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DrawMorseLink

PD Presentation:

X12,1,13,2 X7,16,8,17 X5,1,6,10 X3746 X9,5,10,4 X13,18,14,19 X19,22,20,11 X15,21,16,20 X21,15,22,14 X2,11,3,12 X17,8,18,9

Gauss Code:

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

Jones Polynomial:

- 3q-9/2 + 6q-7/2 - 10q-5/2 + 12q-3/2 - 13q-1/2 + 12q1/2 - 10q3/2 + 6q5/2 - 3q7/2 + q9/2

A2 (sl(3)) Invariant:

q-18 + q-16 + 4q-14 + q-10 + q-8 - 4q-6 + 2q-4 - 2q-2 + 3 + 2q2 + 3q6 - 2q8 - q14

HOMFLY-PT Polynomial:

a-3z-1 + 2a-3z + a-3z3 - 4a-1z-1 - 9a-1z - 7a-1z3 - 2a-1z5 + 6az-1 + 12az + 11az3 + 5az5 + az7 - 5a3z-1 - 8a3z - 4a3z3 - a3z5 + 2a5z-1 + a5z

Kauffman Polynomial:

a-4 - 3a-4z2 + 3a-4z4 - a-4z6 - a-3z-1 + 4a-3z - 9a-3z3 + 9a-3z5 - 3a-3z7 + 3a-2 - 11a-2z2 + 9a-2z4 + 3a-2z6 - 3a-2z8 - 4a-1z-1 + 19a-1z - 37a-1z3 + 36a-1z5 - 10a-1z7 - a-1z9 + 3 - 10z2 + 6z4 + 12z6 - 8z8 - 6az-1 + 29az - 51az3 + 44az5 - 14az7 - az9 + a2 - 3a2z2 + 5a2z6 - 5a2z8 - 5a3z-1 + 21a3z - 29a3z3 + 17a3z5 - 7a3z7 + a4 - a4z2 - 3a4z6 - 2a5z-1 + 7a5z - 6a5z3

Khovanov Homology:

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 10                   1 j = 8                 2   j = 6               4 1   j = 4             6 2     j = 2           6 4       j = 0         7 6         j = -2       6 7           j = -4     4 6             j = -6   2 6               j = -8 1 4                 j = -10 3

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]=	2
In[3]:=	Show[DrawMorseLink[Link[11, NonAlternating, 238]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 238]]
Out[4]=	PD[X[12, 1, 13, 2], X[7, 16, 8, 17], X[5, 1, 6, 10], X[3, 7, 4, 6], > X[9, 5, 10, 4], X[13, 18, 14, 19], X[19, 22, 20, 11], X[15, 21, 16, 20], > X[21, 15, 22, 14], X[2, 11, 3, 12], X[17, 8, 18, 9]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, -4, 5, -3, 4, -2, 11, -5, 3}, > {10, -1, -6, 9, -8, 2, -11, 6, -7, 8, -9, 7}]
In[6]:=	Jones[L][q]
Out[6]=	-3 6 10 12 13 3/2 5/2 7/2 ---- + ---- - ---- + ---- - ------- + 12 Sqrt[q] - 10 q + 6 q - 3 q + 9/2 7/2 5/2 3/2 Sqrt[q] q q q q 9/2 > q
In[7]:=	A2Invariant[L][q]
Out[7]=	-18 -16 4 -10 -8 4 2 2 2 6 8 14 3 + q + q + --- + q + q - -- + -- - -- + 2 q + 3 q - 2 q - q 14 6 4 2 q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 238]][a, z]
Out[8]=	3 5 3 1 4 6 a 5 a 2 a 2 z 9 z 3 5 z ---- - --- + --- - ---- + ---- + --- - --- + 12 a z - 8 a z + a z + -- - 3 a z z z z 3 a 3 a z a a 3 5 7 z 3 3 3 2 z 5 3 5 7 > ---- + 11 a z - 4 a z - ---- + 5 a z - a z + a z a a
In[9]:=	Kauffman[Link[11, NonAlternating, 238]][a, z]
Out[9]=	3 5 -4 3 2 4 1 4 6 a 5 a 2 a 4 z 19 z 3 + a + -- + a + a - ---- - --- - --- - ---- - ---- + --- + ---- + 29 a z + 2 3 a z z z z 3 a a a z a 2 2 3 3 3 5 2 3 z 11 z 2 2 4 2 9 z 37 z > 21 a z + 7 a z - 10 z - ---- - ----- - 3 a z - a z - ---- - ----- - 4 2 3 a a a a 4 4 5 5 3 3 3 5 3 4 3 z 9 z 9 z 36 z > 51 a z - 29 a z - 6 a z + 6 z + ---- + ---- + ---- + ----- + 4 2 3 a a a a 6 6 7 7 5 3 5 6 z 3 z 2 6 4 6 3 z 10 z > 44 a z + 17 a z + 12 z - -- + ---- + 5 a z - 3 a z - ---- - ----- - 4 2 3 a a a a 8 9 7 3 7 8 3 z 2 8 z 9 > 14 a z - 7 a z - 8 z - ---- - 5 a z - -- - a z 2 a a
In[10]:=	Kh[L][q, t]
Out[10]=	7 3 1 4 2 6 4 6 6 7 + -- + ------ + ----- + ----- + ----- + ----- + ----- + ---- + ---- + 6 t + 2 10 4 8 4 8 3 6 3 6 2 4 2 4 2 q q t q t q t q t q t q t q t q t 2 2 2 4 2 4 3 6 3 6 4 8 4 10 5 > 6 q t + 4 q t + 6 q t + 2 q t + 4 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n238

L11n237 L11n239