L11a395

L11a394

L11a396

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a395 at Knotilus!

Link Presentations

[edit Notes on L11a395's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {uvw^{3}-4uvw^{2}+4uvw-2uv-2uw^{3}+5uw^{2}-5uw+2u-2vw^{3}+5vw^{2}-5vw+2v+2w^{3}-4w^{2}+4w-1}{{\sqrt {u}}{\sqrt {v}}w^{3/2}}}$ (db)
Jones polynomial	$1-4q^{-1}+8q^{-2}-11q^{-3}+15q^{-4}-15q^{-5}+17q^{-6}-12q^{-7}+9q^{-8}-5q^{-9}+2q^{-10}-q^{-11}$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$-a^{12}z^{-2}+4a^{10}z^{-2}+4a^{10}-6z^{2}a^{8}-5a^{8}z^{-2}-11a^{8}+4z^{4}a^{6}+9z^{2}a^{6}+2a^{6}z^{-2}+7a^{6}-z^{6}a^{4}-2z^{4}a^{4}-z^{2}a^{4}+z^{4}a^{2}+z^{2}a^{2}$ (db)
Kauffman polynomial	$z^{5}a^{13}-3z^{3}a^{13}+3za^{13}-a^{13}z^{-1}+2z^{6}a^{12}-4z^{4}a^{12}+3z^{2}a^{12}+a^{12}z^{-2}-2a^{12}+2z^{7}a^{11}+2z^{5}a^{11}-12z^{3}a^{11}+13za^{11}-5a^{11}z^{-1}+2z^{8}a^{10}+4z^{6}a^{10}-13z^{4}a^{10}+14z^{2}a^{10}+4a^{10}z^{-2}-10a^{10}+2z^{9}a^{9}+z^{7}a^{9}+3z^{5}a^{9}-16z^{3}a^{9}+21za^{9}-9a^{9}z^{-1}+z^{10}a^{8}+4z^{8}a^{8}-4z^{6}a^{8}-8z^{4}a^{8}+20z^{2}a^{8}+5a^{8}z^{-2}-14a^{8}+6z^{9}a^{7}-9z^{7}a^{7}+4z^{5}a^{7}-6z^{3}a^{7}+11za^{7}-5a^{7}z^{-1}+z^{10}a^{6}+8z^{8}a^{6}-22z^{6}a^{6}+11z^{4}a^{6}+6z^{2}a^{6}+2a^{6}z^{-2}-7a^{6}+4z^{9}a^{5}-4z^{7}a^{5}-8z^{5}a^{5}+6z^{3}a^{5}+6z^{8}a^{4}-15z^{6}a^{4}+8z^{4}a^{4}-2z^{2}a^{4}+4z^{7}a^{3}-10z^{5}a^{3}+5z^{3}a^{3}+z^{6}a^{2}-2z^{4}a^{2}+z^{2}a^{2}$ (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                   5 1   4 -5                 7 4     -3 -7               8 4       4 -9             7 7         0 -11           10 8           2 -13         5 10             5 -15       4 7               -3 -17     1 5                 4 -19   1 4                   -3 -21   1                     1 -23 1                       -1

Integral Khovanov Homology (db, data source)

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

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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