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L11n386 L11n388

Knotscape

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DrawMorseLink

PD Presentation:

X6172 X14,7,15,8 X4,15,1,16 X5,12,6,13 X8493 X11,19,12,22 X21,18,22,5 X9,20,10,21 X17,11,18,10 X19,17,20,16 X2,14,3,13

Gauss Code:

{{1, -11, 5, -3}, {-10, 8, -7, 6}, {-4, -1, 2, -5, -8, 9, -6, 4, 11, -2, 3, 10, -9, 7}}

Jones Polynomial:

q-6 - 4q-5 + 7q-4 - 9q-3 + 11q-2 - 10q-1 + 11 - 6q + 4q2 - q3

A2 (sl(3)) Invariant:

q-18 - 2q-16 + q-14 - 2q-10 + 3q-8 + 5q-4 + 5q-2 + 5 + 7q2 + q4 + 3q6 + q8 - q10

HOMFLY-PT Polynomial:

a-2z-2 - a-2z2 - 2z-2 + 3z2 + 2z4 + a2z-2 - 3a2z2 - 3a2z4 - a2z6 + a4z2 + a4z4

Kauffman Polynomial:

- a-3z + a-3z3 + a-2z-2 - 4a-2z2 + 4a-2z4 - 2a-1z-1 - 4a-1z + 10a-1z3 - 4a-1z5 + 2a-1z7 + 2z-2 + 1 - 11z2 + 22z4 - 13z6 + 4z8 - 2az-1 - 6az + 23az3 - 16az5 + az7 + 2az9 + a2z-2 - 9a2z2 + 25a2z4 - 27a2z6 + 9a2z8 - 4a3z + 20a3z3 - 23a3z5 + 3a3z7 + 2a3z9 - a4z2 + 5a4z4 - 13a4z6 + 5a4z8 - a5z + 6a5z3 - 11a5z5 + 4a5z7 + a6z2 - 2a6z4 + a6z6

Khovanov Homology:

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 j = 7                   1 j = 5                 3   j = 3               3 1   j = 1             8 3     j = -1           6 7       j = -3         5 5 1       j = -5       4 6           j = -7     3 5             j = -9   1 4               j = -11   3                 j = -13 1

Computer Talk. The data above can be recomputed by Mathematica using the package KnotTheory`. Following setup, the sample Mathematica session below reproduces most of the above data (Mathematica system prompts in blue, human input in red, Mathematica output in black):

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	Length[Skeleton[L]]
Out[2]=	3
In[3]:=	Show[DrawMorseLink[Link[11, NonAlternating, 387]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 387]]
Out[4]=	PD[X[6, 1, 7, 2], X[14, 7, 15, 8], X[4, 15, 1, 16], X[5, 12, 6, 13], > X[8, 4, 9, 3], X[11, 19, 12, 22], X[21, 18, 22, 5], X[9, 20, 10, 21], > X[17, 11, 18, 10], X[19, 17, 20, 16], X[2, 14, 3, 13]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, 5, -3}, {-10, 8, -7, 6}, > {-4, -1, 2, -5, -8, 9, -6, 4, 11, -2, 3, 10, -9, 7}]
In[6]:=	Jones[L][q]
Out[6]=	-6 4 7 9 11 10 2 3 11 + q - -- + -- - -- + -- - -- - 6 q + 4 q - q 5 4 3 2 q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-18 2 -14 2 3 5 5 2 4 6 8 10 5 + q - --- + q - --- + -- + -- + -- + 7 q + q + 3 q + q - q 16 10 8 4 2 q q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 387]][a, z]
Out[8]=	2 2 -2 1 a 2 z 2 2 4 2 4 2 4 4 4 2 6 -- + ----- + -- + 3 z - -- - 3 a z + a z + 2 z - 3 a z + a z - a z 2 2 2 2 2 z a z z a
In[9]:=	Kauffman[Link[11, NonAlternating, 387]][a, z]
Out[9]=	2 2 1 a 2 2 a z 4 z 3 5 2 1 + -- + ----- + -- - --- - --- - -- - --- - 6 a z - 4 a z - a z - 11 z - 2 2 2 2 a z z 3 a z a z z a 2 3 3 4 z 2 2 4 2 6 2 z 10 z 3 3 3 > ---- - 9 a z - a z + a z + -- + ----- + 23 a z + 20 a z + 2 3 a a a 4 5 5 3 4 4 z 2 4 4 4 6 4 4 z 5 > 6 a z + 22 z + ---- + 25 a z + 5 a z - 2 a z - ---- - 16 a z - 2 a a 7 3 5 5 5 6 2 6 4 6 6 6 2 z 7 > 23 a z - 11 a z - 13 z - 27 a z - 13 a z + a z + ---- + a z + a 3 7 5 7 8 2 8 4 8 9 3 9 > 3 a z + 4 a z + 4 z + 9 a z + 5 a z + 2 a z + 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	-3 7 1 3 1 4 3 5 4 q + - + 8 q + ------ + ------ + ----- + ----- + ----- + ----- + ----- + q 13 6 11 5 9 5 9 4 7 4 7 3 5 3 q t q t q t q t q t q t q t 6 5 5 6 3 3 2 5 2 7 3 > ----- + ----- + ---- + --- + 3 q t + 3 q t + q t + 3 q t + q t 5 2 3 2 3 q t q t q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n387

L11n386 L11n388