L11a454

L11a453

L11a455

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a454 at Knotilus!

Link Presentations

[edit Notes on L11a454's Link Presentations]

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

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {-t(3)^{2}t(2)^{3}+t(1)t(2)^{3}-t(1)t(3)t(2)^{3}+2t(3)t(2)^{3}-t(2)^{3}+t(1)t(3)^{3}t(2)^{2}-2t(3)^{3}t(2)^{2}-4t(1)t(3)^{2}t(2)^{2}+7t(3)^{2}t(2)^{2}-3t(1)t(2)^{2}+6t(1)t(3)t(2)^{2}-6t(3)t(2)^{2}+2t(2)^{2}-2t(1)t(3)^{3}t(2)+3t(3)^{3}t(2)+6t(1)t(3)^{2}t(2)-6t(3)^{2}t(2)+2t(1)t(2)-7t(1)t(3)t(2)+4t(3)t(2)-t(2)+t(1)t(3)^{3}-t(3)^{3}-2t(1)t(3)^{2}+t(3)^{2}+t(1)t(3)}{{\sqrt {t(1)}}t(2)^{3/2}t(3)^{3/2}}}$ (db)
Jones polynomial	$q^{-6}-q^{5}-3q^{-5}+4q^{4}+8q^{-4}-10q^{3}-14q^{-3}+17q^{2}+21q^{-2}-21q-23q^{-1}+25$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a^{6}-3a^{4}z^{2}+a^{4}z^{-2}-z^{2}a^{-4}-a^{4}+3a^{2}z^{4}+2z^{4}a^{-2}+a^{2}z^{2}-2a^{2}z^{-2}-3a^{2}-a^{-2}-z^{6}+2z^{2}+z^{-2}+4$ (db)
Kauffman polynomial	$2a^{2}z^{10}+2z^{10}+5a^{3}z^{9}+13az^{9}+8z^{9}a^{-1}+5a^{4}z^{8}+13a^{2}z^{8}+12z^{8}a^{-2}+20z^{8}+3a^{5}z^{7}-3a^{3}z^{7}-16az^{7}-z^{7}a^{-1}+9z^{7}a^{-3}+a^{6}z^{6}-9a^{4}z^{6}-39a^{2}z^{6}-20z^{6}a^{-2}+4z^{6}a^{-4}-53z^{6}-7a^{5}z^{5}-10a^{3}z^{5}-8az^{5}-19z^{5}a^{-1}-13z^{5}a^{-3}+z^{5}a^{-5}-3a^{6}z^{4}+5a^{4}z^{4}+41a^{2}z^{4}+16z^{4}a^{-2}-4z^{4}a^{-4}+53z^{4}+5a^{5}z^{3}+14a^{3}z^{3}+19az^{3}+19z^{3}a^{-1}+8z^{3}a^{-3}-z^{3}a^{-5}+3a^{6}z^{2}-3a^{4}z^{2}-27a^{2}z^{2}-10z^{2}a^{-2}+z^{2}a^{-4}-32z^{2}-a^{5}z-8a^{3}z-12az-7za^{-1}-2za^{-3}-a^{6}+3a^{4}+11a^{2}+3a^{-2}+11+2a^{3}z^{-1}+2az^{-1}-a^{4}z^{-2}-2a^{2}z^{-2}-z^{-2}$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 11                       1 -1 9                     3   3 7                   7 1   -6 5                 10 3     7 3               11 7       -4 1             14 10         4 -1           12 14           2 -3         9 11             -2 -5       5 12               7 -7     3 9                 -6 -9   1 6                   5 -11   2                     -2 -13 1                       1

Integral Khovanov Homology (db, data source)

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

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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