L11a402

L11a401

L11a403

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a402 at Knotilus!

Link Presentations

[edit Notes on L11a402's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {uv^{2}w^{4}-2uv^{2}w^{3}+2uv^{2}w^{2}-uv^{2}w-2uvw^{4}+5uvw^{3}-7uvw^{2}+4uvw-uv+uw^{4}-3uw^{3}+4uw^{2}-3uw+u-v^{2}w^{4}+3v^{2}w^{3}-4v^{2}w^{2}+3v^{2}w-v^{2}+vw^{4}-4vw^{3}+7vw^{2}-5vw+2v+w^{3}-2w^{2}+2w-1}{{\sqrt {u}}vw^{2}}}$ (db)
Jones polynomial	$-q^{5}+4q^{4}-9q^{3}+16q^{2}-20q+25-23q^{-1}+21q^{-2}-15q^{-3}+9q^{-4}-4q^{-5}+q^{-6}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$z^{8}-2a^{2}z^{6}-z^{6}a^{-2}+5z^{6}+a^{4}z^{4}-7a^{2}z^{4}-3z^{4}a^{-2}+10z^{4}+2a^{4}z^{2}-8a^{2}z^{2}-3z^{2}a^{-2}+8z^{2}+a^{4}-2a^{2}+1+a^{2}z^{-2}+a^{-2}z^{-2}-2z^{-2}$ (db)
Kauffman polynomial	$a^{6}z^{6}-2a^{6}z^{4}+a^{6}z^{2}+4a^{5}z^{7}-9a^{5}z^{5}+z^{5}a^{-5}+6a^{5}z^{3}-z^{3}a^{-5}-2a^{5}z+7a^{4}z^{8}-15a^{4}z^{6}+4z^{6}a^{-4}+10a^{4}z^{4}-5z^{4}a^{-4}-6a^{4}z^{2}+z^{2}a^{-4}+2a^{4}+6a^{3}z^{9}-3a^{3}z^{7}+8z^{7}a^{-3}-19a^{3}z^{5}-11z^{5}a^{-3}+21a^{3}z^{3}+4z^{3}a^{-3}-7a^{3}z-za^{-3}+2a^{2}z^{10}+17a^{2}z^{8}+11z^{8}a^{-2}-56a^{2}z^{6}-20z^{6}a^{-2}+59a^{2}z^{4}+18z^{4}a^{-2}-28a^{2}z^{2}-9z^{2}a^{-2}+a^{2}z^{-2}+a^{-2}z^{-2}+4a^{2}+14az^{9}+8z^{9}a^{-1}-20az^{7}-5z^{7}a^{-1}-9az^{5}-11z^{5}a^{-1}+25az^{3}+15z^{3}a^{-1}-7az-3za^{-1}-2az^{-1}-2a^{-1}z^{-1}+2z^{10}+21z^{8}-64z^{6}+70z^{4}-31z^{2}+2z^{-2}+3$ (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 11                       1 -1 9                     3   3 7                   6 1   -5 5                 10 3     7 3               11 7       -4 1             14 9         5 -1           11 13           2 -3         10 12             -2 -5       6 12               6 -7     3 9                 -6 -9   1 6                   5 -11   3                     -3 -13 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-1$ $i=1$ $r=-6$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-3$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=-2$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{10}$ $r=-1$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=0$ ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{14}$ $r=1$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=2$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{10}$ $r=3$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=4$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=5$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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