L11n376

L11n375

L11n377

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n376 at Knotilus!

Link Presentations

[edit Notes on L11n376's Link Presentations]

Planar diagram presentation	X6172 X14,7,15,8 X4,15,1,16 X5,10,6,11 X3849 X22,18,19,17 X11,20,12,21 X19,12,20,13 X18,22,5,21 X9,16,10,17 X13,2,14,3
Gauss code	{1, 11, -5, -3}, {-8, 7, 9, -6}, {-4, -1, 2, 5, -10, 4, -7, 8, -11, -2, 3, 10, 6, -9}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(v-1)(w-1)\left(uw^{2}-1\right)}{{\sqrt {u}}{\sqrt {v}}w^{3/2}}}$ (db)
Jones polynomial	$-q^{-9}+q^{-8}-2q^{-7}+4q^{-6}-2q^{-5}+3q^{-4}-q^{-3}+2q^{-2}$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$-a^{10}z^{-2}-a^{10}+a^{8}z^{4}+5a^{8}z^{2}+4a^{8}z^{-2}+7a^{8}-a^{6}z^{6}-6a^{6}z^{4}-13a^{6}z^{2}-5a^{6}z^{-2}-13a^{6}+2a^{4}z^{4}+8a^{4}z^{2}+2a^{4}z^{-2}+7a^{4}$ (db)
Kauffman polynomial	$z^{3}a^{11}-3za^{11}+a^{11}z^{-1}+z^{4}a^{10}-2z^{2}a^{10}-a^{10}z^{-2}+2a^{10}+z^{7}a^{9}-6z^{5}a^{9}+15z^{3}a^{9}-14za^{9}+5a^{9}z^{-1}+z^{8}a^{8}-6z^{6}a^{8}+14z^{4}a^{8}-13z^{2}a^{8}-4a^{8}z^{-2}+11a^{8}+2z^{7}a^{7}-11z^{5}a^{7}+25z^{3}a^{7}-24za^{7}+9a^{7}z^{-1}+z^{8}a^{6}-6z^{6}a^{6}+16z^{4}a^{6}-22z^{2}a^{6}-5a^{6}z^{-2}+16a^{6}+z^{7}a^{5}-5z^{5}a^{5}+11z^{3}a^{5}-13za^{5}+5a^{5}z^{-1}+3z^{4}a^{4}-11z^{2}a^{4}-2a^{4}z^{-2}+8a^{4}$ (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   \ -7 -6 -5 -4 -3 -2 -1 0 χ -3               2 2 -5             1 2 1 -7           2     2 -9       1 1 1     1 -11       4 2       2 -13       2         2 -15   1 2           -1 -17                 0 -19 1               -1

Integral Khovanov Homology (db, data source)

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

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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