L11a251

L11a250

L11a252

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a251 at Knotilus!

Link Presentations

[edit Notes on L11a251's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$\frac{(u-1) (v-1)^3 (u v+1)^2}{u^{3/2} v^{5/2}}$ (db)
Jones polynomial	$-q^{13/2}+4 q^{11/2}-8 q^{9/2}+13 q^{7/2}-18 q^{5/2}+20 q^{3/2}-21 \sqrt{q}+\frac{17}{\sqrt{q}}-\frac{13}{q^{3/2}}+\frac{8}{q^{5/2}}-\frac{4}{q^{7/2}}+\frac{1}{q^{9/2}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^7 a^{-3} -4 z^5 a^{-3} -4 z^3 a^{-3} +z^9 a^{-1} -a z^7+6 z^7 a^{-1} -4 a z^5+12 z^5 a^{-1} -4 a z^3+8 z^3 a^{-1} +a z^{-1} - a^{-1} z^{-1}$ (db)
Kauffman polynomial	$-3 z^{10} a^{-2} -3 z^{10}-7 a z^9-14 z^9 a^{-1} -7 z^9 a^{-3} -7 a^2 z^8-4 z^8 a^{-2} -8 z^8 a^{-4} -3 z^8-4 a^3 z^7+18 a z^7+41 z^7 a^{-1} +12 z^7 a^{-3} -7 z^7 a^{-5} -a^4 z^6+19 a^2 z^6+20 z^6 a^{-2} +12 z^6 a^{-4} -4 z^6 a^{-6} +24 z^6+10 a^3 z^5-15 a z^5-48 z^5 a^{-1} -11 z^5 a^{-3} +11 z^5 a^{-5} -z^5 a^{-7} +2 a^4 z^4-13 a^2 z^4-23 z^4 a^{-2} -4 z^4 a^{-4} +6 z^4 a^{-6} -28 z^4-4 a^3 z^3+6 a z^3+19 z^3 a^{-1} +5 z^3 a^{-3} -3 z^3 a^{-5} +z^3 a^{-7} +3 a^2 z^2+6 z^2 a^{-2} -z^2 a^{-6} +8 z^2+1-a z^{-1} - a^{-1} z^{-1}$ (db)

Khovanov Homology

The coefficients of the monomials $t^rq^j$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$).

\   r    \    j   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 6 χ 14                       1 1 12                     3   -3 10                   5 1   4 8                 8 3     -5 6               10 5       5 4             10 8         -2 2           11 10           1 0         8 12             4 -2       5 9               -4 -4     3 8                 5 -6   1 5                   -4 -8   3                     3 -10 1                       -1

Integral Khovanov Homology (db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=0$ $i=2$ $r=-5$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-2$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-1$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=0$ ${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{11}$ $r=1$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{10}$ $r=2$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{10}$ $r=3$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=5$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=6$ ${\mathbb Z}_2$ ${\mathbb Z}$

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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