L11a488

L11a487

L11a489

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a488 at Knotilus!

Link Presentations

[edit Notes on L11a488's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(u-1)(w-1)^{2}\left(2vw^{2}-vw+v+w^{3}-w^{2}+2w\right)}{{\sqrt {u}}{\sqrt {v}}w^{5/2}}}$ (db)
Jones polynomial	$-q^{11}+3q^{10}-8q^{9}+14q^{8}-18q^{7}+21q^{6}-20q^{5}+19q^{4}-12q^{3}+8q^{2}-3q+1$ (db)
Signature	4 (db)
HOMFLY-PT polynomial	$-z^{2}a^{-10}-a^{-10}z^{-2}-2a^{-10}+3z^{4}a^{-8}+8z^{2}a^{-8}+4a^{-8}z^{-2}+9a^{-8}-2z^{6}a^{-6}-7z^{4}a^{-6}-12z^{2}a^{-6}-5a^{-6}z^{-2}-13a^{-6}-z^{6}a^{-4}-z^{4}a^{-4}+3z^{2}a^{-4}+2a^{-4}z^{-2}+5a^{-4}+z^{4}a^{-2}+2z^{2}a^{-2}+a^{-2}$ (db)
Kauffman polynomial	$z^{10}a^{-6}+z^{10}a^{-8}+4z^{9}a^{-5}+9z^{9}a^{-7}+5z^{9}a^{-9}+5z^{8}a^{-4}+16z^{8}a^{-6}+19z^{8}a^{-8}+8z^{8}a^{-10}+3z^{7}a^{-3}-5z^{7}a^{-7}+4z^{7}a^{-9}+6z^{7}a^{-11}+z^{6}a^{-2}-11z^{6}a^{-4}-51z^{6}a^{-6}-56z^{6}a^{-8}-14z^{6}a^{-10}+3z^{6}a^{-12}-7z^{5}a^{-3}-18z^{5}a^{-5}-29z^{5}a^{-7}-28z^{5}a^{-9}-9z^{5}a^{-11}+z^{5}a^{-13}-3z^{4}a^{-2}+10z^{4}a^{-4}+64z^{4}a^{-6}+71z^{4}a^{-8}+16z^{4}a^{-10}-4z^{4}a^{-12}+4z^{3}a^{-3}+24z^{3}a^{-5}+50z^{3}a^{-7}+39z^{3}a^{-9}+7z^{3}a^{-11}-2z^{3}a^{-13}+3z^{2}a^{-2}-11z^{2}a^{-4}-48z^{2}a^{-6}-48z^{2}a^{-8}-13z^{2}a^{-10}+z^{2}a^{-12}-17za^{-5}-35za^{-7}-23za^{-9}-4za^{-11}+za^{-13}-a^{-2}+8a^{-4}+22a^{-6}+19a^{-8}+5a^{-10}+5a^{-5}z^{-1}+9a^{-7}z^{-1}+5a^{-9}z^{-1}+a^{-11}z^{-1}-2a^{-4}z^{-2}-5a^{-6}z^{-2}-4a^{-8}z^{-2}-a^{-10}z^{-2}$ (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   \ -2 -1 0 1 2 3 4 5 6 7 8 9 χ 23                       1 -1 21                     2   2 19                   6 1   -5 17                 8 2     6 15               10 6       -4 13             11 8         3 11           11 12           1 9         8 9             -1 7       4 11               7 5     4 8                 -4 3   1 6                   5 1   2                     -2 -1 1                       1

Integral Khovanov Homology (db, data source)

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

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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