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DrawMorseLink

PD Presentation:

X6172 X10,3,11,4 X22,16,9,15 X18,12,19,11 X20,14,21,13 X12,20,13,19 X14,22,15,21 X8,18,5,17 X16,8,17,7 X2536 X4,9,1,10

Gauss Code:

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

Jones Polynomial:

q-2 - q-1 + 5 - 5q + 9q2 - 9q3 + 11q4 - 10q5 + 7q6 - 6q7 + 3q8 - q9

A2 (sl(3)) Invariant:

q-6 + 2q-4 + 3q-2 + 7 + 7q2 + 9q4 + 7q6 + 4q8 + 4q10 - 3q12 - q14 - 5q16 - 4q18 - 2q20 - 2q22 + q24 - q26

HOMFLY-PT Polynomial:

- 2a-6z-2 - 5a-6 - 5a-6z2 - 4a-6z4 - a-6z6 + 7a-4z-2 + 18a-4 + 20a-4z2 + 14a-4z4 + 6a-4z6 + a-4z8 - 8a-2z-2 - 20a-2 - 21a-2z2 - 11a-2z4 - 2a-2z6 + 3z-2 + 7 + 5z2 + z4

Kauffman Polynomial:

a-11z3 + 3a-10z4 - a-9z-1 + 3a-9z - 6a-9z3 + 6a-9z5 + a-8 - 8a-8z4 + 7a-8z6 - a-7z-1 + 3a-7z - 3a-7z3 - 10a-7z5 + 7a-7z7 - 2a-6z-2 + 7a-6 - 12a-6z2 + 16a-6z4 - 19a-6z6 + 7a-6z8 + 7a-5z-1 - 21a-5z + 25a-5z3 - 9a-5z5 - 8a-5z7 + 4a-5z9 - 7a-4z-2 + 22a-4 - 37a-4z2 + 51a-4z4 - 34a-4z6 + 5a-4z8 + a-4z10 + 15a-3z-1 - 45a-3z + 44a-3z3 + 2a-3z5 - 18a-3z7 + 5a-3z9 - 8a-2z-2 + 28a-2 - 47a-2z2 + 42a-2z4 - 15a-2z6 - a-2z8 + a-2z10 + 8a-1z-1 - 24a-1z + 23a-1z3 - 5a-1z5 - 3a-1z7 + a-1z9 - 3z-2 + 13 - 22z2 + 18z4 - 7z6 + z8

Khovanov Homology:

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 j = 19                       1 j = 17                     2   j = 15                   4 1   j = 13                 3 2     j = 11               7 4       j = 9             4 3         j = 7           5 7           j = 5         4 4             j = 3       4 8               j = 1     1 1                 j = -1     4                   j = -3 1 1                     j = -5 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, Alternating, 393]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 393]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[22, 16, 9, 15], X[18, 12, 19, 11], > X[20, 14, 21, 13], X[12, 20, 13, 19], X[14, 22, 15, 21], X[8, 18, 5, 17], > X[16, 8, 17, 7], X[2, 5, 3, 6], X[4, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 9, -8}, > {11, -2, 4, -6, 5, -7, 3, -9, 8, -4, 6, -5, 7, -3}]
In[6]:=	Jones[L][q]
Out[6]=	-2 1 2 3 4 5 6 7 8 9 5 + q - - - 5 q + 9 q - 9 q + 11 q - 10 q + 7 q - 6 q + 3 q - q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-6 2 3 2 4 6 8 10 12 14 16 7 + q + -- + -- + 7 q + 9 q + 7 q + 4 q + 4 q - 3 q - q - 5 q - 4 2 q q 18 20 22 24 26 > 4 q - 2 q - 2 q + q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 393]][a, z]
Out[8]=	2 2 2 5 18 20 3 2 7 8 2 5 z 20 z 21 z 7 - -- + -- - -- + -- - ----- + ----- - ----- + 5 z - ---- + ----- - ----- + 6 4 2 2 6 2 4 2 2 2 6 4 2 a a a z a z a z a z a a a 4 4 4 6 6 6 8 4 4 z 14 z 11 z z 6 z 2 z z > z - ---- + ----- - ----- - -- + ---- - ---- + -- 6 4 2 6 4 2 4 a a a a a a a
In[9]:=	Kauffman[Link[11, Alternating, 393]][a, z]
Out[9]=	-8 7 22 28 3 2 7 8 1 1 7 13 + a + -- + -- + -- - -- - ----- - ----- - ----- - ---- - ---- + ---- + 6 4 2 2 6 2 4 2 2 2 9 7 5 a a a z a z a z a z a z a z a z 2 2 15 8 3 z 3 z 21 z 45 z 24 z 2 12 z 37 z > ---- + --- + --- + --- - ---- - ---- - ---- - 22 z - ----- - ----- - 3 a z 9 7 5 3 a 6 4 a z a a a a a a 2 3 3 3 3 3 3 4 4 47 z z 6 z 3 z 25 z 44 z 23 z 4 3 z 8 z > ----- + --- - ---- - ---- + ----- + ----- + ----- + 18 z + ---- - ---- + 2 11 9 7 5 3 a 10 8 a a a a a a a a 4 4 4 5 5 5 5 5 6 16 z 51 z 42 z 6 z 10 z 9 z 2 z 5 z 6 7 z > ----- + ----- + ----- + ---- - ----- - ---- + ---- - ---- - 7 z + ---- - 6 4 2 9 7 5 3 a 8 a a a a a a a a 6 6 6 7 7 7 7 8 8 19 z 34 z 15 z 7 z 8 z 18 z 3 z 8 7 z 5 z > ----- - ----- - ----- + ---- - ---- - ----- - ---- + z + ---- + ---- - 6 4 2 7 5 3 a 6 4 a a a a a a a a 8 9 9 9 10 10 z 4 z 5 z z z z > -- + ---- + ---- + -- + --- + --- 2 5 3 a 4 2 a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	3 3 5 1 1 1 4 q q 4 q 5 7 8 q + 4 q + ----- + ----- + ----- + ---- + -- + - + ---- + 4 q t + 5 q t + 5 4 3 4 3 3 2 2 t t q t q t q t q t t 7 2 9 2 9 3 11 3 11 4 13 4 13 5 > 7 q t + 4 q t + 3 q t + 7 q t + 4 q t + 3 q t + 2 q t + 15 5 15 6 17 6 19 7 > 4 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a393

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