}a^{-1}-z^{3}a^{-1}$ (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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 χ 5                       1 -1 3                     3   3 1                   6 1   -5 -1                 8 3     5 -3               12 7       -5 -5             10 7         3 -7           10 12           2 -9         9 10             -1 -11       4 11               7 -13     4 8                 -4 -15   1 6                   5 -17   2                     -2 -19 1                       1

Integral Khovanov Homology (db, data source)

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

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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