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DrawMorseLink

PD Presentation:

X6172 X10,4,11,3 X12,8,13,7 X18,14,19,13 X9,17,10,16 X17,9,18,8 X22,20,5,19 X20,15,21,16 X14,21,15,22 X2536 X4,12,1,11

Gauss Code:

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

Jones Polynomial:

- q-5/2 + 2q-3/2 - 5q-1/2 + 6q1/2 - 8q3/2 + 8q5/2 - 7q7/2 + 6q9/2 - 4q11/2 + q13/2

A2 (sl(3)) Invariant:

q-8 + q-6 + q-4 + 3q-2 + 1 + 2q2 - q6 + q8 - 3q10 + q12 - q14 + q16 + 2q18 + q22 - q24

HOMFLY-PT Polynomial:

a-7z - a-5z-1 - 4a-5z - 4a-5z3 - a-5z5 + 3a-3z-1 + 9a-3z + 9a-3z3 + 5a-3z5 + a-3z7 - 4a-1z-1 - 9a-1z - 8a-1z3 - 2a-1z5 + 2az-1 + 3az + az3

Kauffman Polynomial:

- a-8z2 + a-7z - 4a-7z3 + a-6 - a-6z2 + a-6z4 - 2a-6z6 - a-5z-1 + 6a-5z - 16a-5z3 + 15a-5z5 - 5a-5z7 + 3a-4 - 4a-4z2 - 3a-4z4 + 10a-4z6 - 4a-4z8 - 3a-3z-1 + 17a-3z - 37a-3z3 + 32a-3z5 - 6a-3z7 - a-3z9 + 3a-2 - 4a-2z2 - 12a-2z4 + 20a-2z6 - 6a-2z8 - 4a-1z-1 + 19a-1z - 34a-1z3 + 22a-1z5 - 2a-1z7 - a-1z9 + 2 - 8z4 + 8z6 - 2z8 - 2az-1 + 7az - 9az3 + 5az5 - az7

Khovanov Homology:

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 14                   1 j = 12                 3   j = 10               3 1   j = 8             4 3     j = 6           4 3       j = 4         4 4         j = 2       4 6           j = 0     1 2             j = -2   1 4               j = -4   1                 j = -6 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]=	2
In[3]:=	Show[DrawMorseLink[Link[11, NonAlternating, 51]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 51]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 4, 11, 3], X[12, 8, 13, 7], X[18, 14, 19, 13], > X[9, 17, 10, 16], X[17, 9, 18, 8], X[22, 20, 5, 19], X[20, 15, 21, 16], > X[14, 21, 15, 22], X[2, 5, 3, 6], X[4, 12, 1, 11]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 3, 6, -5, -2, 11, -3, 4, -9, 8, 5, -6, -4, > 7, -8, 9, -7}]
In[6]:=	Jones[L][q]
Out[6]=	-(5/2) 2 5 3/2 5/2 7/2 9/2 -q + ---- - ------- + 6 Sqrt[q] - 8 q + 8 q - 7 q + 6 q - 3/2 Sqrt[q] q 11/2 13/2 > 4 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-8 -6 -4 3 2 6 8 10 12 14 16 18 1 + q + q + q + -- + 2 q - q + q - 3 q + q - q + q + 2 q + 2 q 22 24 > q - q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 51]][a, z]
Out[8]=	3 3 1 3 4 2 a z 4 z 9 z 9 z 4 z 9 z -(----) + ---- - --- + --- + -- - --- + --- - --- + 3 a z - ---- + ---- - 5 3 a z z 7 5 3 a 5 3 a z a z a a a a a 3 5 5 5 7 8 z 3 z 5 z 2 z z > ---- + a z - -- + ---- - ---- + -- a 5 3 a 3 a a a
In[9]:=	Kauffman[Link[11, NonAlternating, 51]][a, z]
Out[9]=	-6 3 3 1 3 4 2 a z 6 z 17 z 19 z 2 + a + -- + -- - ---- - ---- - --- - --- + -- + --- + ---- + ---- + 7 a z - 4 2 5 3 a z z 7 5 3 a a a a z a z a a a 2 2 2 2 3 3 3 3 4 z z 4 z 4 z 4 z 16 z 37 z 34 z 3 4 z > -- - -- - ---- - ---- - ---- - ----- - ----- - ----- - 9 a z - 8 z + -- - 8 6 4 2 7 5 3 a 6 a a a a a a a a 4 4 5 5 5 6 6 3 z 12 z 15 z 32 z 22 z 5 6 2 z 10 z > ---- - ----- + ----- + ----- + ----- + 5 a z + 8 z - ---- + ----- + 4 2 5 3 a 6 4 a a a a a a 6 7 7 7 8 8 9 9 20 z 5 z 6 z 2 z 7 8 4 z 6 z z z > ----- - ---- - ---- - ---- - a z - 2 z - ---- - ---- - -- - -- 2 5 3 a 4 2 3 a a a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 2 4 1 1 1 -2 4 2 4 q 4 6 q + 4 q + ----- + ----- + ----- + t + ----- + - + ---- + 4 q t + 6 4 4 3 2 3 2 2 t t q t q t q t q t 6 6 2 8 2 8 3 10 3 10 4 12 4 14 5 > 4 q t + 3 q t + 4 q t + 3 q t + 3 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n51

L11n50 L11n52