L11a375

L11a374

L11a376

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a375 at Knotilus!

Link Presentations

[edit Notes on L11a375's Link Presentations]

Planar diagram presentation	X12,1,13,2 X2,13,3,14 X14,3,15,4 X22,7,11,8 X18,10,19,9 X20,16,21,15 X16,6,17,5 X6,18,7,17 X10,20,1,19 X8,11,9,12 X4,21,5,22
Gauss code	{1, -2, 3, -11, 7, -8, 4, -10, 5, -9}, {10, -1, 2, -3, 6, -7, 8, -5, 9, -6, 11, -4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {t(2)^{4}t(1)^{4}-t(2)^{3}t(1)^{4}-2t(2)^{4}t(1)^{3}+6t(2)^{3}t(1)^{3}-5t(2)^{2}t(1)^{3}+t(2)t(1)^{3}+t(2)^{4}t(1)^{2}-6t(2)^{3}t(1)^{2}+11t(2)^{2}t(1)^{2}-6t(2)t(1)^{2}+t(1)^{2}+t(2)^{3}t(1)-5t(2)^{2}t(1)+6t(2)t(1)-2t(1)-t(2)+1}{t(1)^{2}t(2)^{2}}}$ (db)
Jones polynomial	$q^{9/2}-3q^{7/2}+7q^{5/2}-12q^{3/2}+15{\sqrt {q}}-{\frac {19}{\sqrt {q}}}+{\frac {18}{q^{3/2}}}-{\frac {16}{q^{5/2}}}+{\frac {12}{q^{7/2}}}-{\frac {7}{q^{9/2}}}+{\frac {3}{q^{11/2}}}-{\frac {1}{q^{13/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-az^{9}+a^{3}z^{7}-7az^{7}+z^{7}a^{-1}+5a^{3}z^{5}-19az^{5}+5z^{5}a^{-1}+9a^{3}z^{3}-24az^{3}+9z^{3}a^{-1}+6a^{3}z-13az+6za^{-1}+a^{3}z^{-1}-az^{-1}$ (db)
Kauffman polynomial	$a^{7}z^{5}-2a^{7}z^{3}+a^{7}z+3a^{6}z^{6}-5a^{6}z^{4}+2a^{6}z^{2}+5a^{5}z^{7}-7a^{5}z^{5}+2a^{5}z^{3}+6a^{4}z^{8}-9a^{4}z^{6}+z^{6}a^{-4}+7a^{4}z^{4}-3z^{4}a^{-4}-4a^{4}z^{2}+2z^{2}a^{-4}+5a^{3}z^{9}-7a^{3}z^{7}+3z^{7}a^{-3}+9a^{3}z^{5}-8z^{5}a^{-3}-10a^{3}z^{3}+5z^{3}a^{-3}+6a^{3}z-za^{-3}-a^{3}z^{-1}+2a^{2}z^{10}+5a^{2}z^{8}+5z^{8}a^{-2}-17a^{2}z^{6}-13z^{6}a^{-2}+21a^{2}z^{4}+10z^{4}a^{-2}-10a^{2}z^{2}-4z^{2}a^{-2}+a^{2}+10az^{9}+5z^{9}a^{-1}-27az^{7}-12z^{7}a^{-1}+38az^{5}+13z^{5}a^{-1}-32az^{3}-13z^{3}a^{-1}+14az+6za^{-1}-az^{-1}+2z^{10}+4z^{8}-19z^{6}+22z^{4}-10z^{2}$ (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 10                       1 -1 8                     2   2 6                   5 1   -4 4                 7 2     5 2               8 5       -3 0             11 7         4 -2           8 9           1 -4         8 10             -2 -6       5 9               4 -8     2 7                 -5 -10   1 5                   4 -12   2                     -2 -14 1                       1

Integral Khovanov Homology (db, data source)

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

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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