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Knotscape

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DrawMorseLink

PD Presentation:

X6172 X5,12,6,13 X3849 X2,14,3,13 X14,7,15,8 X9,18,10,19 X17,11,18,22 X11,21,12,20 X21,17,22,16 X15,1,16,4 X19,10,20,5

Gauss Code:

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

Jones Polynomial:

q-6 - 2q-5 + 5q-4 - 6q-3 + 9q-2 - 8q-1 + 9 - 6q + 4q2 - 2q3

A2 (sl(3)) Invariant:

q-18 + 3q-14 + 4q-12 + 3q-10 + 7q-8 + 4q-6 + 5q-4 + 4q-2 + 1 + 2q2 - 3q4 - q6 - q8 - 2q10

HOMFLY-PT Polynomial:

- a-2z-2 - 3a-2 - 2a-2z2 + 4z-2 + 12 + 10z2 + 3z4 - 5a2z-2 - 13a2 - 12a2z2 - 5a2z4 - a2z6 + 2a4z-2 + 4a4 + 3a4z2 + a4z4

Kauffman Polynomial:

a-3z-1 - 4a-3z + 3a-3z3 - a-2z-2 + 3a-2 - 4a-2z2 + 2a-2z4 + a-2z6 + 5a-1z-1 - 19a-1z + 21a-1z3 - 9a-1z5 + 3a-1z7 - 4z-2 + 15 - 21z2 + 19z4 - 9z6 + 3z8 + 9az-1 - 33az + 43az3 - 23az5 + 4az7 + az9 - 5a2z-2 + 20a2 - 36a2z2 + 36a2z4 - 22a2z6 + 6a2z8 + 5a3z-1 - 18a3z + 28a3z3 - 20a3z5 + 3a3z7 + a3z9 - 2a4z-2 + 8a4 - 15a4z2 + 15a4z4 - 11a4z6 + 3a4z8 + 3a5z3 - 6a5z5 + 2a5z7 - a6 + 4a6z2 - 4a6z4 + 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                   2 j = 5                 2   j = 3               4 2   j = 1             5 2     j = -1           5 6       j = -3         4 3         j = -5       3 6           j = -7     2 3             j = -9     3               j = -11 1 2                 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, 275]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 275]]
Out[4]=	PD[X[6, 1, 7, 2], X[5, 12, 6, 13], X[3, 8, 4, 9], X[2, 14, 3, 13], > X[14, 7, 15, 8], X[9, 18, 10, 19], X[17, 11, 18, 22], X[11, 21, 12, 20], > X[21, 17, 22, 16], X[15, 1, 16, 4], X[19, 10, 20, 5]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, -3, 10}, {-2, -1, 5, 3, -6, 11}, > {-8, 2, 4, -5, -10, 9, -7, 6, -11, 8, -9, 7}]
In[6]:=	Jones[L][q]
Out[6]=	-6 2 5 6 9 8 2 3 9 + q - -- + -- - -- + -- - - - 6 q + 4 q - 2 q 5 4 3 2 q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-18 3 4 3 7 4 5 4 2 4 6 8 10 1 + q + --- + --- + --- + -- + -- + -- + -- + 2 q - 3 q - q - q - 2 q 14 12 10 8 6 4 2 q q q q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 275]][a, z]
Out[8]=	2 4 2 3 2 4 4 1 5 a 2 a 2 2 z 2 2 12 - -- - 13 a + 4 a + -- - ----- - ---- + ---- + 10 z - ---- - 12 a z + 2 2 2 2 2 2 2 a z a z z z a 4 2 4 2 4 4 4 2 6 > 3 a z + 3 z - 5 a z + a z - a z
In[9]:=	Kauffman[Link[11, NonAlternating, 275]][a, z]
Out[9]=	2 4 3 2 4 6 4 1 5 a 2 a 1 5 9 a 15 + -- + 20 a + 8 a - a - -- - ----- - ---- - ---- + ---- + --- + --- + 2 2 2 2 2 2 3 a z z a z a z z z a z 3 2 5 a 4 z 19 z 3 2 4 z 2 2 4 2 > ---- - --- - ---- - 33 a z - 18 a z - 21 z - ---- - 36 a z - 15 a z + z 3 a 2 a a 3 3 4 6 2 3 z 21 z 3 3 3 5 3 4 2 z > 4 a z + ---- + ----- + 43 a z + 28 a z + 3 a z + 19 z + ---- + 3 a 2 a a 5 2 4 4 4 6 4 9 z 5 3 5 5 5 > 36 a z + 15 a z - 4 a z - ---- - 23 a z - 20 a z - 6 a z - a 6 7 6 z 2 6 4 6 6 6 3 z 7 3 7 > 9 z + -- - 22 a z - 11 a z + a z + ---- + 4 a z + 3 a z + 2 a a 5 7 8 2 8 4 8 9 3 9 > 2 a z + 3 z + 6 a z + 3 a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	6 1 1 2 3 2 3 3 6 - + 5 q + ------ + ------ + ------ + ----- + ----- + ----- + ----- + ----- + q 13 6 11 6 11 5 9 4 7 4 7 3 5 3 5 2 q t q t q t q t q t q t q t q t 4 3 5 3 3 2 5 2 7 3 > ----- + ---- + --- + 2 q t + 4 q t + 2 q t + 2 q t + 2 q t 3 2 3 q t q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n275

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