L11a413

L11a412

L11a414

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a413 at Knotilus!

Link Presentations

[edit Notes on L11a413's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {2uv^{2}w^{3}-2uv^{2}w^{2}+uv^{2}w+uvw^{4}-4uvw^{3}+6uvw^{2}-4uvw+uv+uw^{3}-3uw^{2}+3uw-u+v^{2}w^{4}-3v^{2}w^{3}+3v^{2}w^{2}-v^{2}w-vw^{4}+4vw^{3}-6vw^{2}+4vw-v-w^{3}+2w^{2}-2w}{{\sqrt {u}}vw^{2}}}$ (db)
Jones polynomial	$-q^{-14}+3q^{-13}-7q^{-12}+12q^{-11}-16q^{-10}+19q^{-9}-18q^{-8}+17q^{-7}-11q^{-6}+8q^{-5}-3q^{-4}+q^{-3}$ (db)
Signature	-6 (db)
HOMFLY-PT polynomial	$-a^{14}z^{-2}-a^{14}+4a^{12}z^{2}+4a^{12}z^{-2}+9a^{12}-6a^{10}z^{4}-20a^{10}z^{2}-5a^{10}z^{-2}-19a^{10}+3a^{8}z^{6}+13a^{8}z^{4}+19a^{8}z^{2}+2a^{8}z^{-2}+11a^{8}+a^{6}z^{6}+3a^{6}z^{4}+2a^{6}z^{2}$ (db)
Kauffman polynomial	$a^{17}z^{5}-2a^{17}z^{3}+a^{17}z+3a^{16}z^{6}-5a^{16}z^{4}+3a^{16}z^{2}-a^{16}+5a^{15}z^{7}-7a^{15}z^{5}+4a^{15}z^{3}-2a^{15}z+a^{15}z^{-1}+5a^{14}z^{8}-2a^{14}z^{6}-6a^{14}z^{4}+5a^{14}z^{2}-a^{14}z^{-2}+3a^{13}z^{9}+8a^{13}z^{7}-27a^{13}z^{5}+31a^{13}z^{3}-19a^{13}z+5a^{13}z^{-1}+a^{12}z^{10}+11a^{12}z^{8}-24a^{12}z^{6}+21a^{12}z^{4}-16a^{12}z^{2}-4a^{12}z^{-2}+13a^{12}+7a^{11}z^{9}-2a^{11}z^{7}-29a^{11}z^{5}+50a^{11}z^{3}-35a^{11}z+9a^{11}z^{-1}+a^{10}z^{10}+12a^{10}z^{8}-39a^{10}z^{6}+51a^{10}z^{4}-43a^{10}z^{2}-5a^{10}z^{-2}+22a^{10}+4a^{9}z^{9}-2a^{9}z^{7}-17a^{9}z^{5}+28a^{9}z^{3}-19a^{9}z+5a^{9}z^{-1}+6a^{8}z^{8}-19a^{8}z^{6}+26a^{8}z^{4}-23a^{8}z^{2}-2a^{8}z^{-2}+11a^{8}+3a^{7}z^{7}-7a^{7}z^{5}+3a^{7}z^{3}+a^{6}z^{6}-3a^{6}z^{4}+2a^{6}z^{2}$ (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   \ -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -5                       1 1 -7                     3 1 -2 -9                   5     5 -11                 6 3     -3 -13               11 5       6 -15             8 7         -1 -17           11 10           1 -19         7 10             3 -21       5 9               -4 -23     2 7                 5 -25   1 5                   -4 -27   2                     2 -29 1                       -1

Integral Khovanov Homology (db, data source)

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

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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